A TREATISE
ON THE
MOTION OF VORTEX RINGS.
AN ESSAY TO WHICH THE ADAMS PRIZE WAS ADJUDGED IN 1882, IN THE UNIVERSITY OF CAMBRIDGE.
BY
J. J. THOMSON, M.A.
FELLOW AND ASSISTANT LECTURER OF TRINITY COLLEGE, CAMBRIDGE.
pontoon:
MACMILLAN AND CO.
'1883 [The lUylit of Translation and Reproduction if reserved.

PREFACE.
THE subject selected by the Examiners for the Adams Prize for
1882 was
A " general investigation of the action upon each other of two
closed vortices in a perfect incompressible fluid." In this essay, in addition to the set subject, I have discussed
some points which are intimately connected with it, and I have endeavoured to apply some of the results to the vortex atom theory
of matter.
I have made some alterations in the notation and arrangement since the essay was sent in to the Examiners, in so doing I have received great assistance from Prof. G. H. Darwin, F.R.S. one of
the Examiners, who very kindly lent me the notes he had made
on the essay. Beyond these I have not made any alterations in the first three parts of the essay : but to the fourth part, which treats of a vortex atom theory of chemical action, I have made some additions in the hope of making the theory more complete : paragraph 60 and parts of paragraphs 58 and 59 have been added
since the essay was sent in to the Examiners.
I am very much indebted to Prof. Larmor of Queen's College, Galway, for a careful revision of the proofs and for many valuable
suggestions.
J. J. THOMSON.
TRINITY COLLEGE, CAMBRIDGE. October 1st, 1883.
T.

CONTENTS.

INTRODUCTION

PAOK ix

PART I.

PARAGRAPH

4. Momentum of a system of circular vortex rings

3

5. Moment of momentum of the system

6

6. Kinetic energy of the system .

8

7. Expression for the kinetic energy of a number of circular vortex

rings moving inside a closed vessel

11

8. Theory of the single vortex ring

13

9- Expression for the velocity parallel to the axis of x due to an approxi-

mately circular vortex ring

15

10. The velocity parallel to the axis of y

18

11. The velocity parallel to the axis of z

20

12. Calculation of the coefficients in the expansion of

in the form A Q + AI COB$ + A 2 cos 26+

22

13. Calculation of the periods of vibration of the approximately circular

vortex ring

29

PAET n.

14. The action of two vortex rings on each other

37

15. The expression for the velocity parallel to the axis of x due to one

vortex at a point on the core of the other .

39

16. The velocity parallel to the axis of y

40

17. The velocity parallel to the axis of z

40

The velocity parallel to the axis of z expressed as a function of the

time

41

21. The similar expression for the velocity parallel to the axis of y

43

The similar expression for the velocity parallel to the axis of x

44

The expression for the deflection of one of the vortex rings

46

The change in the radius of the vortex ring

50

The changes in the components of the momentum

52

Effects of the collision on the sizes and directions of motion of the two vortices. 51

Vlll

CONTENTS.

PARAGRAPH
32. The impulses which would produce the same effect as the collision . 33. ) The effect of the collision upon the shape of the vortex ring : calcu-

*AGB 56

34.i

lation of

cos nt . dt
2 _j_ L.2/2\i'"P""'

....

. OD

35. Summary of the effects of the collision on the vortex rings .

. 62

36. Motion of a circular vortex ring in a fluid throughout which the dis-

tribution of velocity is known

63

O ory \
[ Motion of a circular vortex ring past a fixed sphere

. . .67

PAET HI.

39. The velocity potential due to and the vibrations of an approximately

circular vortex column

71

40. Velocity potential due to two vortex columns

74

41. Trigonometrical Lemma
.... 42. Action of two vortex columns upon each other
... 42*. The motion of two linked vortices of equal strength

75 75 78

43. The motion of two linked vortices of unequal strength .

.

.86

44. Calculation of the motion of two linked vortices of equal strength to

a higher order of appproximation

88

45. Proof that the above solution is the only one for circular vortices . 92

.... 46. Momentum and moment of momentum of the vortex ring . .
47. The motion of several vortex rings linked together

92 93

48. The equations giving the motion when a system of n vortex columns

of equal strength is slightly displaced from its position of steady

motion

94

49. The case when n= 3

98

50. The case when w=4

99

51. The case when n- 5

.

100

52. The case when n = 6

103

53. The case when n = 7

105

54. Mayer's experiments with floating magnets

107

55. Summary of this Part

107

PAET IV.

56. Pressure of a gas. Boyle's law

.

.

.

.

.

.

.109

57. Thermal effusion .

.

.

112

58. Sketch of a chemical theory

. .114

59. Theory of quantivalence .

.

60. Valency of the various elements

... .

.

.

.

.

.

.118
121

INTRODUCTION.

IN this Essay the motion of a fluid in which there are circular

vortex rings is discussed. It is divided into four parts, Part I.

contains a discussion of the vibrations which a single vortex

riog executes when it is slightly disturbed from its circular form.

Part II. is an investigation of the action upon each other of two

vortex rings which move so as never to approach closer than by a

large

multiple of

the

diameter of either ;

at the

end

of this section

the effect of a sphere on a circular vortex ring passing near it is

found. Part III. contains an investigation of the motion of two

circular vortex rings linked through each other; the conditions

necessary for the existence of such a system are discussed and the

time of vibration of the system investigated. It also contains an

investigation of the motion of three, four, five, or six vortices

arranged in the most symmetrical way, i.e. so that any plane per-

pendicular to their directions cuts their axes in points forming the

angular points of a regular polygon ; and it is proved that if there

are more than six vortices arranged in this way the steady motion

is unstable. Part IV. contains some applications of the preceding

results to the vortex atom theory of gases, and a sketch of a vortex

atom theory of chemical action.
When we have a mass of fluid under the action of no forces,

the conditions that must be satisfied are, firstly, that the ex-

pressions for the components of the velocity are such as to satisfy

the equation of continuity; secondly, that there should be no
= discontinuity in the pressure ; and, thirdly, that if F(x, yt z,t) Q

be the equation to any surface which always consists of the same

fluid particles, such as the surface of a solid immersed in a fluid or

the surface of a vortex ring, then

dF dF dF dF

w where the differential coefficients are partial, and u, v, are the

velocity components of the fluid at the point x, y, z. As we use in

the following work the expressions given by Helmholtz for the

velocity components at any point of a mass of fluid in which there

is vortex motion ;

and as we have only to deal with vortex motion

which is cfistributed throughout a volume and not spread over a

surface, there will be no discontinuity in the velocity, and so no

discontinuity- in the pressure ; so that the third is the only con-

X

INTRODUCTION.

dition we have explicitly to consider. Thus our method is very

We simple.

substitute in the equation

dF dF dF dF

-ajti

+u

-ajx-

+

v

~j~
ay

+

w-djz-'=0

w the values of w, v, given by the Helmholtz equations, and we
get differential equations sufficient to solve any of the above

problems.
We begin by proving some general expressions for the momen-
tum, moment of momentum, and kinetic energy of a mass of fluid in which there is vortex motion. In equation (9) 7 we get the
following expression for the kinetic energy of a mass of fluid in which the vortex motion is distributed in circular vortex rings,

T where is the kinetic energy; 3 the momentum of a single

vortex ring; *p, d, 9 the components of this momentum along

F the axes of #, y, z respectively ;

the velocity of the vortex ring ;

f,

g,

h

the

coordinates

of

its

centre ;

p

the

perpendicular

from

the

origin

on

the

tangent

plane

to

the surface

containing

the

fluid ;

and p the density of the fluid. When the distance between the

rings is large compared with the diameters of the rings, we prove

in 56 that the terms

for any two rings may be expressed in the following forms ;

dS
,

or

- -f

& /0
(3

cos

6

cos

cos e),

m where r is the distance between the centres of the rings ;

and

m the strengths of the rings, and a and a their radii; S the

velocity due to one vortex ring perpendicular to the plane of the

other ;

e is the

angle

between

their

directions

of

motion ;

and #,

& the angles their directions of motion make with the line joining

their centres.

These equations are, I believe, new, and they have an important
application in the explanation of Boyle's law (see 56).
We then go on to consider the vibrations of a single vortex
ring disturbed slightly from its circular form ; this is necessary for
the succeeding investigations, and it possesses much intrinsic interest. The method used is to calculate by the expressions given

INTRODUCTION.

xi

by Helmholtz the distribution of velocity due to a vortex ring whose central line of vortex core is represented by the equations

+ + p = a 2 (d n cos wjr

n sin ni/r),

where p, z, and -*fr are semi-polar coordinates, the normal to the
mean plane of the central line of the vortex ring through its

centre being taken as the axis of z and where the quantities an ,

A 7n> ^n are small compared with a. The transverse section of

We the vortex ring is small compared with its aperture.

make

use of the fact that the velocity produced by any distribution of

vortices is proportional to the magnetic force produced by electric

currents coinciding in position with the vortex lines, and such that

the strength of the current is proportional to the strength of the

vortex at every point. If currents of electricity flow round an

anchor ring, whose transverse section is small compared with

its aperture, the magnetic effects of the currents are the same as if

all the currents were collected into one flowing along the circular

axis of the anchor ring (Maxwell's Electricity and Magnetism, 2nd

ed. vol. II. 683). Hence the action of a vortex ring of this shape

will be the same as one of equal strength condensed at the central

line of the vortex core. To calculate the values of the velocity

components by Helmholtz's expressions we have to evaluate

f

cosnQ.dO

f

3- , when q is very nearly unity.

.

.

This integral occurs

J V(?-cos<9)'

in the Planetary Theory in the expansion of the Disturbing

Function, and

various

expressions

have

been

found

for

it ;

the

case, however, when q is nearly unity is not important in that

theory, and no expressions have been given which converge quickly

in this case. It was therefore necessary to investigate some

expressions for this integral which would converge quickly in this

case ;

the result of

this investigation is given in equation 25, viz.

1 r 2jr cos nO.de

TTJ O *J(q cos6)

(w _j4)/2? +
1 ^3
1) ('-*)('-)

>- 1 *'

av

-

i)/

v(*n'-f4)/

2
CV 2!)

22

Xll

INTRODUCTION.

^ where gm = 1 + i + 2m _ 1 > and g^l + a;

(

denotes as
)

usual the hyper-geometrical series.

In equations 10 18 the expressions for the components of the

velocity due to the disturbed vortex at any point in the fluid are

given, the expressions going up to and including the squares of

F = the

small

quantities

y an , /3n, n,

8
n;

from these equations, and the

condition that if (x, y, zt t)

be the equation to the surface of

a vortex ring, then

dF
-djtl

+.

u

dF
-djx-

+.

v

dF ~dTu +.

W-ddjz-F=

A
0,

we get

m where is the strength of the vortex, e the radius of the transverse

section, and f(n) = 1

m _ dt ~~ 2-Tra (log

1

(equation 41),

j...

this is the velocity of translation, and this value of it agrees very approximately with the one found by Sir William Thomson :

-*
t

-
(n> 1} log

- ~ 4/(n) l : (equation 42):

We see from this expression that the different parts of the
vortex ring move forward with slightly different velocities, and
F that the velocity of any portion of it is Fa/p, where is the undis^
turbed velocity of the ring, and p the radius of curvature of the central line of vortex core at the point under consideration ; we
might have anticipated this result.

These equations lead to the equation

L\ - 2
n* (n 1)

=
: (equation 44),

we

T

m ~~ (, 64a2 5g

f w ... . _ "" '

INTRODUCTION.

xiii

Thus we see that the ring executes vibrations in the period
27T

thus the circular vortex ring, whose transverse section is small

compared with its aperture, is stable for all displacements of its central line of vortex core. Sir William Thomson has proved that

it is stable for all small alterations in the shape of its transverse

section ;

hence we conclude that

it is stable for

all small

displace-

A ments.

limiting case of the circular vortex ring is the straight

columnar vortex column; we find what our expressions for the

times of vibration reduce to in this limiting case, and find that they

agree very approximately with those found by Sir William Thomson,

who has investigated the vibrations of a straight columnar vortex.
We thus get a confirmation of the accuracy of the work.

In Part II. we find the action upon each other of two vortex

rings which move so as never to approach closer than by a large

multiple of the diameter of either. The method used is as follows:

let the equations to one of the vortices be

+ p = a + 5 (an cos nty

n sin mjr),

= + 2 + Z $

(?B COS tti/r

Sn

sin
711/r)

;

& w then, if be the velocity along the radius, the velocity perpen-
dicular to the plane of the vortex, we have

W= -5?

and, equating coefficients of cos mjr in the expression for &, we
see that dajdt equals the coefficients of cos nty in that expression.
Hence we expand Hi and w in the form

A ^ B ^ + + + + cos

sin

2^ A' cos

B' sin 2>|r . . .

and

express

the

coefficients

A,

B,

A',

B'

in

terms

of the time ;

& and thus get differential equations for n cr , y M 8 n , n. The calcu-

lation of these coefficients is a laborious process and occupies

pp. 38 46. The following is the result of the investigation : If

two vortex rings (I.) and (II.) pass each other, the vortex (I.)

moving with the velocity p, the vortex (II.) with the velocity q,

their directions of motion making an angle e with each other ; and if

c is the shortest distance between the centres of the vortex rings,
m g the shortest distance between the paths of the vortices, and

xiv

INTRODUCTION.

m the strengths of the vortices (I.) and (II.) respectively, a, b
their radii, and k their relative velocity ; then if the equation to the plane of the vortex ring (II.), after the vortices have separated so far that they cease to influence each other, be

+ + = & Z

$

y

COS T/r

sin

where the axis of z is the normal to the undisturbed plane of
vortex (II.)t we have

=?
7'

sin'

. pq

(q - p cos e)

V(c

-
f) (l

-

: (equation 69),
)

---$-- 8 =

2ma"J Q sin" 6 /, 4<f\

(*-&,) ft

........................ (equation 71),

and the radius of the ring is increased by

sm - ~ . 38

... 3

e V(c

2N /,
g) (!

4o2 \
-7- j

,

.

^N

(equation 74),

V where

2
(c

2
g)

is

positive

or negative

according as

the

vortex

(II.)

does or does not intersect the shortest distance between the paths

of the centres of the vortices before the vortex (L).

The effects of the collision may be divided in three parts :

firstly, the effect upon the radii of the vortex rings ; secondly,
the deflection of their paths in a plane perpendicular to the plane

containing parallels to the original directions of motion of the

vortices ;

and, thirdly,

the

deflection

of

their

paths

in

the

plane

parallel to the original directions of motion of both the vortex

= rings.

Let us first consider the effect upon the radii.

Let g

c cos </>,

thus </> is the angle which the line joining the centres of the vortex rings when they are nearest together makes with the shortest

distance between the paths of the centres of the vortex rings;

is
(/>

positive for the vortex ring which first intersects the shortest

distance between the paths negative for the other ring.

The radius of the vortex ring (II.) is diminished by

mcfb

.,

-^^81^6

sin

3<,.

Thus the radius of the ring is diminished or increased accord-

Now ing as sin 3$ is positive or negative.

</> is positive for one

vortex ring negative for the other, thus sin 30 is positive for one

vortex ring negative for the other, so that if the radius of one

vortex ring is increased by the collision the radius of the other

will be diminished.

When is </>

less

than

60

the vortex ring which

first passes through the shortest distance between the paths of the

INTRODUCTION.

XV

centres of the rings diminishes in radius and the other one increases.
When <t> is greater than 60 the vortex ring which first passes

through the shortest distance between the paths increases in radius
and the other one diminishes. When the paths of the centres of

the vortex rings intersect is 90 so that the vortex ring which

first passes through the shortest distance, which in this case is the

point of intersection of the paths, is the one which increases in

When radius.

<j> is zero or the vortex rings intersect the shortest

distance simultaneously there is no change in the radius of either

vortex ring, and this is also the case when </> is 60.

Let us now consider the bending of the path of the centre of

one of the vortex rings perpendicular to the plane which passes

1 1 1 rough the centre of the other ring and is parallel to the original

paths of both the vortex rings.
We see by equation (71) that the path of the centre of the

vortex ring (II.) is bent towards this plane through an angle

this does not change sign with </> and, whichever vortex first passes

through the shortest distance, the deflection is given by the rule

that the path of a vortex ring is bent towards or from the plane

through the centre of the other vortex and parallel to the original

directions of both vortices according as cos3</> is positive or negative,

so that if is less than 30 (j>

the path

of the vortex is bent towards,

and if <f> be greater than 30, from this plane. It follows from this expression that if we have a large quantity of vortex rings uniformly

distributed they will on the whole repel a vortex ring passing by

them.

Let us now consider the bending of the paths of the vortices
in the plane parallel to the original paths of both vortex rings. Equation (69) shews that the path of the vortex ring (II.) is bent in this plane through an angle

^ .

,
^ pq

~ p cos 6^

towards the direction of motion of the other vortex. Thus the
direction of motion of one vortex is bent from or towards the
direction of motion of the other according as sin 3< (q p cos e) is
positive or negative. Comparing this result with the result for the change in the radius, we see that if the velocity of a vortex ring (II.) be greater than the velocity of the other vortex (I.) resolved along the direction of motion of (II.), then the path of each vortex will be bent towards the direction of motion of the
other when its radius is increased and away from the direction of motion of the other when its radius is diminished, while if the

XVI

INTRODUCTION.

velocity of the vortex be less than the velocity of the other resolved along its direction of motion, the direction of motion will be bent
from the direction of the other when its radius is increased and
vice versa. The rules for finding the alteration in the radius were
given before. Equation (75) shews that the effect of the collison is the same
as if an impulse

parallel to the resultant of velocities p ^cose, and q pcose
along the paths of vortices (II.) and (I.) respectively and an
impulse

e cos 3$,

parallel to the shortest distance between the original paths of the

vortex rings, were given to one of the vortices and equal and

opposite impulses to the

other ;

here 3 and

3' are the

momenta

of

the vortices.

We then go on to investigate the other effects of the collision. We find that the collision changes the shapes of the vortices as

well as their sizes and directions of motion. If the two vortices are

equal and their paths intersect, equations (78) and (79) shew that, after collision, their central lines of vortex core are represented by the equations

P == ^

TT& To

^

!

i

8k
(nc/k)*
where Zjr/n is the free period of elliptic vibration of the circular axis. These are the equations to twisted ellipses, whose ellipticities are continually changing ; thus the collision sets the vortex ring vibrating about its circular form.
We then go on to consider the changes in size, shape, and
direction of motion, which a circular vortex ring suffers when placed in a mass of fluid in which there is a distribution of velocity

We given by a velocity potential H.

prove that if -,-7- denotes

differentiation along the direction of motion of the vortex ring, I, m, n the direction cosines of this direction of motion, and a the
radius of the ring,

INTRODUCTION. da

=

dt

dh* dxdh

dm d'Cl <FH

_..

^Z >Y1

-
-

-
_,- _

r//

dh* dydh

(equation 80).

XV II

The first of these equations shews that the radius of a

vortex ring placed in a mass of fluid will increase or decrease

according as the velocity at the centre of the ring along the

straight axis decreases or increases as we travel along a stream

We line through the centre.

apply these equations to the case of

a circular vortex ring moving past a fixed sphere, and find the

alteration in the radius and the deflection. .

In Part III. we consider vortex rings which are linked through

We each other.

shew that if the vortex rings are of equal strengths

and approximately circular they must both lie on the surface of an

anchor ring whose transverse section is small compared with its

aperture, the manner of linking being such that there are always

portions of the two vortex rings at opposite extremities of a diameter

of the transverse section. The two vortex rings rotate with an

angular

velocity

2
2m/7rd

round

the

circular

axis

of the

anchor ring,

whilst this circular axis moves forward with the comparatively slow

velocity ^

m log

-
2,

where

is the strength and e the radius of

the transverse section of the vortex ring, a is the radius of the circular axis of the anchor ring and d the diameter of its trans-
verse section.
We begin by considering the effect which the proximity of the
two vortex rings has upon the shapes of their cross sections; since the distance between the rings is large compared with the radii of their transverse sections and the two rings are always nearly parallel, the problem is very approximately the same as that of two parallel straight columnar vortices, and as the mathematical work
is more simple for this case, this is the one we consider. By means of a Lemma ( 33) which enables us to transfer cylindrical harmonics from one origin to another, we find that the centres of the
transverse sections of the vortex columns describe circles with the
centre of gravity of the two cross sections of the vortex columns as centre, and that the shapes of their transverse sections keep changing, being always approximately elliptical and oscillating about the circular shape, the ellipticity and time of vibration is given by

XV111

INTRODUCTION.

We equation (89).

then go on to discuss the transverse vibrations

of the central lines of vortex core of two equal vortex rings linked

We together.

find that for each mode of deformation there are

two periods of vibration, a quick one and a slow one.

If the equations to the central line of one of the vortex rings be

cos n^r + pn sin wy,

+ cos mfr Sn sin nty,

and the equations to the circular axis of the other be of the
form with an', j3n\ 7,', 8n', written for an , /?, 7n , 8B , we prove

= + B + an

J. cos (i>

e)

cos (yu,

e')

= + + + '

ctn

ul cos (vt e) J5 cos (/A

e')

same

=i= ry

n

A. SI

- (equation 96),

where

eJ-.Bsm^ + e')

m /f

/o

^v-, !

= VK - v

(n 1)] log

Thus, if the conditions allow of the vortices being arranged in
this way the motion is stable. In 41 we discuss the condition
necessary for the existence of such an arrangement of vortex rings ;
the result is, that if / be the momentum, T the resultant moment
of momentum, r the number of times the vortices are linked through each other, and p the density of the fluid, then /, F are constants
determining the size of the system, and the conditions are that

F=

2
rmrprad .

These equations determine a and d\ from these equations we get

Now

2
c^/a

must

be

small, hence

the

condition

that

the

rings

should be approximately circular and the motion steady and stable,

We is that F (4<m7rp) h/rP should be small.

then go on to consider the

case of two unequal vortex rings, and in (43) we arrive at results

similar in character to those we have been describing; the chief

difference is that the system cannot exist unless the moment of

momentum has a certain value which is given in equation (105),

and which only depends on the strengths and volumes of the

INTRODUCTION.

xix

vortices, and the number of times they are linked through each

other.

In the latter half of Part III. we consider the case when n

vortices are twisted round each other in such a way that they all lie on the surface of an anchor ring and their central lines

of vortex core cut the plane of any transverse section of the

anchor ring at the angular point of a regular polygon inscribed in

We this cross section.

find the times of vibration when n equals

3, 4, 5, or 6, and prove that the motion is unstable for seven or

more vortices, so that not more than six vortices can be arranged

in this way. Part IV. contains the application of these results to the vortex
atom theory of gases, and to the theory of chemical combination.

ON THE MOTION OF VORTEX KINGS.

1. THE theory that the properties of bodies may be

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mtrld? V * of

8 translation

kinlle1tiec;

and ,n this way

lk

an

P

ssess '

in

virtue

energy; it can also vibrate about its

possess internal energy, and thus it

otion circular
affords

radEn

matenals for explaiuing the phenomena of heat and

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1

2

ON THE MOTION OF VORTEX RINGS.

which is most convenient, nor can it hope to explain any property

of bodies by giving the same property to the atom. Since this theory is the only one that attempts to give any account of the
mechanism of the intermolecular forces, it enables us to form much

the clearest mental representation of what goes on when one atom influences another. Though the theory is not sufficiently de-

veloped for us to say whether or not it succeeds in explaining

all

the

properties

of bodies,

yet,

since

it

to gives^

the

subject

of

vortex motion the greater part of the interest it possesses, I shall

not scruple to examine the consequences according to this theory

of any results I may obtain.

The present essay is divided into four parts : the first part, which is a necessary preliminary to the others, treats of some general propositions in vortex motion and considers at some length
the theory of the single vortex ring ; the second part treats of the

mutual action of two vortex rings which never approach closer

than a large multiple of the diameter of either, it also treats of the effect of a solid body immersed in the fluid on a vortex^ ring

passing near it; the third part treats of knotted and linked

vortices ;

and the

fourth part contains a sketch

of

a vortex theory

of chemical combination, and the application of the results

obtaining in the preceding parts to the vortex ring theory of

gases.

It will be seen that the work is almost entirely kinematical ;

we start with the fact that the vortex ring always consists of the

same particles of fluid (the proof of which, however, requires dynamical considerations), and we find that the rest of the work is kinematical. This is further evidence that the vortex theory

of matter is of a much more fundamental character than the

ordinary solid particle theory, since the mutual action of two

vortex " clash

rings

can
"

of atoms

be in

found by kinematical principles, whilst the the ordinary theory introduces us to forces

which themselves demand a theory to explain them.

PAKT I.

Some General Propositions in Vortex Motion.

WE We the

2.
formulae

shall, for
we shall

convenience
require.

of reference, begin by quoting shall always denote the com-

ponents of the velocity at the point (x, y, z) of the incompressible

fluid by the letters, u,v,w; the components of the angular velocity

of molecular rotation will be denoted by f, 77, f

Velocity.

f 3.

The elements of velocity arising from rotations

'
7?'

in the element of fluid dxdy'dz are given by

2^.3 {? (# #0
1

'
(z /)} dxdy'dz ...(1),

where r is the distance between the points (x, y, z) and (x't y', /).

Momentum.

4. The value of the momentum may be got by the following

Considerations : Consider a single closed ring of strength m, the

~ velocity potential at any point in the irrotationally moving fluid

- due to it is

times the solid angle subtended by the vortex

nng at that point, thus it is a many-valued function whose cyclic

constant
we shall

is 2m. render

If we close the opening of the ring by a
the region acyclic. Now we know that the

barrier
motion

any instant can be generated by applying an impulsive pressure

12

4

ON THE MOTION OF VORTEX RINGS.

to the surface of the vortex ring and an impulsive pressure over the barrier equal per unit of area to p times the cyclic constant, p
being the density of the fluid. Now if the transverse dimensions
of the vortex ring be small in comparison with its aperture, the
impulse over it may be neglected in comparison with that over the barrier, and thus we see that the motion can be generated by a
normal impulsive pressure over the barrier equal per unit of area
to 2m/?.

Resolving

the

impulse

parallel

to

the

axis

of

x y

we

get

momentum of the whole fluid system parallel to x = %mpx

(projection of area of vortex ring on plane yz),

with similar expressions for the components parallel to the axes of y and z.

Thus for a single circular vortex ring, if a be its radius and X, fj,, v the direction-cosines of the normal to its plane, the com-
ponents of momentum parallel to the axes of x, yy z respectively
are

The momentum may also be investigated analytically in the
following way:
P Let be the x component of the whole momentum of the fluid
which moves irrotationally due to a single vortex ring of strength m.
H Let be the velocity potential, then
P.

Integrating with respect to x,

H H where ilj and

are the values of
2

at two points on opposite

Now sides of the barrier and infinitely close to it.

the solid angle

subtended by the ring increases by 4-Tr on crossing the boundary,

thus

H - 11 = 2m

t

2

;

therefore

P = 2m ffp dy dz,

where the integration is to be taken all over the barrier closing the vortex ring ; if X, fi, v be the direction-cosines of the normal to
this barrier at any point

where dS is an element of the barrier.

ON Till: MOTION OF VORTEX I:
Now
where ds is an element of the boundary of the barrier, i.e. an element of the vortex ring, thus
*/("$
and if we extend the integration over all places where there is vortex motion, this will be the expression for the a? component of the momentum due to any distribution of vortex motion.
R Thus, if P, Q, be the components of the momentum along
x, y, z respectively,
(2).
- y& dx dy dz dP
Again
V But where a force potential exists,
du

where

V x = /

+

+2 i (vel -)

(Lamb's Treatise on the Motion of Fluids, p. 241) ; therefore
dP^=
dt

Since v is single-valued and vanishes at an infinite distance,

Again ,

/jj (vf

(Lamb's Treatise, p. 161, equation 31) ; therefore

-

dt

P We or is constant.

may prove in a similar way that both Q and

R are

constant ;

thus

the resultant

momentum

arising from

any

distribution of vortices in an unlimited mass of fluid remains

constant both in magnitude and direction.

ON THE MOTION OF VORTEX RINGS.

Moment of Momentum.

N 5. Let L, M,

be the components of the moment of

momentum about the axes of x, y, z respectively ; let the other

notation be the same as before; then for a single vortex ring

L p ff/(wy vz) doc dy dz

i

-

&J

dxdy-z

(1,

-

H a)

dx

dz}

= 2wp dS ff(z/Jt> yv)

;

this surface integral is, by Stokes' theorem, equal to the line
integral

So

and if we extend the integration over all places where there
is vortex motion, this will be the expression for the x component
of the moment of momentum due to any distribution of vortices. Thus

dL f/ dw
m

............... (3).
xdydz}
dv

as before, thus

g
^dt
= -j- 2 ///{y (uij vf)

z (w% u%)} dx dydz

Since % is a single-valued function, the last term vanishes, and

K - - = W tft * +
Xf/

*\ ^ 7 7

fff f (dw

dxdydz

dv\ -

dv

du

ON THE MOTION OF VORTEX RINGS.
Integrating this by parts, it
= ff(zw*dxdz zwvdxdy zuvdydz + zu*dxdz) dw dw du du

The surface integrals are taken over a surface at an infinite

R distance

from the origin; now we know that at an infinite

w distance u, v, are at most of the order -^, while the element of

R surface is of the order R*, and z is of the order ; thus the surface

R integral

is

of

the

order
-^

at

most,

and

so

vanishes

when

is in-

definitely great.

Integrating by parts, similar considerations will shew that

zw

dw
-r-

,j ,
dxdydz

=

0,

dy

zu -j- d&dydz = ;
so the integral we are considering becomes dw du

or, since

du dv dw

it
since

= fffvwdxdydz,
0.

Similarly 2 fffy (UTJ

v)dx

dy

dz

=

fffvw

dx

dy

dz y

and thus -^ = 2p ffj{y (urj -v$-z (w% - u)} dxdydz = ;

M L thus is constant. "We may prove in a similar way that

and

N are also constant, and thus the resultant moment of momentum

arising from any distribution of vortices in an unlimited mass of
fluid remains constant both in magnitude and direction. When

there are solids in the fluid at a finite distance from the vortices,

then the surface integrals do not necessarily vanish, and the mo-

mentum and moment of momentum are no longer constant.

8

ON THE MOTION OF VORTEX RINGS.

Kinetic Energy.

6. The kinetic energy (see Lamb's Treatise, 136)

+ w x) + = 2pfff{u (y% zrj) v (z%

(xr) yg)} dxdydz ;

this may be written, using the same notation as before,

dx dz\

f dy

c

where S means summation for all the vortices.

We shall in subsequent investigations require the expression

for the kinetic energy of a system of circular vortex rings. To

evaluate the integral for the case of a single vortex ring with any

origin we shall first find its value when the origin is at the centre

(7 then we shall find the additional term introduced when we

;

P move the origin to a point on the normal to the plane of the

PO vortex through C', and such that

is parallel to the plane of the

vortex ; and, finally, the term introduced by moving the origin from
Pto 0.

When the origin is at C", the integral = 2pm jVads,
V where is the velocity perpendicular to the plane of the vortex.
V If be the mean value of this quantity taken round the ring, the
integral

When we move the origin from Cr to P, the additional term

introduced

= - 2pm fp 9lds,

where 9^ is the velocity along the radius vector measured outwards,

and p the perpendicular from

on the

plane of

the

vortex ;

thus

the integral

P When we change the origin from to
introduced
= 2pm fc cos Vds,

the additional term

where c is the projection of OC' on the plane of the vortex ring, and <f> the angle between this projection and the radius vector drawn from the centre of the vortex ring to any point on the circumference.

Let us take as our initial line the intersection of the plane of the vortex ring with the plane through its centre containing the normal and a parallel to the axis of z.

ON THE MOTION OF VORTEX RINGS.

9

Let be
-^r

the

angle

any

radius

of

the

vortex

ring

makes

with

this initial line, o> the angle which the projection of 0(7 on the

plane of the vortex makes with this initial line ; then

V Let be expanded in the form

V- V + Acosilr + B&m'*lr + (7 cos 2^ + Dsin 2^ + &c.,

then

= + B / cos <f> Vds ira (A cos o>

sin o>).

V Since is not uniform round the vortex ring, the plane of the We A B vortex ring will not move parallel to itself, but will change its aspect.
must express and in terms of the rates of change of the
direction-cosines of the normal to the plane of the vortex ring.

Let the perpendicular from any point on the vortex ring at the
time t + dt on the plane of the ring at the time t be

+ ^ + fy Sa cos

Sj3 sin >|r ;

thus the velocity perpendicular to the plane of the vortex

d) dz

dj3 .

Comparing this expression with the former expression for the velocity, we get

Fig.l.

We must now find r- , -j- in terms of the rates of change of the at at direction-cosines of the normals to the plane of the ring.

10

ON THE MOTION OF VOKTEX RINGS.

Draw a sphere with its centre at the centre G' of the vortex

C ring. Let A, B, be the extremities of axes parallel to the axes

MN P x, y, z. Let / be the pole of the ring determined by e and 6 as

shewn in the figure. Let

be the ring itself and any point

on it defined by the angle ty. The displaced position of the plane

.of the vortex ring may be got by rotating the plane of the ring

M through
^ the ring

an angle 80/a for which

about the radius

= 0,

and

through

vector
an angle

in the plane of Sa/a about the

N = radius vector

for which ty

.

The first rotation leaves

z

un-

changed and diminishes e by 8/3/a sin 6 ; the second rotation diminishes 6 by Sa/a and leaves e unaltered, thus

a

s.

'

a sin 6

7 If Z, 77i, n be the direction-cosines of it is clear that

= m = = Z sin 6 cos e,

sin 6 sin e, n cos 0, and

-- + .-. Z ==

fc
cos 6 cos e a

Op a sin e,

-- -- bm =

SOL
cos 6 sm e
a

80 a cos e,

= rs
on

sm -Sa . /j 0. a

It follows at once that

da.

a dn

d/3

fdl .

dm

therefore

A + B = + --dm cos a

.
sm co

(dn cos co
a <-T; ^7,
at sin u
[

fdl -^

s.m

e

\at

\.

}

=- cos e sin co } .

at

J

)

Now if X', ft,', v be the direction-cosines of the projection of 00' on the plane of the vortex ring, and^ a, h the coordinates

of (7,

V = cos e cos 6 cos &> sin 6 sin ,

= + //

sin e cos cos co cos e sin &>,

= i/

sin cos co.

It is also easily proved that

ON THE MOTION OF VORTEX RINGS.

11

h np
c

So

COS

ft)

=

-

-v sm

6B

=

(h

np) r-4r

=

l-*nf+

mnq^

sin* 6 . h

c sin 6

c sin 6

= sin ft> fi cos

V sin e = -*& (jtl

csm

thus

A + = ^ m cos w

.
-B sm w

a (dn

/eW

w + -: 5 < -T7 cos sm eft

,
(
V

sm dmj\ . ]

-djt-

1} )

o> . [

This, after substituting for cos ft) and sin co the values given

above,

dn dm .

Thus

+ B sm = /A

. r>

\

(A cos ft)

ft))

9
ZTrpma

f f-dl
I f~n+ff

dm
-ji

+

,
h

dn

Thus the kinetic energy of the vortex ring

If

I

be

the

momentum

of

the

vortex

ring,

viz.

2
27rpma ,

and

+ *P, d, 9t
this may

the be

components of
written, since p

7 =

along the
/*+ mg

axes nh,

of #, y,

z

respectively,

and thus the kinetic energy of any system of circular vortex rings

This expression for the kinetic energy is closely analogous to Clausius' expression for the virial in the ordinary molecular

theory of gases.

We 7.

have in the preceding investigation supposed that

the bounding surfaces were infinitely distant from the vortices,

so that the surface integrals might be neglected; we shall,

however, require the expression for the kinetic energy when

this is not the case.

12

ON THE MOTION OF VORTEX RINGS.

The expression
2P fff{u (y%- *v) + v(z- x%) + w(a?q- y%)} dxdydz
becomes, on integrating by parts and retaining the surface integrals, supposing, however, that the boundaries are fixed so that
lu + mv 4- nw 0,

if I, m, n are the direction-cosines of the normal to the boundary

surface,

+ iP

///

u*
(

v* +

w *)

dxdydz

%p

Jf

2
(w

+

2
v

+

w2 )

(x

dydz+y

dxdz

+

z

dxdy],

or if dS be an element of the surface and p the perpendicular

from the origin on the tangent plane

- ^ = + + ^ 2

*

iP /// ( ? ^

v

2
) dxdydz

+ 2
// (w

v2

But by the preceding investigation it also equals

Thus T, the kinetic energy, is given by the equation

-0).

MOTION OF A SINGLE VORTEX.

8. HAVING investigated these general theorems we shall go

on to consider the motion of a single approximately circular vortex

We ring.

shall suppose that the transverse section of the vortex

We core is small compared with the aperture of the ring.

know

that the velocity produced by any distribution of vortices is pro-

portional to the magnetic force produced by electric currents coin-

ciding in position with the vortex lines, and such that the strength

of the current is proportional to the strength of the vortex at
every point. Now if currents of electricity flow round an anchor

ring, whose transverse section is small compared with its aperture,

the magnetic effects of the currents are the same as if all the

currents were collected into one flowing along the central line of

the anchor ring (Maxwell's Treatise on Electricity and Magnetism,

2nd edition, vol. II., 683). Hence the action of a vortex ring of

this shape will be the same as one of equal strength condensed at

the central line of the vortex core.

Let the equation to this central line be
= + + p a an cos n-^r y?n sin n^r,
cos n + S sin n,
when z, p, ^r are the cylindrical coordinates of a point on the
central line of the vortex core, the normal of the vortex ring being taken as the axis of z, the axis of x being the initial line from
which the angle ty is measured, a is the mean radius of the central
line of the vortex core, 3 the perpendicular from the origin on the
mean plane of the vortex, and an, /Sn, yn , Bn quantities which are
m very small compared with a. Let be the strength of the vortex
ring, e the radius of the transverse section of the core. Now, by equations (1), the velocity components due to a vortex of this

14

ON THE MOTION OF VORTEX RINGS.

strength, situated at the central line of the vortex core, are
given by

a(*-*>- "t

l (.

dz'

dx

,x

-* ,

,

(8

7,
'

where r is the distance between the points (x, y, z] and (x, y ', z'), and the integrals are taken all round the vortex ring.

Now

= ^ + + x p cos ty = a cos

an cos nty cos ty /3n sin nty cos ty,

= = + + y'

p sin T/T

a sin tjr

au cos WT/T sin ty

fin sin w^Jr sin ->|r,

therefore

= a sn ir sn a cos w

Ady' = a cos + cos -v/r

= - n (7

- COS n

cos nyr)
n cos ifr (ctn sin mjr

w ftn cos

sin
7i^|r),

n simjr (an sin rc/\|r /5n cos wi

w In calculating the values of u, v,

we shall retain small

quantities up to and including those of the order of the squares of

7 an, ftn) 8 n , n . Although, for our present purpose, which is to. find the time of oscillation of the vortex about its circular form, we only

require to go to the first powers of an, &c., yet we go to the higher order of approximation because, when we come to consider the

question of knotted vortices, we require the terms containing the

squares of these quantities.

If R, <f>, z be the cylindrical coordinates of the point x, y, z,

r*

=

+ 2
{p

E*-<2pfi

cos

(<

- +) + (z - zj}\

now when we substitute for p its value it is evident that -3 can be

expanded in the form

+ B + 2 (s) (At

C s cos nty

s sin n-^r

cos

sn

x coss (^--<),

A B G where

s contains terms independent of w ...,

and are of the

s

s

D E first, and

and
&

t of the second order in these quantities.

MOTION OF A SINGLE VORTEX.

15

A The part of t which is independent of an ... is evidently
COS 80 dd

but we shall investigate the values of all these coefficients later.

Velocity parallel to the axis of x.

0. In the equation

m f2"! (.

,.dy'

,. dz\ ,

^) * "-SFj. ?{<'-*>4-fc- y) rf

the expression to be integrated becomes, when the values for

y', z',

, -7-7 are substituted and the terms arranged in order of

-jj~,

magnitude, f being written for z $,

+ n^ 8 (fa cos ty ny (yu sin

S
n

cos

nty)

^ + + + + - J {(n 1) an (n 1) oyn} cos (n 1)

- + + - ^ J {(TI

1 )

bcn

(TI

1
)

ayn}

cos

(n

1
)

^ +

K(n

+1 )

+ {ft,

-
(n

1)

08J

sin

(n

+

1)

n cos

f - - # + i ( 7

S

)

cos 2w

l

(

!)

+ cos (2n -

^ - i (. * + 7n A) {sm (2n + 1) + sin (2n - 1)

Let

us consider the term

771

2 ""
f

fa

COS

1T/r

,
,

I

.

a^|r.

^TTJo

Expanding -5 this equals

cos n + sn n>r e
E , cos 27ii|r -f t sin STII/T) cos s (ijr -</>)}.

Remembering that
r27r
= I cos 7?i^/r cos nty d^jr
Jo
this equals

m if does not equal w,

16

ON THE MOTION OF VORTEX KINGS.

5I
LIT 1 o

dty fa COS -Jr

sin {^ + (8

sn -
1

+ - ., cos for (2n

^U - 1) *}

^n for - (2n +

cos
l

5 cos ( w + 1 )* +

- ^ *-i

cos (

1
)

+ ^-1 sin ( 7i-

+ n+l COS ^

+ - n-1 COS ?l
+ ^ + +l sin (2n l) </>]]

Similarly, we may prove that

ij

^ f ny (7. sin

- s cos w^) ^^

w) - (7n sin n<

8n cos

+ 7n - 5,3^

- - + 7n 0* J ^n 2n

J

(52n

S
tt

(72n 7n) cos 2n (/>),

and that

+ - + 5 + x {A n+l cos (w 1) $ i (Si cos ^

sin
(7j

2n+1 cos (2n +1) ^

and that

+ (72H+1 sin

m4^

27r
T
Jo

1
P

^n

"

^

&'**1

CS ~

x (J.^ cos (n - 1)
and that

+ i (Bl cos

^^ + (7 sin +

t

<f>

cos (2n -1

0^ +

sin (2n -1)0)},

+ + + ~ 1} ?/s- (n 1) aS-! sin (n

+ (-l)a3n}

+ + + B x [-A^, sin (n +1) <^ 1 (Bl sin ^>

<7 cos

t

<f>

UH sin (2

MOTION OF A SINGLE VORTEX.

17

and

x - (yl (,_ l sia (>i

1)

+ 7?^ - + - + $ J ( 7^ sin

(7, cos <

sin (2n 1) </>

-^ cos (2* -!)<)}

The

integral

of

the

terras

involving

the

products

a
n,

&,,...

^ sn <> -

+ 7 t

Bn

cos (2/i + 1) ^ +

cos
cos (2n -

^ (2n + !)< +

sin (2 -

Thus t* = terms not containing an + terms containing an ... to + the first power terms containing an ... to the second power.

The term not containing H
= %m&A l cos<f> .................. (10).

The terms containing an ... to the first power

= A Jra [2ny n (yn sin n<j> Sn cos n(j>)

3 & + {&
^ + {?

A^} + + + - n+l [(

1) n (n 1) ayr]

cos (n + 1)

- [(n - 1) ft. + (n + 1) 7n] ^n-J cos (n - 1) *

^ + + + - + K + (ft C^i

1)

. (n 1) aSJ n+1 ] sin (/i 1) <

^ + {^

- K" ~ 1) J3. + (n + 1) oSJ ^.J sin (n - 1) fl

(11).

The terms containing an ...to the second power

B + (2 <?

sin 27i> - C cos

sn
2n

^ A J - + t (ft. 7J

+ ~ (68. aS.) C7J cos </>

(A A + {2

-

/3n7j

^ x

+

(?/9.

+

aSJ

+ - (fa.

a7n )

CJ sin <j>

^ + { - i (^7n - AA) + + + J?^ - 2n+1 i [(n 1) tn (n 1) a7n]

aD - + + - + i [(

1) &5n (

1) aSJ (72n+1

+ ^ Zn+l ] cos (2n 1)

M + ( - i K7. - ^n) ^ ~ 2n-! i [(n - 1) ?i. + (n + 1) a7J

+ H(w - 1) fi8 + (n + 1) aSJ (72n_t + afD^} cos (2rz - 1) ^

A aJ B^ ^ + {. - + { i (a

+ + ft7.) 2n+: t [(

1)

+ (n - 1)

+ i [(w + 1) ft, +

(-!)

ayj

+ af^^} (7 2tt+1

sin (2

+ 1)

A 5M + ( ~ i (

^ + ^.7J

+ - - + H( 2 -t

1) Jft, (w 1 ) aSJ

C^ - - + + i [(n 1) ft,, (n 1) a7n]

+ affi^J sin (2/i - 1) 0] (12)

T.

18

ON THE MOTION OF VORTEX RINGS.

C 1A

The expression to be integrated becomes on substitution

-3 [fa sin ty nx (<yn sin nty Sn cos n

The term

m -1

.

^ + - + + 1) fi8n (n 1) aSJ cos (n 1)
+ - 1) Sn ] cos (w

(n- 1) a7n} sin (n

- + - ^ + i {(

1) frn+ (

1) ayn] sin (TO 1)

f -
n

A) cos

- J (aw7n + /?A) sin

A) -

^ {sm (2n + 1) - sin (2n -

(^ ^ f i

- - + n -I- &<yJ {cos (2

1)

cos (2n 1) }].

ma ^ + + B a^j sin

{5 -J-

n+1 sin (w 1) (/>

n_^ sin (w-

1) <

- + + - C (7n+1 cos (n l)(f>

n^ cos (w 1) <}

E + - D,n_, sin (2w -!)</>- 2n+l cos (2n !)</>
O H-^^ cos -

The term

m

= \ mnx {A n (yn sin n(j> Sn cos n

The term

+ + - ^ + B + x (J w+1 cos (n 1) <

i (5j cos (/>

sin

2n+1 cos (2n 1) <

The term

- - + + {(n

1) 63.

(

1) ^}

cos (n -

J^ ^ x {

- + + ^ cos (w 1)

i (5j cos (/>

sin

+ -B^ cos (2w - 1) ^

0^ +

sin (2n -

MOTION OF A SINGLE VORTEX.

19

The term

. + (n- I)*/.}

+ 5^ A + + + $ ^ n sin (n 1) <f> i (-S, sin

0, cos <

+ sin (2n 1) <

-<U !)) + COS (211

The term

- + + + 5^ - sin (rc

1) <

{-.#! sin <

(7 cos

t

</>

sin (2n 1) </>

The

integral

of the

terms

involving

the

products

a
a,

/3n . . .

A = $m[n O9.y. - .8.) cos - J (./3. + /9.8J 4, sin ^

M - 1 (.7. - /3A) (^ M si (2 + 1) * - ^ - sin (2n 1)

(A MM i

+ /3n7J

cos (2* - 1) ^ - 4 W1 cos (2n

Thus v = terms not containing + an ... terms containing aw ...to
the first power 4- terms containing an ... to the second power.

= Jmfa^ The term not containing an ...

sin <f>

............... (13).

The terms containing a n . . . to the first power

= m J [ 2na;A n (yn sin ncf> 8n cos n<f>)

^! ^ + + - -

{[(

1) S3. (

1)

J

+

cos (n

^M - {[(n - 1) fiS. + (+!) o8J

- - arc..,} cos (

1)

+ - + - M - + {[(n 1) fa. (n 1) 07.] 4,., afB J sin (n 1) 0] .. . (14).

The terms containing a.. . .to the second power

B = \m[-nac [7. (2 G, + M sin 2n^> - (?. cos 2n</>)

- + 5 + S. (2

to cos 2n<

<? sin 2<^)J

+ n [2 (^n7s - a.8.) A, - (fiS..+ aSJ 5, + (fa. + ay.) C,} cos

A A + 1- (.7. + /9.8J

-

(ft,

y.)

+ - (58.

^ .) 0, j sin

^, + !4 (.. + /8.7.)

-*[(* + 1) 68. + ( - 1) "J ^.

af^J -$[(n + 1) ft, + (n - 1) aSJ - (7M+1

cos (2n + 1) ^

(A 5M + !- I

+ J - - + /9.7n) sn_, i [(

+ 1) fi8. (

!) aSJ

M a^.J + - J [(n - 1) fa. + (n 1) yj (7 +

- cos (2n 1) <(,

22

20

ON THE MOTION OF VORTEX RINGS.

-
!- 4 (,.%,

+ ( - 1) 08J M+, + ar

su

M -
(4 <.T.

A) A-, + i [(

- 1) f . + (

+ 1) *yj 5

C^ J [(

- 1) 89. + (

+ 1) o8J

- arZ^J sin (2n + 1) <. . .(15)

The expression to be integrated becomes after substitution

- + + + ^) 2 3 [a

a (y sin i/r

cos

2 a (an cos n>|r /Sw sin wi/r)

+ J (n- + 1) (y&n - xz^ cos (n + 1) -^
+ - ^ - + 4 (w 1) (^a n 7//5w) cos (n 1)
- 4 (* + 1) (2/^ + xfr) sin (n + 1 ) ^

^ - 4 (n - 1) (y7n - #/3M) sin (w - 1)

+ + 2 (a n

2
cos ?i>|r

z

+ cos ^>r sn ?i\r

sn

~ ^ The term r 27rJo

= + 5 2
Jma (2J.

+ n cos ncj) (7n sin n

sn

The term

9?z r27T /

51 + ZTTJo ~r$ {y sin ty

x cos -\Jr) c?>/r

R H = = putting x

y cos <j>,

sin < becomes

maR 2- 1

+ B + 1 (Bn+l

+ + C^) n_^ cos nj> (Cn+l

sin n<f>

cos

sn

+ 5 a ?7i r27r

The term I ^J o

-3 (an cos ?^^|r
*

B sin n

+ 5 + = ma + [A n (an cos 7i<^) /Sn sin nfy

aw Cj3

cos

sn

#0 The term

m -;<+!)

(y/8,

-

+ 27r
f

-1
3

cos

(n

T*7T

Jo *

+ + (^ C x {A n+l cos (n 1) </> J

cos (f>

si: l

sn

MOTION OF A SINGLE VOBTEX.

21

The term T (n 4>TT
m,

r 1) (x* n -f y/3 ') I -^ cos (n ]o

cos n - + <

^ + cos <

sn <

v

cos (2n -!)< +

- sin (2

1) </>)J

The term - (n + 1) (y*H + aft)

sin (n +

+ x {^1 B+1 sin (n 1) <

+ Q [J?4 sin <f>+

cos </>

M + sn n + 0-

2n+1 cos

^ ^ The

term

-

4?r

(n

-

1)

(ya,

-

xfo I*" Jo

^

sin

(n

-

x {-A,,.! sin (n

+ 5 1) ^ | (

t sin <f>+ C^ cos

+ - ,_, sin (2n 1) - a - 2n.t cos (2n

The term containing the second powers of an ...

= + ^ A + - + A^ 2

2

i rn ((a n

n)

2
i (a n

2
/3 n)

J.
2n

cos

2nc/>

*J3n

sin

Thus w = terms not involving an + terms containing aw ... to the + first power terms containing an . . . to the second power.

The terms not involving an

=

Jm(2a2A

-a RJ J

1)

(16).

The terms involving an ...to the first power become after substi-
R R tuting for x and y, cos < and sin </> respectively
M im [(alB. - laR (B + B^) + 2a,n4.
A + iJJa, {(n - 1) n.t -( + !) J,^)) cos <
+ C^,) + 2^X
(-l)4 . )l 1 -(n +l)^ n.1})sm^] (17).

The term involving an . . . to the second power

[^. m

- ^ + - + + 1
00.0. \ 8.B, i/3n O, i (a . /3'.)

+ i { B*n [(n - 1) A.-, -( + !) ^,,]

- - -
M + + sA B/8. [(

- 1) C

(n

1)

C7
2

,

l+1

]

4a (fl

<79l/3.)

+

+2 - ?

cos 2

22

ON THE MOTION OF VOKTEX KINGS.

.[(n-i)i(t.l -(+l),CLJ

B J + - - [(

1) sn+l (n 1) m

+ 4 (0.

(18).

We 12.

must now proceed to determine the values of the

B A quantities which we have denoted by the symbols n, C n) n , &c.

We have, in fact, to determine the coefficients in the expansion of

__1 f - 2RP cos (0 - 0)}
or, as it is generally written for symmetry, of

1

in the form

{l

+

2
a

f
-2acos(<9-<)}

'

-
(0

) +... 8H cos w(0 -

This problem also occurs in the Planetary Theory in the expan-

sion of the disturbing function, and consequently these coefficients

have received

a good

deal

of

attention ;

they have

been

considered

by, amongst others, Laplace, in the Mecanique Celeste, t. I.
Pontecoulant, Du Systeme du Monde, vol. ill. chap. II.

49 ;

These mathematicians obtain series for these coefficients pro-

ceeding by ascending powers of a. The case we are most concerned

with

is

when

the

point

whose

coordinates

are

R,

z t

</>

is

close

to

the

R vortex ring, and then

is very nearly equal to p and ? is very

small, so that a is very nearly equal to unity, and thus the series

given by these mathematicians converge very slowly, and are almost

We useless for our present purpose.

must investigate some' expres-

sion which will converge quickly when a is nearly unity.

Our problem in its simplest form may be stated as follows, if

= + c
I

cos
Cj

+. . .cn cos n6 +. . . j

-(q cos 6y
we have to determine cn in a form which will converge rapidly if q
be nearly unity.

Let

=&
i

+&i

+ cos 6+...T}n cos n6

2

(2-cos0)

c= - - Then by Fourier's theorem,

1- [*" I

cosn0 3 a,6Q,

= c

T

fl

- 1 27r

*

[

cos nd

$e '

de

TTJo ^/7-rns/?^

MOTION OP A SINGLE VORTEX.

23

Now

d sin n0

ncosnfl

^(?-COS0) i=S (?-COS0)*

- + {cos (n 1) 6 - cos (n 1)0} .--.
f
(g-C080)

Integrating both sides with respect to
and 2-7T, we have

between the limits

or

= 4n6n c..,- C.., ........................ (20).

Reducing the right-hand side of equation (19) to a common denominator, we have
d sin n0

^(?-COS0)*
- - + - + + = 47i? cos nO ((2?i 1) cos (n 1) 6 (2n 1) cos (n 1) 0}

(#-cos0)* .

Integrating both sides with respect to 6 between the limits

and 2-77, we get

0=4 + + (2-l)c 2ca -{(2re l)cn.I

nH} ......... (21).

By means of this and equation (20), we easily get

and thus, if we know the values of the 5's, we can easily get those of the c's, and as the 6's are easier to calculate we shall determine
them and deduce the values of the c's.

Let F=

=b
^

+b l

cos0+...bn cosn0+....

By differentiation we have
dV
,

d*V

F hence, substituting for the value just written and equating the
coefficients of cos nd we have

Let

where

</>

(q)

and i/r

(q)

are rational and integral algebraic functions

of q.

Substituting in the differential equation, we find

24

ON THE MOTION OF VORTEX RINGS.

Let us change the variable from q to x, where x = q
equations then become

\, the

Let

< = a + !#+

Substituting in the differential equation for <, we find

therefore

~
m+l

2
2(m+l)

"

,

2*

27

(3!)

~

~

'-'

,

or, with the ordinary notation for the hypergeometrical series,

Let

=
-f (a?)

Substituting in the differential equation for -fy (a?), we find

a
~ tt

ra.ra + l

2

w+1

2

(m

+

2
1)

So

= - - ^r (x) a ^(J n, } + w, 1, Ja?)

where the general term inside the bracket
=2

To complete the solution we have to determine the values of a

We and .

shall do- this by determining the value of bn when q is

very nearly unity, or when x is small.

We may prove, in exactly the same way as we proved equation
(20), that

or

6^ - = + 4>n (1 x) bn (2n 1)

+ (2w 4- 1)

MOTION OF A SINGLE VORTEX.

25

By the help of this sequence equation we can express bm in
terms of 6 and in the form

+ b
.)

(A'

We only want now to determine o

and

a ,

i.e.

the

parts

of

i|r

A and (/> independent of a?, thus we only want the coefficients and

A' in the the same

equation
as if we

just put

written;
x = in

now evidently A. and A' will be the sequence equation and then

determine bn in terms of b

and b lt

The sequence equation becomes, when x 0,

the solution of this is

where C and C' are arbitrary constants.

Determining the arbitrary constants in terms of b

and b we lt

find

6. = 26

for

in

the

sequence

equation

involving

6 ,

26

must be written

instead of 6A.

Now

dO

do

de

where

Now, when k is very nearly unity, we know that

[$*

dd>
:

=,

4

10"

= where &,

*J(I

^2 ),

in

our

case =

^/ f

,

.

J

26

ON THE MOTION OF VORTEX RINGS.

Therefore, when q is very nearly unity

2b =

log J4 y^l^Y)! approximately
cos# dO

When k is very nearly unity

rl*
I V(l
Jo

= ^2

2
sin ^)

d(f>

l approximately ;

therefore

Therefore, when q is very nearly unity,

comparing this with our former solution for 6n, we find

If

Thus

t

,

TTJ o

where

^Tm = 2

(l +
\

i

+...m-l)

so

that

W K,

=2,K,

=

3,

K,

=

#,Kt

= -V,

Z. =

&c.

MOTION OF A SINGLE VORTEX.

27

If (5n denote the sum of the reciprocals of the natural numbers up to and including n, then

Now

@n = . 577215 +]ogn + -

+ , ...,

see Boole's Finite Differences, 2nd edition, p. 93.

Thus n

We only want the value of bn when x is very small, and thus
we have approximately

_^ (i _n).

By equation (22)

=

c.,

j

i (Qbn

(01)

b , .).

If we substitute for bn and 6n+1 their values, as given above, we
find that approximately

W -4/

-(^ +f) ...(27).

The integrals we have to evaluate are of the form cosnd.dd

which may be written

cos nO . dO

where
therefore
and

- 28

ON THE MOTION OF VORTEX RINGS.

and the integral we wish to find =

- cn, if the value

_
be put for x in equation (27).

2Rp

Let us denote -

-r c .
"

when

x

has

this

value

by

S'.

Then

= + S'

5' t

COS

Ojr

-()+..

./Sr
tt

COS

tt

OJr -<)+.

Now in S'n, p and f are functions of -^,

= + + /5 a an cosfti/r /3n sin 71-^,

and

f= ^

= z' (<2r

3)

n^ + (ryn cos

S tt

sin

w'^r).

Now let 8n be the value of S'n when p = a and g=(z ).

By Taylor's theorem,

= S + + n

n

(a cos n^r

ySM sin w\|r)

5-
-y-

(y cos n

+ + p i (orw cos n^r

n

sin

2
w^)

-- - + + ( w cos n^r Pn sin n^r) (7, cos nty Sn sin n^)

+ + ^ J (yB cos ?i^|r

Sn

sin

2
ni|r)

+ terms involving the cubes and higher powers of n, &c.

+ - cos , ,fr

.
cr.-"

7,

n^ - + .
sm

, _ rfS.
ft

,, dS
8.

cos 2^ (- - ftf - 2 K7n -

2f ^? A + [ 1

BU.

o

r

a,aA

- * , (a

s

+

a\ A,7)

+ terms of higher orders.

n

+

s
7.A

MOTION OF A SINGLE VORTEX.

29

Hence, comparing these equations with 8, we see that

dS

We 13.

can now go on to find the motion of a vortex ring

disturbed slightly from its circular form. It will here be only

necessary to retain the first power of the quantities ..., so that

we shall neglect all terms containing the squares of those

quantities.

e

Fig. 2.

3C

Let fig. 2 represent a section of the vortex ring by the plane
of the paper. Let be the origin of coordinates, and let C be

the centre of the transverse section of the vortex core let the

;

CP radius

of this section

e ;

let

CP make an angle ^ with 00

produced.

Then the equations to the surface of the vortex ring are

smn^ p = a + an cos nty + /3n

+ ecosx ......... (29),

= + + ^ + z 3 7n cos wfy $n sm W

% e sin ......... (30).

F Now = if

(x, y, z, t)

be an equation to a surface which as

it moves always consists of the same particles of the fluid, then we

know that

dF dF dF dF .
S+ Stt + *3 +t**-'

w where tbe differential coefficients are partial, and where u, v, are
the x, y, z components of the velocity of the fluid at the point
4 (*> V>

The

surface

of a

vortex

ring

is

evidently

a

surface

of this

kind ;

we may therefore apply this result to its equation.

If we apply this theorem to equation (29), we find

~ ~ cosmjr 4-

n sin n\jr

(7 n sinni/r -/:?,, cos ?n/r)

ctt

(it

& X- esin ^.

= 0,

30

ON. THE MOTION OF VORTEX RINGS.

& V where is the velocity of the fluid along the radius vector, the Xangular velocity of the fluid round the normals to the vortex ring,
the angular velocity round a tangent to the central line of the vortex core.

^ Now if the

vortex

be

truly circular,

SP

vanishes ;

thus

con-

tains an and j3n to the first power ; and a^P will be of the second

order in otn, and may for our present purpose be neglected. Neglect-

ing such terms, the equation becomes

~ X & -~ cos wfr +

= ^ sin mfy e sin .

(31).

But

+ = u cos ^r

v sin *fy

i&.

R Since and f are now the coordinates of a point on the surface

of the vortex ring,

we have

R = + + a an cos nty fin sin nty -t- e cos %,

f=

+ + yn cos nty Bn sin n-fy e sin %,

and writing i|r instead of $ in equations (11) and (14), we find,

neglecting

terms

of

the

order

a

8 n,

+ = + + A ^ma u cos

v sin

^Jr

>|r

(7,, cos rnjr Sn sin n^r e sin ^) 1

4 + Jm {(w. - 1) M+1 - (n + 1) A^} ayn cos w^

A + lm{(n-l]

A n+l

-

(n

+

1)

n_J a n sin TII/T

=

^meA 1

sin

^

+

i

7?za

A
{2 l

+

(nl)

A n+l

- + wf + (w

1)

-4
W+1 J

(7n cos

8M sin tnp).

But

= + Ul

y2 cos w*|r

w sin n>^

Ctu

ti/t

X ^ e sin .

;

therefore, equating coefficients of sin ^, cos mjr, sin nty, we get

(32),

A Now as we neglect the squares of an ..., we may put n = Sn

2

R and

=+ = g a e cos ^, f ^ e sin ; that is, x -^-^ in the quantity

2idj

denoted by $n.

Making these substitutions in equation (27), we get

MOTION OF A SINGLE VORTEX.

31

thus therefore

1

a
(4a

4'nwlT-

X w -- + = --2 V V ?re

3m /, 64a9

?-*

5(lg~ '

5

167ra*V

K\
4

or, if &) bo the angular velocity of molecular rotation, so that

?*'-} ............ (36),

and

since

e-

is

small,

e*

64aa

2 log 5-

will

be

small ;

thus

we

have

approximately

X= ft>,

which agrees with the result given by Sir William Thomson in a note to Professor Tait's translation of Helmholtz's paper, Phil. Mag.
1867.

A A^ A Substituting in equation (33) the values of v

lt

n_v i.e.

S ^ in this case

lt

n_lt

S n+l

given

in

equation

(35), we find

"Yw ~ . ., .

1

/0 ^. N ......... (37)>

where we have neglected terms of the form Af(n) + C, where -4 and C are numerical coefficients, since when n is small f(n) is
64a2 small compared with ?r log 5- , and when n is large it is small
6
compared with ny(n).

Now

unless

n

be

very large,

64a2
log 5

is

very large

compared

with/(n), and the equation becomes

......

dt

*

a
7ra

'
e*

(38).

But if/(ft) be so large that/(n) is comparable with log 5- ;
then, since approximately
= + - f(ri) -288607 log 2n J log n (Boole's jFYm'fe Differences, p. 93)
equation (37) becomes

5n' (log JJ- 21544)

(39).

32

ON THE MOTION OF VORTEX RINGS.

This formula must be used when n is so large that ne is com-
parable with a.

We have exactly the same relation between d/3n/dt and 8n as
between dajdt and yn.

If we make the second of the equations to the surface of the
vortex ring satisfy the condition necessary for it to be the equation
to a surface which always consists of the same particles, we get, using the same notation as before,

7

J

*J

+w

-

-j- -J cosn-^+

sinfti/r

cit

ctt

u/t

n (yn sin nty

W X ^ SM cos nty) -f e cos . -w = 0;

or, neglecting

(yn sin n^r 8n cos nty) M/* as before

77 we find

j\

-^ f + X = w -J 4-

cos n^r +

^ sin n-ty e cos .

...... (40).

Cill

tit

Cit

But we know by equations (16) and (17) that

cos

where

+ 2aA @n n] sin n
R + + = a an cos w^r ySM sin wi|r 4- e cos ^, &c.

R A Substituting this value for

and the values of n, J3n , &c.

given in equation (28), we find

w=

cos ^ + sn

R Where in /SM, after differentiation is put equal to a + e cos %,

2

=e

and x

-x2

,

Zfl

Equating in the two expressions for w, the term independent
^ of and x, the coefficient of cos ^ and the coefficients of cos nty
and sin n^r, we get

MOTION OF A SINGLE VORTEX.

33

= 5=

Jm a*,
[a

-
{S.

U + 2-SL -

with a similar equation between dSJdt and y&n.
before differentiation

where

When n <8> has not to be differentiated, it equals

2

"/

I

The first equation gives the velocity of translation of the

S vortex ring, substituting the values for $ and

we find
t

In a note to Professor Tait's translation of Helmholtz's paper
on Vortex Motion, PAi7. Mag., 1867, Sir William Thomson states
that the velocity of translation of a circular vortex ring is
8a

This agrees very approximately with the result we have just obtained, and Mr T. C. Lewis, in the Quarterly Journal of Mathematics, vol. xvi. obtains the same expression as we have for the
velocity of translation.
X The second expression gives the same value for the angular
velocity as we had before.
-- The third equation gives on substitution and differentiation 1 .......... (42) >

T.

3

34

ON THE MOTION OF VORTEX RINGS.

A neglecting as before terms of the form Af(n) + C, where and G
are numerical coefficients.

We have a similar equation between dSJdt and /3n.

7

7

Substituting these values

for

-dt^ and

-~ in
ut

equation

(40),

we

W find that the velocity of translation

at any point on the ring

is given by

w- + iw ^r jt

*-
5 (w 1} log {

- 4/(n) - l (a cos wi }

or, neglecting 4/(n),

If p be the radius of curvature at any point of the central line
of vortex core, we can easily prove that

-=- +

+ a
(

cos nty

Pn sin n$)>

so that the velocity of translation of any point of the vortex ring
= c^ a
dt p"
thus those portions of the axis which at any time have the greatest curvature will have the greatest velocity.

Returning

to

the

equation

for

-~
,

we

have

as

before

,
where

m L = T -

, 64a2

2 log 2

(43),

except when n is so large that ne is at all comparable with a, then

L = m -- T

-

- f,

4<a?

4A ?ra2

log V

we

2n-1,5~4A4A \ .
/

L approximately ; the accurate value of is

this is the same coefficient as we had in the equation giving dzjdt
so that our equations are

MOTION OF A SINGLE VORTEX.

35

~ Differentiating the first of these, and substituting for

from

the second, we find

the solution of which is
an = ^ cos
and therefore

8in [L v {nt(nt ~

A B where and are arbitrary constants.

We can shew by work of an exactly similar kind, that

A VK &, = cos [L

-
(n* 1)}

These equations shew that the circular vortex ring is stable for
all small displacements of its central line of vortex core. Sir
William Thomson has proved, that it is stable for all small alterations in the shape of its transverse section, hence we conclude that it is stable for all small displacements. The time of vibration
2?r

27T

47m2

''

where

-T / (n) = 1 +*4 J +

Thus, unless w be very large, the time of vibration
2?r

V or, if be the velocity of translation of the vortex ring

2?r

a

Thus for elliptic deformation the time of vibration is "289 times

the time taken by the vortex ring to pass over a length equal to its

circumference.

32

36

ON THE MOTION OF VORTEX RINGS.

_ When ne is at all comparable with a, the time of vibration is

approximately

27T

27m2

'
V- m I (log

-1-0772)'

or, since we may write, as n is large, nz instead of n*

-- if I be the wave length

,

1, it equals,

27T

Now this case agrees infinitely nearly with the transverse vibra-
tions of a straight columnar vortex which have been investigated by Sir William Thomson.

In the sub-case in which l/e is large, he finds that the period
of vibration

27T

(Phil Mag., Sept
this equals

2o>7rV (log
1880, p. 167

+
eq. 61) ;

or, since

loge 2 = '62314,

27T

and thus agrees very approximately with the value we have just
found.

Since the amplitudes of <xn and /3n when n is large are approximately the same as those of <yn and 8n, we can represent a displacement of this kind by conceiving the central line of the vortex core to be wound round an anchor ring of small transverse section so as to make n turns round the central line of the vortex ring, and

this form

to

travel

along

the

anchor ring

with

velocity

,

where

r

is the time of vibration just found and I the wave length.

PART II.

To find the action of two vortices upon each other which move so as never to approach closer than a large multiple of the diameter
of either.

14. The expressions for the velocity due to a circular vortex

AB ring, which we investigated in the previous part, will enable

us to solve this problem. If we call the two vortices

and

AB CD CD, then in order to find the effect of the vortex

on

we must find the velocity at CD due to AB. Now, since

the vortices never approach very closely to each other, they

will not differ much from circles; hence in finding the velocity

due to one of them at a point remote from its core, say at the

surface of the other, we may, without appreciable error, suppose

the vortex ring to be circular.

Let the shortest distance between the directions of motion

of the vortices be perpendicular to the plane of the paper ; thus the plane of the paper will be parallel to the directions of motion of both vortices.

AB m Let the semi-polar equations to the central line of the vortex of strength (fig. 3) be

= + + 2 p a

(aB cos n<j>

n sin n$),

= + S + $

(7,, cos n<f> Bn sin n<f>),

AB when z is measured perpendicularly to the plane of the vortex

and </> is measured from the intersection of the plane of the vortex
AB with the plane of the paper ; y an, /3n, n) Bn are all very small m compared with a. Let be the strength of the vortex AB.

CD Let the equations to the central line of the vortex

of

m strength be

= + 2 + p b

(a'n cos nty' f?n sin nty'),

n' n' = *' ' + 2

cos

+ S' sin

38

ON THE MOTION OF VORTEX RINGS.

where z' is measured perpendicularly to the plane of the vortex CD, and ty' from the intersection of the plane of this vortex with the plane of the paper; a'n, fi'n, y'n, S'n are all very small in comparison with b.

We y shall have to express orn, y M, w, 8B, a'n, /3'M , y'n> S'n as

functions of

the time ;

we

shall then have found the action of the

two vortices on each other.

AB Z To find the action of

CD on

let us take as our axis of

AB the perpendicular to the plane of the vortex

through its centre,

XZ the plane of

parallel to the plane of the paper and the axis of

Y drawn upwards from the plane of the paper.

Let e be the angle between the direction of motion of the two

vortices ;

I,

m,

n

the

direction-cosines

of

a

radius

vector

of

the

CD vortex

drawn from the centre of that vortex.

X Z X Let Z, (fig. 4) be the points where the axes of and

cut

Flg.4.

K a sphere whose centre is at the origin of coordinates,

the point

CD where a parallel to the direction of motion of the vortex

cuts

P this sphere, and the point where a parallel to the radius vector of

KP the vortex CD cuts the sphere :

will be a quadrant of a circle.

Then we easily see, by Spherical Trigonometry, that

I=

COS COS -vjr,

m = sin -fy,

= 7i

sin e cos ^Jr.

Now w by equations (10, 13, 16) the velocities u, v, parallel to

Z AB the axes of X, F, due to the vortex

supposed circular are

given by the equations

MOTION OF TWO VORTEX RINGS.

39

_ _ where Since 1

R = X*+Y*.

__ ___ 1

a
(o -2aflco80)

8
(a

+

tf

+

Z*

-

2a#

cos

0)*

where, since R* + Z* is very great compared with a, the terms
diminish rapidly,

and

= -4, SaR

f Now if t g, h be the coordinates of the centre of the vortex Z CD, and X, Y, the coordinates of a point on the central line of

that vortex,

Xf+bl =f+ 6 cos e cos -|r,

Y = g + bm = g + b sin ty,

h+ Z =

= In h b sin e cos ty ;

therefore

^ Z E2 + Z2 = F 2 + 2 +

/ + f + + + + 2

^2

2& (/cos e cos i|r

g sin i/r

/i, sin cos ^r)

2
6.

15.

w = ;r^ maXZA. = | ma2

Z Substituting the values given above for X, Y, and writing d*

+ 2

2

for/ -f #

h*

+

2
6

we

find

that

approximately

+ - f sin e cos e -~ (h cos e -/sin e) (/cos e ^ sin e)

-

h

sin

2 e)

-
/j

cos

sin e - h cos e + -^fg (/cos e - h sin e)J |^ sin 2^ +...... (47).

40

ON THE MOTION OF VORTEX RINGS.

When in these expressions we have a coefficient consisting of
several terms of different orders of small quantities we only retain
the largest term.

=1
16. v

Substituting as before we find
v = fma2
~ sn

+ f f cos eg (/sin e

h

cos

e)+

Y~J*

{

(/cose-

Asm

e)

2

2
-#

}J

-j-7

cos

a

- + n - - //5flr

,\ . sin e

^ -^ 1J

5^,,
(/cos e

hj

sin

^
e)

3
+ y(/cose-Asin ))|-6 sm2^] ......... (48).

17.

w^im^cfA.-aEA^

1

)

i -

2
(/

+

2
^)

+

3

- (A sin e 2/cos e)

[2

^2

+

2
J(/

+/)

(/cose

-Asine))^

cos

(ft^ + - - 2 sin e

{(/cos e h sin e) (3/cos e h sin e) g*}

-

+ - - 8
(/

2
<7 ) (( /cos e

A

Bin

2
e)

2 </ }

cos

+ 15 f 3/cos e h sin e

-

(/' + /) (/cos 6 -'1 sin e)

2f + sin

...

. . . (49).

MOTION OF TWO VORTEX RINGS.

41

18. In using these expressions to find the effect of the vortex
AB on CD, we have to find the velocity perpendicular to the

CD plane of

and along the radius vector. Then, as in the case of

-~ the single vortex, we have equations of the type

= coefficient

of cosnijr in the expression for the velocity along the radius
vector of CD.

To solve these differential equations, we must have the
quantities on the right-hand side expressed in terms of the time.
Hence we must express the value for w u, v, which we have just
obtained in terms of the time.

19. In the small terms which express the velocity at the

CD vortex

due to the vortex AB, we may, for a first approxi-

mation, calculate the quantities on the supposition that the motion

is undisturbed.

Let us reckon the time from the instant when the centres of

the vortices are nearest together.

AB Let p and q be the velocities of the vortices

and CD

+ respectively ;

k

the

relative

velocity,

viz.

2
V(p

2*

2p<? cos e) ;

c the shortest distance between their centres.

CD Then, since /, g, h are the coordinates of the centre of

at

the time t,

/=f -f qsme.t,

9=&

= + h j)

(q cos e p) t,

where f, g, J are the values of /, g, h when t =

distance between the centres of the vortices, viz.

a minimum when t 0,

therefore

= iq sin e -f |) (q cos e p)

;

" f = -*-

q cos e p qsm

since the
;

therefore if |j be positive, we have

_
f

and

f w 20. If we substitute for t g, h in the expression for their
values in terms of the time, we find that as far as the term inde-

pendent

of i|r

goes,

42

ON THE MOTION OF VORTEX RINGS.

1 2

^i ma*

[f3 (

2
(<r

f2 sin e
p g) :?

2\ c)

H

pf<f 7; - (q cose p)q sine. t+ (2 (gr cose

sin2 2
e}

...(52).

The coefficient of cos i/r

= f raa2 2 f^ j

~% 2 + - {q (sin e

2

2
cos

e)

2p cos e}

-sin e (q cose+p)t}

^ l

, + (L + Mt + Ne + Pt5)- 5

l
T

gg

where

(53),

4
'Of
(^> cos e

- + / 2
q) I c (p

q cos e) 2

eftf

sin2

\ e
J

2
- +pq - + (jp

q

cos

e)

2
(p

cos e

2
2^*) -f Q ^ (pg' (3

8
cos e)

-
A/

2

2

sin e . g {2p

P = sin3 e q*p.

gp cos e

The coefficient of sin

_ -_-- '
r J1f /0^2* -

2

R

(c
_^

i)

T

2

g*__')

xvt* (/

bBri.alTlk2* ^C

+ q sin e (p

q cos e) t

2
(5^

sin2

e

2
4P) j

(54).

The coefficient of cos ty may be written

-g 72 f mab p \

2

2

sin e

/c

1 +(

2
x
-(pcose <?)(3cose(_p

^cose)

^sin e)

H ^r
A/

+ sin e [2p (p cos e ^) q (p q cos e)}

+ +p sin ejp (^ sin 2e

sin e) f J

-^j

2
(c

+

;(
W)^J

(55),

MOTION OF TWO VORTEX RINGS.

43

where

L' =

-pf + gV - ( c* (q cos 6

sin' e) (c* (p cos e q)*

2 Jc* n*
JF

- - + cos e q) (2pq cos e (p* q*)) + (q cos e

+ 2
p) (p (f

--
a

cos e -)

cos ~

2pq cos e))},

P,3, 2 sin'

(p cos e-^)J,

The coefficient of sin 2-fr may be written

* ma'g f j

~

- + ^ {3p cos e

2
5 (3 cos e

2
sin

e) J

(56) -
P where L, M, N, have the same values as in equation (53).
21. The velocity parallel to the axis of y.
^ The term independent of

^ The coefficient of cos

S

~ .

+

5

- sin 6 . 2 (p cos e q)

The coefficient of sin

44

ON THE MOTION OF VORTEX RINGS.

The coefficient of cos 2\/r

= ma2 2

f

6 fi

e

~* - (p sin e q sin 2e)

*

jcos

where

L" =

^

2

2

,3

{(c Q ) (p sin e q)*

M" = - + + 2 Ta- {(c

2
g ) (p cos e

q) [(2

2
sin e) pq

cos

e

2
(p

q

N" =

~P + - + sin e {pg (3

2
cos e)

2 (p*

^) cos e},

= P"

sin2 e

2
.p

(q cos

e

^).

The coefficient of sin 2-r

^

~

-

+ + J *

9

9
{2p2- cos e (p

2

-p) 2 )J * -1- sin e .p (qcos e

t*} /

2

(c

A;V) 2 J

............... ...... (61).

22. The velocity parallel to the axis of x.

The term independent of i/r

| ma2 \

, ^ sin e . q (p q cos e)

~%
^

[q*

cos

2e-2p2

cos

e

2
+p }

i

+sin

e.

(62).

The coefficient of cos ty

f wa26 ][

-^ -p n , ^ fe sin2e

sine) -1- (#cos 2e

cos e)

^ g-

2

MOTION OF TWO VORTEX RINGS.

45

where

L'" = --

-p sin . q (q cos c-p)(q

cos e),

M'" =

. (gcos2e-p2*cos (008* e + 2) +/?(2 + cos'e) -p'cos e),

= N"' sin e .

T^ + + + - 9 (- q cos e

<fp (1

sin* e)

2
gp

cos

e

8
p ),

P'"

-

a
sin

e.pq(q

cos

e

/?).

The coefficient of sin ty

^^ = - J# ma'5g I' ^8-2 sin 6 . q (p -^ cose) -

(<f cos 2e

- + 2
2pg cose+p ) t

-p) f sin e.g' (q cos e

h

...(64).

* '. Qcr "t*

j
^J

The coefficient of cos 2^|r

2,2of sin e cose

+ +/ -pq '"'/)

sin

e

2
{2 2

cos

e

(1 2 cos2 e)

cos e}

2

-2^cose+p )^+sme.p(^cos2e-pcose)n-^

^

v (A + J#+ JV + Pf +

where

sin e . g (5 cos e

p)

2
{c

(p

cos

e

cose -

-.
^

2
~- - pf + {(c

2
g)

(p

cos

e

q)

9
(q

cos

2e

cos e (cos 2e coss e)

K>

.

- 8 +p + -p + g (1

2 cos 2e)

p 3 cos e)

2
tf&

9
(q

cos

2e

2
2pj cos e -f )},

= ~ - + ^C

^

^

sin

e

[q

(q

cos

e

-p)

2
(p

cos

2e

2pq cos e

7*)

AJ

+ - + 2p (p cos e

q)

9
(q

cos

2e

<2pq cos e p*)},

2

2

3

p - - sin e . (3p9 p

s
23

cos

e),

Q = - l

3
sin

e

. p*q

(q

cos

e

p).

46

ON THE MOTION OF VORTEX RINGS.

The coefficient of sin 2 ty

+ (q cos 2e

= -p ^~ J 7/za262
1- (

g2 (g

sin

2e

sin e)

'---* /? cos e ) t )

+ -.

7 ( ^

~*7f (pcose -q) (p-q cos e)

AB 23. To find the effect of the vortex

on CD we require

the expressions for the velocity perpendicular to the plane of the

CD vortex ring

and along its radius vector.

The velocity perpendicular to the plane of CD = w cos e + u sin e.

Now in this expression, the most important terms are the coeffi-

We cients

of

cos T/T

and

sin ty,

because

these

terms,

as

we

shall

see,

determine the deflection of the vortex.

shall therefore pro-

ceed to find these terms first.

The coefficient of cos ty in the expression for the velocity per-
pendicular to the plane of CD may be written as

where

A=

i

- 2
{c (pcos 2e

#cos e)

5

a
sin

- pq ^
,2

(p cos e <?)},

5= 2 c

(p

sin

2e

q sin e)

2

2

5 (c

f"l )

5

+ - ,2

{q

<f .p cos e

qp* (1 +sin2 e) +p* cos e),

C=

r

+ 2
{4A;

(g'

cos

e

p)

2

2

sin e .^? (8p

2
7^

^^ cos e)},

rC

D = + 2

2

sin e }5^) g sin e

k* (q

3p cos e)J.

The coefficient of sin ty

where

K /"^2

*21 \

, 2 j^sih'e,

sn e cos e - sn e

= p (7' ^2 cos e 5 (q cos e p) (q

cos e).

MOTION OF TWO VORTEX RINGS.

47

Now, since the equation to the vortex CD is

= + 2 + z j'

(y'n cos n-\fr S' n sin n-ty}.

The velocity perpendicular to the plane of the vortex

since as 8',,, 7',, and

are all small quantities we may neglect

n tf cos n\r

sin n V.

= Thus -7^ coefficient of cos -Jr in the expression for the velodt
city perpendicular to the plane of the vortex CD.

A CD reference to equation (43) will shew that the vortex

con-

tributes nothing to this term, so that

d

ma*b

&

Integrating we find

where the arbitrary determined so as to

constant
make y\

=ariswihngenfrto=m

the
oo

integration
.

has

been

If we substitute for A, B, C,D the values given above, we shall

get the value for 7', at each instant of the collision ; but at present

we shall only consider the change in 7^ when it has got so far away

AB We from the vortex

that its motion is again undisturbed.

can

find this change in 7^ by putting t = oo in the above formula, on

doing this we find

A C or substituting for and their values,

, 2ma?bpq ,

= -^ ~ - 7 1

fe

. /cos e) I 1

-4(f\
)

., 2
V(c

sm' 2 .

.

fi )

6.. . .(69).

We 24.

have similarly

~ = CLO
\ coefficient of sin 1^ in the expression for the velocity perpen-

dicular to the plane of the vortex

48

ON THE MOTION OF VORTEX RINGS.

Integrating we find

t ...... (70) '

AB where the arbitrary constant arising from
determined so as to make = when 8'j

the
=

integration has been
oo . The change in

when
S'j

the vortex

CD is

so far away from

that its motion is

undisturbed is given by

Substituting we find

We 25.

have in paragraph (6) investigated the changes in

the direction cosines of the direction of motion of the vortex ring
due to changes in the coefficients <y\ and S\. From that investigation we find that the direction cosines of the direction of motion
of the vortex CD after the impact are

i

sin e f4y-1 cos e,

4

+ 4 COS

ry'1 Sin 6,

or

substituting

for

7^ and

S' the values t

just found, the

direction

cosines become

-- V - sin e

2
2wift
^-

- 2

/

(c

(f) f 1

4o2 \ -4j

2
sin e

cos

e.pq

(q-p

cos

e),

cos e +

2

2

V - - (c

S) l

sin3e. pq(q-p cos e).

P Thus if C A, B, (fig. 5) be the points where the axes of x, y, z

cut a sphere with the origin for centre and

the point where

a parallel through this centre to the direction of motion of the

CD vortex

before the collision cuts the sphere.

CD Then if the vortex

be the first to intersect the shortest

MOTION OF TWO VORTEX RINGS.

40

distance between the directions of motion of the vortices, P' will be the point where a parallel to the direction of motion after impact

Fig.G.

AB cuts the sphere, supposing g to be positive and < \c and the

CD velocity of

greater than the velocity of

resolved along the

We direction of motion of CD, i.e. if q p cos e be positive.

may

^

describe this by saying that the direction of motion of the vortex

ring is altered in the same way as it would be if the vortex ring

received an impulse parallel to the shortest distance between the

directions of motion of the vortices and another impulse perpen-

dicular both to its own direction of motion and the shortest

distance ;

the first impulse being from and the second towards the

vortex AB. In this case the angle between the direction of motion

AB of CD and the original direction of motion of

is diminished by

the impact.

AB If the vortex

be the first to intersect the shortest distance

then

we

must

change

the

sign

V of

2
(c

tf) in the expressions

for f and j; this will change the sign of <y\ but will leave S\
unaltered, and consequently P" the point where the direction of

CD motion of

after the impact intersects the sphere of reference

will be situated as in the figure ; in this case the angle between

CD the direction of motion of

and the original direction of

AB motion of

is increased by the impact. The angle through

CD which the direction of motion of

is deflected

If the becomes

paths

..................... (72).

= of the vortices intersect so that (J

0, this

2

sin e ,

._.,.

pq(q-pcose) ..................... (73),

T.

4

50

ON THE MOTION OF VORTEX RINGS.

or the deflection is cceteris paribus inversely proportional to the cube of the shortest distance between the vortices.
If the paths of the vortices do not intersect, but the vortices
= move so as to come as close together as possible, then c g,
and the deflection

This is again inversely proportional to the cube of the distance.

If in the two cases above, c be the same, then the deflection

when the paths of the vortices intersect will be greater, equal

to or less than when they do not, according as 8 (q

2
_pcose) is

greater,

equal

to, or

less

than

2

p* sin

e ;

thus,

unless

the

relative

velocity of the vortices perpendicular to the direction of motion of
CD is great compared with that along CD, the deflection will be

greater when the directions of motion of the vortices intersect than

when they do not.

The expression for the deflection simplifies when the line

joining the vortices at the instant when they are nearest

= = together is inclined at an angle of 30 to the shortest distance
between their directions of motion, in this case g c cos 30 c |\/3,

thus 8' = as 1

~2
4ft
oC2 vanishes, and the deflection

(qp 2wm2

2
sin e

.

pq

cos e)

c*k*

which, if c be the same, is the same as when the vortices intersect.

We 26.

have next to consider how the vortex CD is

altered in size by the collision.

We know that if a' be the alteration in the radius of the
vortex CD that

~ = coefficient of the term independent of -^ in the expression
cLu
for the velocity along the radius vector of CD.

Now a reference to equation (38) will shew that the vortex CD
contributes nothing to this term itself, so that

= 5
-y-

coefficient of the term independent of ty in the expression

(Lt

CD for the velocity along the radius vector of

due to the vortex

AB.

Since X, /JL, v, the direction-cosines of a radius vector, are by 6 given by the equations

MOTION OF TWO VORTEX RIN

X=
=
fju
v=

COS COS >/r,
sin ^,
sin cos yfr,

-- = coefficient of the term independent of ^r in
(it

+ w u cos e cos ^r

v sin i/r

Bin cos 1

Hence by equations (53), (59), (63),

where

p= _ _ - - * ,

sin
V(c^j^)

p g c

j-

j

(4

cos*e)

2/cos e

8
g}

2

2

og sin * . jfy],

G=c<{(,cose-,o(2-^

H & = ^ fc* ~

- + sin

e

8
(8p

cos

e

-jfq

cos2

e

lljfq

3
4j ),

K = Jf (2 (3 cos

+ p) 3p sin* e}

Integrating, we find

5pg sin'e (?

p cos e).

t

/&P 2ff
U6 +

where the arbitrary been determined so

constant arising
as to make = a'

from the
when t

=integration cc. If

has
we

K substitute for F, G, H,

the values just written we shall get

the change in the radius at any instant, but at present we shall

CD only consider the change in the radius of

when it has got

AB so far away from the vortex

that its motion is again

We undisturbed.

can find this change in the radius by putting

t

oo

in the above formula ;

doing this we find

ma?b 4F
,

5*

F H Substituting for and their values, we find

ma?b

3
sin

e

.

Thus we see that the radius of the vortex which first passes

through the shortest distance between their directions of motion is

AB increased, provided c> 2g. If

had first intersected the shortest

42

52

ON THE MOTION OF VORTEX RINGS.

distance we should have had to change the sign of *J(c* (f), then
a' would be negative, and the radius of CD would be diminished.

If the directions of motion of the vortices intersect, so that

g = 0, then

,

maz

3

*

b sin e .

or the increase in radius is cceteris paribus inversely proportional to the cube of the shortest distance between the vortices.

If the directions of motion of the vortices do not intersect, but

the vortices move so as to come as close together as possible, then

= c

and a'
g,

0, and the radius of the vortex in this case is not

altered by the collision.

= If c 2o;, or if the line joining the vortices when they are

nearest together be inclined at an angle of 60 to the shortest

distance between the directions of motion of the vortices, then

= '
a o

0, or in this case again the radius of the vortex is not altered

by the collision. Thus we see for our present purpose we may

divide collisions into two classes. In the first class the line joining

the centres of the vortices when they are nearest together is in-

clined at an angle greater than 60 to the shortest distance between

the directions of motion of the vortices. In this case the vortex

which first passes through the shortest distance increases in radius, and consequently decreases in velocity and increases in energy, while the other vortex decreases in radius and energy and increases

in velocity.

In the second class of collisions the line joining the centres of
the vortices when they are nearest together is inclined at an angle
less than 60 to the shortest distance between the directions of
motion of the vortices. In this case the vortex which first passes through the shortest distance decreases in radius, and consequently increases in velocity and decreases in energy, while the other vortex increases in radius and energy and decreases in velocity.

27. Having found the change in the radius and the change in the direction of motion of the vortex, we can find the changes
in the components of the momentum of the vortex referred to any

axes.

F Let

be the momentum of the vortex

CD ;

1$, (&',

Hi' its com-

ponents

parallel

to

the

axes

of x,

y,

z

respectively,

I',

m' }

ri

the

direction-cosines of the normal to the plane of the vortex.

Thus

5'

so

8'

similarly,

MOTION OF TWO VORTEX RINGS.

= + ar

8&V 2 -~*',, <S'

'

b

Now It remains to find SZ', 3m', 8n' in terms of y' and

' .

if

AB I, ^P, (01, HI denote the same quantities for the vortex

as the

same letters accented do for the vortex CD, then it is easy to

prove that the direction-cosines of the old axes referred to the new

are as follows.

The direction-cosines of the old axis of a? are

E.Fsine

are -an? cose an -air cose

5.5'sine '

I.Fsine

The direction-cosines of the old axis of y are

I. r sine

I.l'sine

The direction-cosines of the old axis of z are

''''

g"

'

5J

'

3E

Thus if X, /^, v be the direction-cosines of the normal to the

CD plane of the vortex

referred to the old axes, then

gy,_

5.$' sine

.
I.* sine

\vith symmetrical expressions for Bmf and Sri.

Now by 6

SX =

1 cos e,

&/ =

T1 sin e.
6

Substituting for y\ and S'j their values, we find 2ma?pq sin e f^ p cose //3

with symmetrical expressions for Sm' and 8'.

54 Thus

ON THE MOTION OF VORTEX EINGS.

/- 4\

(

(?)

/_
\

-jp)j +9(1

......... (75),

with symmetrical expressions for 8(fH' and 8<Kt'.

If < be the angle which the line joining the centres of the
vortices when they are nearest together makes with the shortest

distance between the paths of the centres of the vortex rings,

then

_

= g

c cos <f>,

so

= - = - - f 1

4 2 7c2 -g?

2
c sin < (4 sin ^> 3)

c sin

and Thus

- cos
with symmetrical expressions for 8(0)1' and
^ ^ Since + r is constant throughout the motion

similarly

8(& =

We 28.

can now sum up the effects of the collision upon the

AE We vortex rings

and CD.

shall find it convenient to express

them in terms of the angle </> used in the last paragraph : < is the angle which the line joining the centres of the vortex, rings when

they are nearest together makes with the shortest distance between

the paths of the centres of the vortex rings, < is positive for the

vortex ring which first intersects the shortest distance between the

paths, negative for the other ring, so that with a given may ft, <f>

be regarded as giving the delay of one vortex behind the other.

29. Let us first consider the effect of the collision on the radii of the vortex rings.

MOTION OF TWO VORTEX RINGS.

55

The radius of the vortex ring CD is diminished by
ma*b .

Thus the radius of the ring is diminished or increased accord-

Now ing as sin 30 is positive or negative.

is positive for one

vortex ring negative for the other, thus sin 30 is positive for one

vortex ring negative for the other, so that if the radius of one

vortex ring is increased by the collision the radius of the other
will be diminished. When is less than 60 the vortex ring which

first passes through the shortest distance between the paths of the

centres of the rings diminishes in radius and the other one increases.
When is greater than 60 the vortex ring which first passes

through the shortest distance between the paths increases in radius
and the other one diminishes. When the paths of the centres of

the vortex rings intersect is 90, so that the vortex ring which

first passes through the shortest distance, which in this case is the

point of intersection of the paths, is the one which increases in
radius. When is zero or the vortex rings intersect the shortest

distance simultaneously there is no change in the radius of either

vortex ring, and this is also the case when is 60.

30. Let us now consider the bending of the path of the

centre of one of the vortex rings perpendicular to the plane through

the centre of the other ring and parallel to the original paths of

both the vortex rings.

We see by equation (71) that the path of the centre of the

CD vortex ring

is bent towards this plane through an angle

this does not change sign with 0, and whichever vortex first passes
through the shortest distance the deflection is given by the rule that the path of a vortex ring is bent towards or from the plane through the centre of the other vortex and parallel to the original directions of both vortices according as cos 30 is positive or negative, so that if is less than 30 the path of the vortex is bent towards, and if be greater than 30 from this plane. It follows from this expression for the deflection that if we have a large quantity of vortex rings uniformly distributed they will on the whole repel a vortex ring passing by them.

31. Let us now consider the bending of the paths of the

vortices in the plane parallel to the original paths of both vortex

CD rings. Equation (69) shews that the path of the vortex ring

is

bent in this plane through an angle

.o .
e sm

,
^ pq

(q

"~

p

cos

e)

56

ON THE MOTION OF VORTEX RINGS.

towards the direction of motion of the other vortex. Thus the

direction of motion of one vortex is bent from or towards the

direction of motion of the other according as sin 3(/> (q p cos e) is

positive or negative. Comparing this result with the result for

the change in the radius,, we see that if the velocity of a vortex

AB CD ring

be greater than the velocity of the other vortex

resolved along the direction of motion of CD, then the path of

each vortex will be bent towards the direction of motion of the

other when its radius is increased and away from the direction of motion of the other when its radius is diminished, while if the

velocity of the vortex be less than the velocity of the other resolved
along its direction of motion, the direction of motion will be bent
from the direction of the other when its radius is increased and

vice versa. The rules for finding the alteration in the radius were

given before.

32. Equation (75) shews that the effect of the collision is

the same as if an impulse

'

sm. 2

.

,

e sin 36,

parallel to the resultant of velocities p q cos e and q p cos e
along the paths of vortices (CD) and (AB) respectively, and an
impulse

parallel to the shortest distance between the original paths of the

vortex rings, were given to one of the vortices and equal and

opposite

impulses

to

the other ;

here

5 and

5' are the momenta of

the vortices.

We 33.

have so far been engaged with the changes in the

magnitude and position of the vortex ring CD, and have not

considered the changes in shape which the vortex ring suffers from

the collision. These changes will be expressed by the quantities

We a a 2, /32 ,

s, /33, &c.

must now investigate the values of these

quantities.

Now we know

-~ coefficient of cos 2i|r in the expression for the velocity along
Cut

the radius vector.

A reference to equation (38) will shew that the vortex ring
CD itself contributes to this coefficient the term

2m' . 86 ,

MOTION OF TWO VORTEX RINGS.

57

AB The vortex ring

contributes, as we see from equations (53),

(59), and (63), a term

ma'b

where

8

F = + c j t sin e [p*q (2

cos'e)

4>pq* cos e

S<f

8

2p

cos
e] ,

ff =

= 3^ JfT

p sin2e

2
5pq sin e (q

cos e),

where, in order to make the work as simple as possible, we

have

put

=
Q

; so that the undisturbed paths of the vortices

intersect.

Thus

say

d*

2m', Sb ,

-~ Now

= the coefficient of cos 2>|r in the expression for the

velocity perpendicular to the plane of the vortex CD.

CD The vortex

itself contributes to this coefficient the term

. m'

86 ,

f^log^-.a,

AB The vortex

contributes, as we see from equations

and (65), the term

ma*b

G H R L ,

,,

,, , t

f ,,

,y

(55)

= Say for brevity F(f), where if, as before, we put g 0,

4

c

n F"

(p cos e q) { (p cos e q) (Spq sin2 e &2 cos e)

+ p 5&2 ($p sin2 e

^ cos e)),

+

( p cos

e

q)

p

(V

p? sin' e

-

58

ON THE MOTION OF VOETEX KINGS.

^,, =

(p cos e qf] (q cos e p)

F O - + + - 5F J

}f cos e .

2

2
? ) cos e

2pg)],

^ ^ ^ ^ gcsine K

_^ ^2
sin2 g

_^

g+ ^

= + L" J^9 sin2 e {2lp (q p cos e) (q cos e -p) kz (5p cos e 6q}}.

Thus

^ differentiating this equation, and substituting for

from the

other equation, we find

^ W W ^ w dV

+3Q

/ m/ i 1
(7rF

g

2

+ 2 /

u\

m/

q

T

72==jP

t-p.^

= % y (*) sa ;

m /

, 26\ 2

or writing w for 3 ( 9 log -r ) ,

j-

The solution of this differential equation is
= + -4 cos TI -S sin nt

cos ni
n

-- .

sin r^ f*

. ,,

n

-| J

%(
'

or choosing the arbitary constants so that <y'z and -

both

vanish when t

oo we find ,

,
= - p ^ sm -- ^J-oo % 7

cos n-t n j _ a,

..

.

,

(f)

n ., ,,,
Ji

sin ?i-^ /"*

,,

, ,,

(0 cos wi dt .

The integral

/
v(lT) want dt'
J -00

involves integrals of the form

, {

I have not been able to evaluate these integrals except
when = oo .

MOTION OF TWO VORTEX RINGS.

59

AB AB In the expression for 7', the terms under the integral express

the effect of the vortex

on CD. Now the vortex

will

CD only exert an appreciable effect on

during the time the

vortices are in the neighbourhood of the place where they are

nearest together ; and thus, after the collision, we may, without

appreciable error, write the equation for y't as

, _ Pcosnt Qain nt

where

f+co

P= I

% sin nt . (t) dt,

J 00

r+ca

Q=

x cos nt . (t) dt.

J -oo

Thus the vortex rings are thrown by the collision into vibration, and after the collision is over the period of the vibration is

, the same as the period of the corresponding free vibration of

the vortex CD.

To find Pand Q we have to find

cos nt . dt

or if we write q for kT ,
/::

cos nt . dt

Now q is the time taken by the vortices to separate by a

distance c, while --

is
(

13) of the same order as the time taken

by the vortex CD to pass over a length equal to its diameter;
but, since c is large compared with the diameter of the vortex,
~27T or nq is large.

n

i-

.

Let

Jo

nt cCoOsu dt it. i/ . ix/i/
1|0 .^
(a*

=n

p v.

By differentiation we find

Vl = ~ A>

Hence we find

f~d

" p--ll\(d

~~

P

-2\ ~

d
~

n

I T^
J \dn

I
n J dn

CO

OF THE MOTION OF VOKTEX RINGS.

This may be written

We can

easily

verify

that

v
p

satisfies

the

differential

equation

l*

A

^dn-*

n

p
dn

_/^z \n

*

1

J

Let us assume

If we substitute this expression for vp in the differential equation, and equate to zero the various powers of n, we get the
equations

-qAt -

xA -p*A =0,

+ a? 2) 4, + (a + !)(* + 2) .4 -qA 2 - (x + l)A l -p*A l =0,

these give therefore

m_-qAm-(x+m- 1)A^-tfA^=

ZqA t + +

(i-pV. =0,

2
(|-P

)A

=0,

M +

[i

(2m

-

2
I)

-/]

^

= ;

~

-'

A and A alone remains to be determined; if we can determine for

any value of p, we
(76). Now when p

can find
= 0,

it for

any other by means of

equation

'"cosnt.dt

tf + 0*

and

cosnt.dt

,,. \

-i=K(t.nq)

MOTION OF TWO VORTEX RINGS.

61

K (Heine, Kugelfunctionen, vol. II. 50), where

is the second kind

J of Bessel's function of zero order and i =

1.

When nq is large,

(Heine, vol. I. 61); hence

and, by equation (76), we find on comparing the coefficient of
i that

therefore

cos < .

and this series converges rapidly when nq is large. The other integrals in Q are of the form

and these evidently vanish.

P The integrals in are of the forms

sinnt.dt

m

*

tsinnt.dt

The first of these evidently vanishes, and the second
cos nt . dt

and we have just found the value of the integral.

62

ON THE MOTION OF VORTEX RINGS.

We 34.

can

now

find

the

values

of <y' z

and

a'2 .

By28,

7> *5!*_g-E-,

where

/* -f-oo

P= I

sin nt .

J oo

r+ao

=

cosnt.y (t) dt.

J oo

If we substitute for ^ () its value, and evaluate the integrals by means of formula (77), and retain only the largest terms, we
shall find

(qp + 2

- p 4>p (<f

)

cos e) cos e (<f

p*)*} . n5

,

(nc/k)

-
x,

.
- (V - - - sm e

(p cos e

2
g)

2
(g

n - .

22

5

j? ) } .

.

2

(we/A;)

If the vortices move with equal velocities these expressions simplify very much and become

-- m j-DP=

V

(2ir) 8QATjJ

o'6V

e- wc/A

-cose-

yh ,

(nc/k)

-W- n
<2=

m
"

V (2w)

sm

~W8Ar

so that

a>- therefore

Vo . %K

+ sin (trf

e) ...... (79).

(nc/k)

These equations represent twisted ellipses whose

ellipticity is

m V(27r) a2Zm4 e" c/*

5
V3.2A;

(nc/]jf

The time of vibration is the corresponding free period.

greatest

We 35.

can now sum up the effects of the collision of two

AB vortices

and CD.

The collisions must be divided into two classes, (1) those in which the shortest distance between the vortices is greater than twice the shortest distance between the directions of motion of

the vortices ;

(2) those in which it is less.

MOTION OF TWO VORTEX RINGS.

63

Class I.

CD If the vortex

be the first to intersect the shortest distance

between the directions of motion of the vortices its radius is

increased, and if its velocity is greater than the velocity of AB,

resolved along the direction of motion of CD, it is bent towards the

direction of motion of AB, and away from the plane containing

the path of AB, and a parallel to that of CD. If its velocity is

less than the value stated above it is bent from the direction of

AB motion of

and away from the plane containing the path of

AB the centre of

and a parallel to that of CD. This is the direction

CD AB in which the path of

is deflected if

first intersects the

shortest distance between the directions of motion of the vortices,

CD but in this case the radius of

is diminished.

Class II.

CD If the vortex

be the first to intersect the shortest

distance between the directions of motion of the vortices its

radius is diminished by the collision. It is bent from or towards

AB the direction of motion of AB greater or less than the velocity of

according as its velocity is resolved along the direction

AB of motion of CD, and away from or towards the plane containing

the path of

and a parallel to that of CD, according as the

shortest distance between the vortices is greater or less than

2

^ times the shortest distance between their directions of motion.

V3

AB The deflection of

with reference to this plane is the same

AB CD AB whether

or

first intersect the shortest distance. If

be

CD the first to intersect the shortest distance, the radius of

is

CD increased, and the deflection of the path of

relative to the

AB direction of motion of

is the opposite of that when CD was the

first to intersect the shortest distance.

When the directions of motion of the vortices intersect, these
results admit of much simpler statement, and, though included in Class I., it may be worth while to restate them. In this

case the result is that the vortex which first passes through the

point of intersection of the directions of motion of the vortices

is deflected towards the direction of motion of the other; it

increases

in

radius

and

energy, and

its velocity

is

decreased ;

the

other vortex is deflected in the same direction, it decreases in

radius and energy, and its velocity is increased.

36. Very closely allied to the problem of finding the action of two vortices on each other is the problem of finding the motion of one vortex when placed in a mass of fluid throughout which

64

ON THE MOTION OF VORTEX RINGS.

We the distribution of velocity is known.

proceed to consider

this problem, using the notation of 14. Let 1 be the velocity potential of that part of the motion which is not due to the vortex

ring itself. Let the equations to the central line of the vortex

core be

= + S p a

(arn cos nty -f /3n sin mfr),

= + 5 z

+ (yn cos n^jr Sn sin nty).

Let

2
Trwe

be

the

strength

of the

vortex ;

let I, m, n be the

direction-cosines of the normal to its plane, X, /^, v the direction-

cosines of a radius vector of the vortex then

;

(

6)

I=
m=
n=

sin 6 cos e, sin 6 sin e, cos 0,

X= =
li
v=

cos e cos d cos -fy

sin e sin ty,

+ sin e cos 6 cos ty

cos e sin 1^*,

sin cos -fy.

Let a?, y, be the coordinates of the centre of the vortex;
w if u, v, be the velocities parallel to the axes of x, yt z at a point

on the vortex ring, then, by Taylor's theorem,

+ d\dl = tt

dl
-djx^+ O

fd I\X-dJx-

d ^ dx ay I* -J- -f- IT -d5z-) -j

r ^ a [X t '* f\'

~|~ ij

~^

dx dy dx \

c?v c?n

v -j- // ~j~~ ~T~ ~T~ I

j

dz)

|~ . . ,

with symmetrical expressions for v and w.
The velocity along the radius vector = \u + fjiv + vw

d .

d d\ .

d i .

d .

= -=- term in the expression for the velocity along the radius vector,
etc

which is independent of ty.

As

X, p,

v

all

involve

the
i/r,

first

powers

of these

quantities

furnish nothing to this term.

X2 =

1(1

-I2 )

+

2
Jcos2^(cos

0cos2

e~sin2

e)-sin2^smecosecos0,

2=
/A
v* =

m -J (1

2

2
)+^cos2i/r(cos

^

2
sin e

- + 2^ i (1

w2 )

i cos

2
sin ^,

+ 2
cos e) sin 2i/r sine cose cos 6,

X/*=

J

Zm

+|cos

2
2i|r(l+cos 0)sin ecos e+-|

sin

2i|r cos

6

cos

2e,

+^ + Xi^= fo -J

cos 2-v/r (

sin

cos 6 cos e)

-| sin 2\^ sin 6 sin e,

= + /^

J m?z

cos 2i|r ( sin cos sin e) -J sin 2^- sin 6 cos e.

MOTION OF TWO VORTEX RINGS.

65

^ The vortex itself contributes no term independent of to the
expression for the velocity along the radius vector; thus if the
radius of the ring be small, we have approximately
da

-T j
cur ay

207m -j -
cfoflte

Sinn -j T ; dydz)

or smce '

^do

or, if -rj- denote differentiation along the normal to the plane of

the vortex nng,

da =
-^

\ a, ^p

From this equation we see that the radius of a vortex ring

placed in a mass of fluid will increase or decrease according as the

velocity along the normal to the plane of the vortex ring at the

centre of the ring decreases or increases as we travel along a

A stream line through the centre.

simple application of this result

is to the case when we have a fixed ring placed near a fixed

barrier parallel to the plane of the ring. The effect of the barrier

is to superpose on the distribution of velocity due to the vortex ring a velocity from the barrier which decreases as we recede from

the barrier ;

it is this superposed

velocity which affects the

size of

the ring, and, since the velocity decreases as we go along a stream

line (which flows from the barrier), the preceding rule shews that

the vortex will increase in size, which agrees with the well-known

result for this case.

Let us now find how the vortex ring is deflected.

The velocity perpendicular to the plane of the vortex

=

dl
~dThT

+

/d

d

^(\A'-drx-+A'r'd^-y

+

d\
^1~} dz)

dl
~dTh7~

d

d d\*dl

d:rx

+ar-jdy

+

i>d-Tz-))

-d,hr +

>

The coefficient of cos i/r

-- d\dl =

a

/ cos
{\

e

cos

6n

d djx

+

sm

e

d
-=
dy

sm

6

-=-
dzj)

-d^h

+

terms

in

a

.

The coefficient of sin ty

d\dl = + + a.(

.

d

sin e -=-

{
\

dx

sin

e

cos

6Q 7 dy)

-drhr

terms

.
in

as .

T.

5

66

ON THE MOTION OF VORTEX RINGS.

= -~ coefficient of cos ty in the expression for trie velocity perpen-

dicular to the plane of the vortex.

The vortex itself contributes nothing to the coefficients of
either cos-^r or sini/r in the expression for the velocity perpendicular to the plane of the vortex (see equation 43).

Thus

= d-y^^
at

a{(cos
\

e

cos

6Q

d
-=-
dx

-f

sin

d
e -j-
dy

sm

6Q

d\dl

-7-
dzj

d-^hr

approximately,

d\dl d^ = + -ajtj

af [ \

.

d

sin e -dyx-

sm e

cos

6Q -y- I dy)

-dyvh-

.

Now by

6,
dl =
at -j-.

IdS, .
--r-'sine

lcZ7l -

cos

6

cose,

a dt

a dt

dm = - - -TT
a

-1 a

dS, -TT dt

cos

e

sm 1 c?7,

. .,

a--idjt-cos

e,

dn
^r=
dt

1 dy. . >.
a-djt}tmO.

Substituting the values just found

expressions, we find

^?_ = dt

dh J'

z

^ 2I1 dhdx

2& -^ for

,.

in

these

m dm dz L

= 7

"" ^JL2

ac

afi

dn

2
cZ !!

= -7- n

2

^ (ZA'

_7Z. _y
a/i dy
(Z/i C?x

.(80).

These equations enable us to find the orientation of the plane of the vortex at any time.

To find the change in the shape of the vortex, we have
= -y^ coefficient of cos 2i|r in the expression for the velocity along

the radius vector.

Now the vortex itself contributes to this coefficient the term

, 8a

.

... OON

5- log .72 (see equation 88).

MOTION OF TWO VORTEX RINGS.

67

And if we pick out the coefficient of cos 2-f arising from the velocity potential H, we shall find that it reduces to

where

denotes differentiation along an axis coinciding in

;

cue

^ direction with the radius of the vortex ring for which

= \ir.

Thu3 d*t 7tf"

Again,
y = coefficient of cos 2>Jr in the expression for the velocity
perpendicular to the plane of the vortex.
Now the vortex itself contributes to this coefficient the term

~-

f

log

Cb

6

.

a s

(see

equation

43).

And if we pick out the coefficient of cos 2-^ arising from the velocity potential fl, we shall find that it reduces to

Thus

^and and this, with the preceding equation connecting

yz , enables

us

to

find

a 2

and

%.

We have two exactly analogous equations connecting dftjdt and

S 2,

the

only

difference

being

that we

substitute

-77-, for

-77-,

where

(i/J

Q/fC

-p denotes differentiation with respect to an axis passing through
dfc

the centre and coinciding in direction with the radius of the vortex
^ ring for which = 0.

We 37.

can apply these equations to find the motion of

We a vortex ring which passes by a fixed obstacle.

shall suppose

that the distance of the vortex from the obstacle is large compared

with the diameter of the vortex, and that the obstacle is a

sphere.
Let the plane containing the centre of the fixed sphere J5, the centre of the vortex A, and a parallel to the direction of
m motion of the vortex be taken as the plane of xy. Let the axis of
x be parallel to the direction of motion of the vortex. Let be
the strength of the vortex, and a its radius.

68

ON THE MOTION OF VORTEX RINGS.

P The velocity potential due to the vortex at a point

>

*
dx

'

(if)

aPProximately-

Now

BP AB if

<

AB,

and

Q iy

Q . ..are 2

spherical

harmonics

with

for axis.

At the surface of the sphere the velocity parallel to x

m m = + . ^

= a, , d* ( 1 \ ~dx* (API

,
^

3cos2 l9-l
,2
~AW

smaller terms >

AB where 6 is the angle

makes with the axis of a?.

-- The velocity parallel to the axis of y

1\

, 2 3 cos 6 sin

Now at the surface of the sphere the velocity must be entirely
tangential, hence we must superpose a distribution of velocity,
giving a radial velocity over the sphere equal and opposite to the radial velocity due to the vortex ring, i. e. equal to

- - + /YY)

W \

2

i -TVS a* IT (3cos

Of
1) 1 3 cos 6 sin

Li

jO.x5

[O

j

AB if x and y be the coordinates
radius of the sphere. Let

of
=

a R.

point H,

on the

the sphere, b the velocity potential

which will give this radial velocity, is given by the equation

where r = BP.

+ -k ,_
(v 3

cos92/(19-l^)vd-Cay-?; l-r

3

cos

~.
^ sm

~d ^ -7-

1)

(iyr-J

II is approximately the value of the velocity potential which produces the disturbance of the motion of the vortex.

MOTION OF TWO VORTEX RINGS.

The equation

da

.

3r~

becomes in this case

^cti

=

i

^RJ*(3PcosV-l')dUaf3rcos0sin0

* 'I. dtfdyr}

Now

-TT = d8 /IN

- - 3 (5 cos'0 3 cos 0)

5?(f)

>

'

~~- 3 sin 0(1 5 cos'fl)

We must express the quantities on the right-hand side of the
equation in terms of the time.

Let us measure the time from the instant when the line joining

the centre of the sphere to the centre of the vortex is per-

pendicular to the direction of motion of the vortex. Let u be the

velocity

of

the

vortex ;

then

we

have,

accurately

if

the

motion

were

undisturbed, and very approximately as the motion of the vortex is

only slightly disturbed,

OOS0

sin 0>

where c is the shortest distance between the centre of the vortex and the centre of the sphere.

Substituting we find da
dt~

s9

" nU*?m'a b

*

a
(C

+

uVf

'

thus the vortex expands until it gets to its shortest distance from the centre of the sphere, after passing its shortest distance it ceases to expand and begins to contract.

Integrating the differential equation, we get

where a is the value of a before the vortex got near the sphere.

Thus we see that the radius is the same after the vortex

has passed quite away from the sphere as it was before it got

R = near to it, since in both cases

oo ;

in

intermediate

positions

it

is always greater.

70

ON THE MOTION OF VORTEX RINGS.

The greatest value of the radius is

'

the greatest increase in the radius is thus proportional to the volume of the sphere, and inversely proportional to the sixth power of the shortest distance between the vortex and the

sphere.

38. To find the way in which the direction of motion of

m the vortex is altered we have, if I,

are the x and y direction

cosines of the normal to its plane,

dm = cPQ cm

'

dt

dx* dxdy

m Now in the undisturbed motion

= 0,

so we may write

this

equation
dm

'

dt

dxdy

m dm
-

=

1

/6V-f /0
D3-l v

.Zr

(

NT

- +3 --- d 28 ,

_ N

3 - /-lA

d3

-.

-

cos 6 sin

/!-'

7

efaVfy Vf/

a/oa ^

3
cZ fl\'_

dx*dy (rr)

r*

d*

Substituting these values, we find

thus -=dt

is

always

negative,

or

the

vortex

moves

as

if

attracted

by the sphere; expressing the right-hand side in terms of the time, we get
dm

& Integrating both sides from t =

to = + oo we find that ,

m, the whole angle turned through by the vortex, is given by the

,.
equation

m=-

and this effect varies inversely as the sixth power of the shortest distance between the vortex ring and the sphere, and directly as the volume of the sphere. Sir William Thomson shewed by
general reasoning that a vortex passing near a fixed solid will appear to be attracted by it ("Vortex Motion," Edinburgh Transactions, vol. xxv. p. 229) ; and this result agrees with the
results we have obtained for the sphere.

( 71 )

PART III.

Linked Vortices.

WE 39.

must now pass on to discuss the case of Linked

We Vortices.

shall suppose that we have two vortex rings linked

one through the other in such a way that the shortest distance

between the vortex rings at any point is small compared with the

radius of the aperture of either vortex ring, but large compared

with the radius of the cross section of either of them. Thus, the

circumstances in this case are the opposite to those in the case we

have just been considering, when the shortest distance between

the vortices was large compared with the diameter of either.

In the present case it is important to examine the changes

in the shape of the cross section of the vortices, in order to see

We that they remain approximately circular.

shall, therefore,

discuss this problem first.

Since the distance between the vortices is very small compared with the radii of the apertures of the vortices, the changes in their cross sections will be very approximately the same as the changes in the cross sections of two infinitely long straight cylindrical vortex columns placed in the same mass of fluid in such a manner that the distance between them is great compared with
the radius of either of their cross sections.

We shall prove that if the cross sections of two such vortex
columns are at any moment approximately circular they will
always remain so.
We must first find the velocity potential due to such a vortex
column.

Let the equation to the cross section be

p

=

a

+

an

cos

nO

+

j3n

sin

nQ y

vhere an and f$n are small compared with a, the mean radius
of the section. Let o> be the angular velocity of molecular
rotation.

72

ON THE MOTION OF VORTEX RINGS.

The

stream function due i|r

to this

distribution

of

vorticity

is

given by the equation

=

\\ ay log r dxf dy'

(Lamb's Treatise on the Motion of Fluids, 138, equation 33),

where

r

is

the

distance

of

the

points

x }

y

from

the

points

x',

y'.

Thus *\fr is the potential of matter of density
over the cross section.
At a point outside the cylinder let
At a point inside the cylinder let

distributed
an rn '

a

.. (82).

Thus, since i|r is continuous, these two values must be equal at the surface of the cylinder; thus, if we substitute
+ + r a an cos n 6 j3n sin nO,

we may equate the coefficients of cos n6 and sin nd in the two exr
pressions for ty.

Doing this we get, neglecting powers higher than the first of

an and &,

A=

The

differential

coefficients of

ty are

continuous ;

thus

the

two

values of -^- must be the same at the surface of the cylinder;

differentiating both expressions for i|r with respect to r, putting

r = a + a.n cos n6 + @n sin nO,

and

equating the coefficients of

cos nO

and

sin n6 }

we

find

-- = o>an

nA-n
a

nA'-n a

COT ,

nB nB'

Solving these equations, we find

A _~ <Wn T>

n

'

"

n

I.IN'KED VORTICES.

73

Thus at a point outside the cylinder,
^r = (7-waMogr + ~(an co8n^ + /9n 8m7i^)^...(83).

We can now find the time of vibration of a single vortex
column whose section differs slightly from the circular form.

+ " For if p = a an cos nd 4- ft sin nd be the equation to the cross

section, then, since the surface always consists of the same particles

F = of the fluid, using the theorem that if (x, y, z, t)

be the

equation to such a surface,

dF dF dF dF

-dJtT

+

U-dTx-

+

V

-j
dy

hw-djz-=0,

we get

where 3& is the velocity of the fluid at the surface of the cylinder

along the radius vector and

its angular velocity round the axis

of the cylinder.

*

Thus, when r = 3& =

e 1rfdtr.
a + an cos nd + ft sin nO,
co (a n sin nO ft c

neglecting squares of an and ft.

Hence substituting in equation (84) and neglecting all powers of n and ft above the first, we get

-^ - co (an sinnO ft cos nd) = -^ cos nQ +

sin nd

nco (an sin nd - ft cos nd) :

equating coefficients of cos nd and sin nd, we get

therefore

74

ON THE MOTION OF VORTEX RINGS.

or

= A + an

cos {(n 1) cot /3},

4 &, = sin {(w-1) <* + },

A where and /3 are arbitrary constants.

Thus

r = a + Acos[{n6-(n-l)a)t}-j3]

(86).

Thus the section never differs much from a circle, and the
disturbance in the shape travels round the cylinder in the time

27T
w'
(n 1)
These results agreed with those stated by Sir William Thomson in his paper on "Vortex Atoms" (Phil. Mag. 1867), and
proved in his paper "On the Vibration of a Columnar Vortex." Proceedings of the Royal Society of Edinburgh, March 1, 1880;
reprinted in Phil Mag., Sep. 1880.

40. Let us now consider the case when there are two vortex
columns in the fluid (fig. 7).

Fig.7.

Let

p = a + 2 (o^ cos n6 + /3n sin n&)

A be the equation to the cross section of the one with as centre,

and let

& p = I + 2 (' cos n& + n sin nP)

B be the equation to the cross section of the one with A being measured from and p from B.

as centre, p

AB Let c be the distance

between their centres, and e the

AB angle

makes with the initial line.

Then the

stream

function -fy

due

to

the

two

vortex

columns

at

P a point is given by the equation

ty = C coo? log r + 2

+ (an cos nO /3n sin n6) n

- a/6log /+ 2 71

^ cos nP+ff* sin n^) tt , 7*

LINKED VORTICES.

75

where r=AP,r' = BP, and 6, & are the angles AP and BP make
with the initial line, <u and o>' are the angular velocities of molecular rotation of the two vortex columns.

We shall want to use the current function at the surface

of both the cylinders, thus it will be convenient to find a method

of transforming that part of the stream function where the

A coordinates used are measured from

as origin to coordinates

B with

as origin, and vice versd. To do this we shall use

the following lemma, which may be easily proved by trigo-

nometry.

Fig.8.

Lemma.

41.

If AP= r, BP = r,

< PAB = ty,

< PBC = x,

ThenifV<c

sin

cosn>/r =

< if r

c,

1.2.3
1.2.3
c.
1.2 r'

log r = | log r2

= + + 72
i log (r

2
c

2cr' cos x)

V

76

ON THE MOTION OF VORTEX RINGS.

If C<r',

2

3

= + - ^ ^ log r log r

c

c

c

% + - cos

2^ J 2 cos

J 8 cos 3^; . . , .

We can now find the effect of the vortex columns on each

other.

& For if be the radial velocity of a point Q on one of the

B vortex columns relative to

the centre of that vortex column,

and b the velocity of Q relative to B, perpendicular to BQ,

then as before

&=

cos nB + -

- - sin nff n (a' n sin nO {3'n cos n0) . . . (87).

& B Now, the part of due to the vortex column with as centre

= a)' (a.'n cos nff f n sin nff),

& the part of

due to the term

2
o>a

log

r

in

the

stream

function

4 sin 2 (<9'-e) --]sin 3 (^ - c) + ~sin4(^4) .

c

c

c

the term

@ (an cos w^ -f n sin w^) -^ ,

+ gives aw -fci (an cos we 4- /5n sin we) (w 1)

+ aw n (ftn cos we a.n sin Tie) (w 1)

Since aw, /5M, and -- are all small quantities, as we are c

neglecting the squares of small quantities, we may neglect these

terms

which

involve

quantities

of

the

order

of

a

2
n

;

and

for

the

same reason, we may in equation (87) put

= w! ',

since

it

only

differs

from

it

by

small

quantities

of the

order

an

and - and ,

in

c

that equation is multiplied by quantities of this order.

& @ Substituting these values for

and

in equation (87), and

equating the

coefficients

of cos 6 ',

and

sin &

on

each

side

of the

equations, we get

U, ""' dff,

dt

dt

= ^ = or,

as

and
a.\

ft'^

are zero

initially we get

0^

0,

0, and

LINKED VORTICES.

77

= similarly

a t

0,

, = 0; and thus the motion of the centre of gravity

of either vortex column is not disturbed. If we equate the

coefficients of cos 20' and sin 20' on each side of equation (87), we get

dot.'

and

Now ^15 travels round approximately uniformly with an

^~- angular velocity n, where n =

, this value of n follows at

once if we remember that the centre of gravity of the two vortex
columns remains at rest.

AB Thus taking the initial position of

as the initial line from

which to measure our angles, we have = e nt.

Thus

da'

,,

cocfb .

therefore

--

2 - -j-cos 27i^;

therefore

= Now,

let

of s

,

fBz

M - + cocfb (2n a)') cos 2nt

2

m*_

= initially, then dz'Jdt

initially, and we get

Thus the cross section at any instant is an ellipse. This

ellipse does not, however, remain of the same shape, but vibrates

about the circular form ;

the maximum ellipticity is proportional to

a . , p . , and thus varies inversely as the square of the distance

between the vortex columns.

long

one

n

and

a

short

one

&)

The vibration has two periods, a

The terms in as , fis will involve -3, and thus will be relatively

78

ON THE MOTION OF VORTEX RINGS.

unimportant, as

2,

/32 only

involve the square of -;
c

the

same

reasoning applies a fortiori to an and J3n when n is greater than three.

42. Our investigation of the motion of two infinite cylindrical vortices shews that to retain an approximately circular cross section the vortices must be at a distance from each other large compared with the diameter of the cross section of either. If we consider a
portion of two linked vortices near each other, and regard them as straight, which we may do if the distance between them is small compared with the radius of the aperture of either, we see that the
-- vortices will spin round each other with an angular velocity
m when and m' are the strength of the two vortices, and d the
shortest distance between the two parts of the vortices we are considering; thus, if the motion is to be steady, we must have this
angular velocity approximately constant all round the vortices, and therefore c? must be approximately constant all round the
vortices.

To get a clear conception of the way the vortices, supposed for the moment of equal strength, are linked, we may regard them as linked round an anchor ring whose transverse section is small compared with its aperture, the manner of linking being such that there are always portions of the two vortices at opposite extremities of a diameter of a transverse section of the anchor ring. The
shortest distance between pieces of the two vortices is then approximately constant, and equal to the diameter of the transverse
section of the anchor ring.

Let us suppose that the vortex is linked r times round the anchor ring, then the equation to the central line of vortex core
may be written

= + + ^ + p

a

a cos 6 t

l sin 6 +. . .ar cos r6 /3r sin rO

-f... aw

% + z

= j

YI cos

0.+

S t

ntt

0+...

+ cos rO & sinY/9 r

+... cos n

Let the equations to the second vortex differ from these only

in having accents affixed to the letters. Here av $,; yv ^; V OL j3\-,

y v S\, &c.

are

all

small

in

comparison with

a

and

a, but

a @ r,

r ',

yS r, r ;

f a'

r,

r;

y'r , S'r are

large

compared with

the others, so that

in the expression for the velocities due to the vortex rings we shall

go to the squares of these quantities, but only retain the first

powers of the other quantities denoted by the Greek letters. Let

m be the strength of the vortex whose equation was first written,

m which we shall call vortex (I),

the strength of the other, which

LINKED VORTICES.

79

we shall call vortex (II). Let e and e be the radii of the cross
sections of vortices (I) and (II) respectively.
A Let n H denote the value of the quantity we denoted in 13 A by n , due to the vortex (I) at a point on the surface of itself.
A A iy n the value of the quantity n due to the vortex (I) at a
point on the surface of vortex (II).
A A 9l n the value of the quantity n due to the vortex (II) at a
point on the surface of the vortex (I).
A ^A n the value of the quantity n due to the vortex (II) at a
point on the surface of itself.

Now, from equations (11) and (14) the terms of the first order

in

a
n,

&c.,

in

the

expression

for

the

velocity

along

the

radius

vector

due to the vortex (I) at the surface of the vortex (II) are

ma 131 y n cos n + n sn n

A-J + H( - 1) iA-H -( + !) !

n^ + (7* cos

8B sin n^r)}.

If we suppose the two vortices wound round an anchor ring, of diameter d, in such a way that there are always portions of the two
vortices at opposite extremities of a diameter of the transverse
A section, then in the expression for n given in equation (35) we
A must put x ^ 2 . Substituting this value of n and retaining Ci
only the most important terms, we find that the velocity along the radius vector of the vortex (II) due to the vortex (I)

+ - ,
(7 cos nty

a

N
S' n sin nty

-

} log

f ^- + (y, cos n* + 8n sin

/ A=n-* -

) (-

(

-

fi4rt sV)

J) log

.

jf

By equation (38) we see that the velocity along the radius
vector of the vortex (II) due to this vortex itself

m ,

^ + Smn ^,

.

"

,x

*i

g

But from the equation

= + 2 + p a'

(a'M cos nty 4- P'n sin n-^r)

e cos (f>,

^ we see that if we only retain the first powers of the quantities

a 'n>

/^n>

e velocity along the radius vector

equating the coefficients of cos n*jr and sin n-ty in this expression for the velocity and in the expression just, found, wo find

80
da'n

ON THE MOTION OF VORTEX RINGS.

m<

'*-*

m' ,
n 4-Tra

2
64a'
e

From equations (16) and (17) the terms of the first order
in aw, &c., in the expression for the velocity perpendicular to the plane of vortex (II) due to vortex (I)
cos n + sn n

2 (a'n cos n^r + #. sin n^) i

+

cos

+ sin

^ A + + 2a 12

n

- a {(n 1)

- + ^ (n 1) 12 n+1 ) ,

where, before differentiation, the A'a are to be regarded as
functions of / and R, and after differentiation we put

r = a + ar cos rir + f sn r*,

R + ^ CL'-^T a r cos njr

sin
ri/r,

and

retain the

largest

terms ;

the quantities ar, y8r, a r, f$'r, have

each JcZ for their maximum value. If we substitute in these

expressions the values for the quantities denoted by the ul's

= given in equation (35), and put x

flP
^ , we find that the

aOt

velocity perpendicular to the plane of vortex (II) due to
vortex (I)

8a

2

m ^ ^ ~~ + + ~ . 8 n C S n

*&

-
Sm

,\ /4a

n

n. 2
(

4N ,

64<

g

if we go to the first powers only of the quantities denoted by tbe Greek letters.

The velocity perpendicular to the plane of the vortex (II)

due to this vortex itself, is by equation (43)

mA ^m \

8a g ~?

~

+ - \

m' ,

/ 47rZ2 ^

2~

., 64a'2
^ g ~e^

C SW

LINKED VORTICES.

81

But from the equation

- + 2 z i

(y n cos n>/r -f &'n sin w>|r),

we see, as in equation (40), that the velocity perpendicular to
the plane of the vortex, is

Hence, equating the constant terms and the coefficients of

cos

and sin

n^jr

nty, in

this

expression, and

the

expression we

have just found for the same quantity, we get

m

2
4-Tra

+ - - - + 4a"

. ,f

,

(
(^ -JT

64(A i) log -gr)

, /4a' a.

.

1 log

64a*

64a

m

.2

-
x

,

64a'2

(B

In the case equal, thus

we are
= a a'.

considering

the

mean

radii

of

the

vortices

are

If we write for the sake of brevity,

64a2

W p 4a2

, 64a2

* g ~<F

s\t

/y

\i

O^r'v

Then our equations become
T.

(90).
,(91),
6

82

ON THE MOTION OF VORTEX RINGS.

If we go to the vortex (I), we get

d

m'

m 8a

, Sa

M M where and Q are what

and Q' become when e is written

r

for e .

J Equating the two values of

we must have
,

m , 8a m' , 8a

m' , 8a

m , 8a
7 S '

or

We shall first

= e

e'.

=/

-

mlog- m 6

log 6 ........................ (93).

m consider the case when

=
%

m,

and

therefore

In this case our equations are

Adding the first and third of these equations, we get
i(' + ^~(L-M-Nn
adding the second and fourth, we get

Hence

-^K + O = 0;

LINKED VORTin

therefore
where

+ = A a'w

or,,

cos (vt -f e),

= *'

-

(Q + R-F)(M+N-L),

and ^1 and e are arbitrary constants.
Substituting the values of the quantities involved in the ex
pression for v, we find

therefore

{( JIL^

log^' _!))

................. (94),

u.6

F or if

be the velocity of translation of the vortex ring we have

very nearly

+ sin (rf e)
Subtracting the third from the first of the four equations giving
-I* &c., we get

Subtracting the fourth from the second of these equations,
we get

Hence

N- |, (a'. -.0 + (^i)' (L +

10 (R + P-Q) ('. - =.) = ;

therefore

= + a'n an .B cos (/it e) ,

where

^=f

N P (L +

- I/) (12 +

Q),

B and and e' are arbitrary constants.

Substituting the values of the quantities involved in the ex-

pression

for

2 /i ,

we

find

Sa2 9 -( -l)log 62

ON THE MOTION OF VORTEX KINGS.

2

2 '4T<</U7<

i

8Ot/t7'

a,

and where

7

-= 7.

80-

...... (95) >

Combining the expressions for a'n + # and a! n

a
n>

and

doubling

A B the arbitrary constants and for convenience, we find

= a'n
an

A + B + cos (vt e) -f- cos (/*

e')

A B + cos (^ e)

+ cos (pi e')

</(n* -I) n

(06).

sn

- + e) 5' sin (^* e')

Since exactly the same relation exists between fin and and SM , as between a'n and 7' , an and 7^, we shall have

C + + D + cos (vt e)

cos (/jut

e')

+ D + C cos (z/

e)

cos (pt e')

'
n , /3n

+ (7 sin (vt -\-e) D' sin

>

(97),

=
n

^ ^ c sin

+ _ e) jp, gin

+

-4a2

8a

where

.Z/ = -

OCv
-dT-

91 vv
^log&-

As consequences of these equations we see (1) that the motion of the kind we have been considering is possible and stable; (2) that for each mode of displacement there are two periods of

.,

>..i

. 2?r

, 2?r

vibrations, viz.

and

.

v

p.

7

o

Now,

if - be a

of

the

same

order

d as -^ , ct

will

be

of the

order

-;
6

and when

x

is

large, x

is very great

compared with

log#, thus