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Full text of "On the Capillary Phenomena of Jets"

1879.] On the Capillary Phenomena of Jets, 71 



VI. " On the Capillary Phenomena of Jets." By Lord Rayleigh, 

F.B.S. Received May 5, 1879. 

When water issues under high pressure from a circular orifice in a 
thin plate, a jet is formed whose section, though diminished in area, 
retains the circular form. But if the orifice be not circular, the section 
of the jet undergoes remarkable transformations, which were elabo- 
rately investigated by Bidone,^ many years ago. The peculiarities of 
the orifice are exaggerated in the jet, but in an inverted manner. 
The following examples are taken from Bidone's memoir. 

Fig. 1. 




Fig. 1, orifice in the form of an ellipse (A), of which the major 
axis is horizontal, and 24 lines long ; the minor axis is vertical, and 17 
lines long. The head of water is 6 feet. 

Near the orifice the sections of the vein are elliptical with major 
axis horizontal. The ellipticity gradually diminishes until at a distance 
of 30 lines from the orifice the section is circular. Beyond this 
point the vertical axis of the section increases, and the horizontal axis 
decreases, so that the vein reduces itself to a fiat vertical sheet, very 
broad and thin. This sheet preserves its continuity to a distance of 
6 feet from the orifice, where the vein is penetrated by air. 

B represents tne section at a distance of 30 lines from the orifice. 
It is a circle of 16 or 17 lines diameter. 

is the section at a distance of 6 inches from the orifice. It is an 

* " Experiences sur la Forme et sur la Direction des Yeines et des Courans d'Eau 
lances par diverses Ouvertures." Par Q-eorge Bidone. 



72 



Lord Rayleigh on 



[May 15, 



elliptical figure, whose major axis is 22 lines long and minor axis 
14 lines long. 

D is the section at 24 inches from the orifice. It also is an ellip- 
tical figure, whose vertical axis is 45 lines long and horizontal axis 



about 12 lines long. 



In fig. 2, the orifice (A) is an equilateral triangle, with sides 2 
inches long. The head of water is 6 feet. The vein resolves itself 
into three flat sheets disposed symmetrically round the axis, the planes 



Fig. 2. 




of the sheets being perpendicular to the sides of the orifice. These 
sheets are very thin, and retain their transparence and continuity to a 
distance of 42 inches, reckoned from the orifice. The sections repre- 
sented by B, 0, D, E are taken at distances from the orifice equal 
respectively to 1 inch, 6 inches, 12 inches, and 24 inches. 

Similarly, a vein issuing from an orifice in the form of a regular 
polygon, of any number of sides, resolves itself into an equal number 



1879.] the Capillary Phenomena of Jets. 73 

of tliin sheets, whose planes are perpendicular to the sides of the 
polygon. 

Bidone explains the formation of these sheets, in the main (as it 
appears to me), satisfactorily, by reference to simpler cases of meeting 
streams. Thus equal jets, moving in the same straight line, with equal 
and opposite velocities, flatten themselves into a disk, situated in the 
perpendicular plane. If the axes of the jets intersect obliquely, a 
sheet is formed symmetrically in the plane perpendicular to that of the 
impinging jets. Those portions of a jet which proceed from the out- 
lying parts of an unsymmetrioal orifice are considered to behave, in 
some degree, like independent meeting streams. 

In many cases, more especially when the orifices are small and the 
heads of water low, the extension of the sheets in directions perpen- 
dicular to the jet, reaches a limit. Sections taken at greater distances 
from the orifice show a gradual shortening of the sheets, until a com- 
pact form is attained, similar to that at the first contraction. Beyond 
this point, if the jet retains its coherence, sheets are gradually thrown 
out again, but in directions bisecting the angles between the directions 
of the former sheets. These sheets may, in their turn, reach a limit 
of development, again contract, and so on. The forms assumed in 
the case of orifices of various shapes, including the rectangle, the 
equilateral triangle, and the square, have been carefully investigated 
and figured by Magnus. # Phenomena of this kind are of every-day 
occurrence, and may generally be observed whenever liquid falls from 
the lip of a moderately elevated vessel* 

Admitting the substantial accuracy of Bidone' s explanation of the 
formation and primary expansion of the sheets or excrescences, we 
have to inquire into the cause of the subsequent contraction. Bidone 
attributes it to the viscosity of the fluid, which may certainly be put 
out of the question. In Magnus's view the cause is "cohesion;" but 
he does not explain what is to be understood under this designation, 
and it is doubtful whether he had a clear idea, upon the subject. The 
true explanation appears to have been first given by BufE,f who refers 
the phenomenon distinctly to the capillary force. Under the operation 
of this force the fluid behaves, as if enclosed in an envelope of constant 
tension, and the recurrent form of the jet is due to vibrations of the 
fluid column about the circular figure of equilibrium, superposed upon 
the general progressive motion. Since the phase of vibration depends 
upon the time elapsed, it is always the same at the same point in space, 
and thus the motion is steady in the hydro -dynamical sense, and the 
boundary of the jet is a fixed surface. 

In so far as the vibrations may be considered to be isochronous, the 

* " Hydraulische TIntersuchungen." " Pogg. Ann.," xcv, 1855. 
t " Pogg. Ann.," Bd. c, 1857. 



74 Lord Rayleigh on [May 15, 

distance between consecutive corresponding points of the recurrent 
figure, or, as it may be called, the wave-length of the figure, is directly 
proportional to the velocity of the jet, i.e., to the square root of the head 
of water. This elongation of wave-length with increasing pressure 
was observed by Bidone and by Magnus, but no definite law was 
arrived at. As a jet falls under the action of gravity, its velocity 
increases, and thus an augmentation of wave-length might be ex- 
pected ; but, as will appear later, most of this augmentation is com- 
pensated by a change in the frequency of vibration due to the 
attenuation which is the necessary concomitant of the increased velo- 
city. Consequently but little variation in the magnitudes of successive 
wave-lengths is to be noticed, even in the case of jets falling vertically 
with small initial velocity. In the following experiments the jets 
issued horizontally from orifices in thin plates, usually adapted to a 
large stoneware bottle, which served as reservoir or cistern. The 
plates were of tin, soldered to the ends of short brass tubes rather 
more than an inch in diameter, by the aid of which they could be 
conveniently fitted to a tubulure in the lower part of the bottle. The 
pressure at any moment of the outflow could be measured by a water 
manometer read with a scale of millimetres. Some little uncertainty 
necessarily attended the determination of the zero point ; it was 
usually taken to be the reading of the scale at which the jet ceased to 
clear itself from the plate on the running out of the water. At the 
beginning of an experiment, the orifice was plugged with a small roll 
of clean paper, and the bottle was filled from an india-rubber tube in 
connexion with a tap. After a sufficient time had elapsed for the 
water in the bottle to come sensibly to rest, the plug was withdrawn, 
and the observations were commenced. The jet is exceedingly sensi- 
tive to disturbances in the reservoir, and no arrangement hitherto 
tried for maintaining the level of the water has been successful. The 
measurements of wave-length (\) were made with the aid of a pair of 
dividers adjusted so as to include one or more wave-lengths ; and as 
nearly as possible at the same moment the manometer was read, The 
distance between the points of the dividers was afterwards taken from 
a scale of millimetres. The facility, and in some cases the success, of 
the operation of observing the wave-length depends very much upon 
the suitability of the illumination. 

The first set of observations here given refers to a somewhat elon- 
gated orifice of rectangular form. The pressures and wave-lengths 
are measured in millimetres. The third column contains numbers 
proportional to the square roots of the pressures. 



1879.] the Capillary Phenomena of Jets, 75 

Table I.-— November 11, 1878. 



sssure. 


Wave-length.. 


V (Pressure) 


253 


104 




91 


216 


91 




84 


178 


81 




76 


144 


70 




69 


113* . . . 


61* . . . , 




61* 


83 


51 ... 




52 


58 


42 




43 


39 


33 




36 


21 


24 


- 


26 



The agreement of the second and third columns is pretty good on 
the whole. Small discrepancies at the bottom of the table may be 
due to the uncertainty attaching to the zero point of pressure, and 
also to another cause, which will be referred to later. At the higher 
pressures the observed wave-lengths have a marked tendency to in- 
crease more rapidly than the velocity of the jet. This result, which 
was confirmed by other observations, points to a departure from the 
law of isochronous vibration. Strict isochronism is only to be expected 
when vibrations are infinitely small, that is, in the present application 
when the section of the jet never deviates more than infinitesimally 
from the circular form. Daring the vibrations with which Table I is 
concerned, however, the departures from circularity are very con- 
siderable, and there is no reason for supposing that such vibrations 
will be executed in exactly the same time as vibrations of infinitely 
small amplitude. Nevertheless, this consideration would not lead to 
an explanation of the discrepancies in Table I, unless it were the fact 
that the amplitude of vibration increased with the pressure under 
which the jet issues. 

As a matter of observation the increase of amplitude is very ap- 
parent, and was noticed by Magnus. It is also a direct consequence 
of theory, inasmuch as the lateral velocities to which the vibrations 
are due vary in direct proportion to the longitudinal velocity of the 
jet. Consequently the amplitude varies approximately as the square 
root of the pressure, or as the wave-length. The amplitude here 
spoken of is measured, of course, by the departure from circularity, 
and not by the value of the maximum radius itself. 

The law of the square root of the pressure thus applies only to small 
amplitudes, and unfortunately it is precisely these small amplitudes 
which it is difficult to experiment upon. Still it is possible to approach 
theoretical requirements more nearly than in the experiments of 
Table I. 

The next set of measurements (Table IT) refer to an aperture in the 
form of an ellipse of moderate eccentricity. Two wave-lengths were 



76 



Lord Rayleigh on 



[May 15, 



included in the measurements ; in other respects the arrangements 
were as before. 



Pressure. 

262 
208 
182 
158 
129 
107* 

86 

69 

56 

42 

M 

27 

21 

In this case the 



Table II. — November 12. 

Wave-length. 

401 



361 

34 

31 

28i 

251* 

221 

20 

18 

15 

3 31 

12* 

10 



V (Pressure.) 

40 

351 

331 

31 

28 

251* 

23 

201 

181 

16 

141 

13 

11* 



law is fully verified, the discrepancies being de- 
cidedly within the limits of experimental error. 

On the other hand, the discrepancies may be exaggerated by the 
use of higher pressures,. Table III relates to the same orifice f as 
Table I. Instead of the stoneware bottle, a tall wooden box was nsed 



as reservoir. 












Table III. — December 20. 




Pressure. 


Wave-length. 


V (Pressure). 


757 


200 


155 


672 






184 


145 


587 






171 


136 


497 






152 


125 


442 






141 


118 


365 






123 


107 


289 






106 


95J 


234 






93 


86 


189 






79 


77 


154 






70 


70 


123 






62 


62 


107 






58 


58 


89* 






53* 


53* 


74 




• • • • 


48 


48 


61 




. . . . 


441 





The wave-lengths at the high pressures very greatly exceed those 
calculated from the lower pressures according to the law applicable to 
small vibrations. 

f Its condition may have changed a little in the interval. 



1879.] 



the Capillary Phenomena of Jets, 



11 



It is possible, however, to observe in cases where the amplitude is 
so small, that the discrepancies are moderate even at higher pressures 
than those recorded in Table III. The measurements in Table IV are 
of a jet from an elliptical aperture of small eccentricity. The ratio of 
axes is about 5 : 6. The wooden box was used. Two wave-lengths 
were measured. 



Table IV. — December 18. 



'ressure. 


Wave-length. 


1287 


79f 


1195 ..... 


82 


1117 


/y-g- 


1023 


73 


947 


70| 


852 ..... 


66| 


770 ..... 


641 


695 ..... 


61± 


620 ..... 


58 


532 


54| 


451 


48i 


371 


. . . 45| 


290* 


39J* 


248 


36! 


192 ..... 


. . . 31! 


158 


28f 


133 


26! 


Ill 


24! 


94 


21} 


85 ..... 


*1 
. . • — X 



V (Pressure). 



• T» • • 



• • • 9 • 



831 
80 
771 
74£ 

68 

641 

61 

58 

49J 

44| 

39^* 

36| 

32 

29J 

241 
22i 
21| 



The following experiments relate to an orifice in the form of an 
equilateral triangle, with slightly rounded corners. The side measures 
about 3 millims. In this case the peculiarities of the contour are re- 
peated three times in passing round the circumference. Two wave- 
lengths were measured. 



Pressure. 
215 
166 
127 

92* 

66 

43 

27 



Table Y. — November 16. 
Ware-length. 



f This is, doubtless, an error 



36 

Ox-g" 

27 
23* 

JL t/ 

14! 

11* 



V (Pressure). 

35 
31 

27 
23* 

X *J~n 

xOo* 

-1^2* 



At these high pressures the observation is difficult. 



78 



Lord Rayleigh. on 



[May 15, 



Here again we observe the tendency of the wave-length to increase 
more rapidly than the square root of the pressure. 

At higher pressures the difference is naturally still more marked. 
With the same aperture, and the wooden box as reservoir, the results 
were : — 



Pressure. 

1072 

992 
888 
827 
762 
702 
619 
539 
468 
415 
337 
292 
251 
213 

189 
163 
140 
111 

90 

70 
57 
45 



Table VI. — December 17. 

Wave-length. 



V (Pressure). 



• «••<*<»•• 



102 
94 
89 
86 
81 
77 
70 
66 
591 
541 
47 
42 
38| 
341 

33 
31 

28i 
24| 
22| 
19f 

m 

16i 



• • • • « • 



80 


•4 


77 


■5 


73 


•2 


70 


•7 


67 


•8 


65 


•o 


61 


1 


57 


•o 


53 


■1 


50 





44 


6 


42 





38 


•9 


35 


8 


33' 


7 


31- 


3 


29 


1 


25* 


9 


23" 


3 


20 


6 


18 


5 


16 


•5 



The wave-lengths down to 34Jr are immediate measurements ; those 
below are deduced from measurements of two wave-lengths. 

Similar experiments were made with jets from a square hole (side = 
2 millims.), the peculiarities of which are repeated four times in pass- 
ing round the circumference. Two wave-lengths were measured. 



Pressure. 

447 
377 
312 
269 

247 
218 



Table VII. — December 14. 

V (Pressure). 

.... 30 -2 

27 7 



Wave-length. 
32 



291 

27 

241 

23 

211 



25-2 
23*4 
22-5 
21-1 



Corrected. 
29-9 
27*4 
24*9 
23 1 
22 1 
20 7 



1879.] the Capillary Phenomena of Jets. 79 

Pressure. Wave-length. v (Pressure) . Corrected. 

192 20 19*8 19-3 

167* ■ . 18J* 18*5* 18-0 

136 16J ...... 16-6 16 1 

107 14 14-8 14-2 

87 13 13-3 12 7 

65 lOf 11-5 ...... 10-8 

47 81 9'8 8-9 

The third column contains numbers proportional to the square roots 
of the pressures. In the fourth column a correction is introduced, 
the significance of which will be explained later. 

The value of X, other things being the same, depends upon the 
nature of the fluid. Thus methylated alcohol gave a wave-length 
about twice as great as tap water. This is a consequence of the 
smaller capillarity. 

If a water jet be touched by a fragment of wood moistened with 
oil, the waves in front of the place of contact are considerably drawn 
out ; but no sensible effect appears to be propagated up the stream. 

If a jet of mercury discharging into dilute sulphuric acid be 
polarized by an electric current, the change in the capillary constant 
discovered by Lipmann shows itself by alterations in the length of 
the w r ave. 

When the wave-length is considerable in comparison with the 
diameter of the jet, the vibrations about the circular form take place 
practically in two dimensions, and are easily calculated mathematically. 
The more general case, in which there is no limitation upon the mag- 
nitude of the diameter, involves the use of Bessel's functions. The 
investigation will be found in Appendix I. For the present we will 
confine ourselves to a statement of the results for vibrations in two 
dimensions. 

Let us suppose that the polar equation of the section is 

r=a + a n cos n0 (1), 

so that the curve is an undulating one, repeating itself n times over 
the circumference. The mean radius is a ; and, since the deviation 
from the circular form is small, a n is a small quantity in comparison 
with a . The vibration is expressed by the variation of a n as a har- 
monic function of the time. Thus if a n occos (]pt — e), it may be proved 
that 

_p=jr*TV-*A-V(» 3 — rc) ..... (2). 

In this equation T is the superficial tension, p the density, A the area 

of the section (equal to w# 2 ), and the frequency of vibration is p~27r. 

For a jet of given fluid and of given area, the frequency of vibration 

varies as */(n s —n) or \Z(n — l)n(n-\-l). The case of n~l corre- 



n== 


*> 


n-~ 


3, 


n— 


4 


w=. 


5, 


n = 


6, 


n<= 


7, 


n— 


8, 


w= 


9, 



80 Lord Rayleigh on [May 15, 

sponds to a displacement of the jet as a whole, without alteration in 
the form of the boundary. Accordingly there is no potential energy, 
and the frequency of vibration is zero. For n=2 the boundary is 
elliptical, for n—3 triangular with rounded corners, and so on. With 
most forms of orifice the jet is subject to more than one kind of 
vibration at the same time. Thus with a square orifice vibrations 
would occur corresponding to %=4, w=8, w=12, &c. However, the 
higher modes of vibrations are quite subordinate, and may usually be 
neglected. The values of */(wP — n) for various values of n are shown 
below. 

F =V6x y 4= v / 6x 2 
p=</6x 7 10= -v/6 x 3 16 

p= v/6 x [/ 20= ^6 x 4-47 
p=V6x y 35= v / 6x 5*92 
p= v / 6x y 56=V6x 7-48 
■^v^x y 84=v/6x 9-17 
p= 76 x 7120= v/6 x 1 "95 
u=12, p= v/6 x ^286= v/6 x 16 '95 

It appears that the frequency for to =3 is jast double that for n=2 ; 
so that the wave-length for a triangular jet should be the half of that 
of an elliptical jet of equal area, the other circumstances being the 
same. 

For a given fluid and mode of vibration (n\ the frequency varies 
as A~\ the thicker jet having the longer time of vibration. If v be the 
velocity of the jet, \ = 2 7rvp~ l . If the jet convey a given volume 
of fluid, v oc A" 1 , and thus X oc A"*. Accordingly in the case of a 
jet falling vertically, the increase of X due to velocity is in great 
measure compensated by the decrease due to diminishing area of 
section. 

The law of variation of p for a given mode of vibration with the 
nature of the fluid, and the area of the section, may be found by con- 
siderations of dimensions. T is a force divided by a line, so that its 
dimensions are 1 in mass, in length, and — 2 in time. The volume 
density p is of 1 dimension in mass, —3 in length, and in time. A 
is of course of 2 dimensions in length, and in mass and time. Thus 
the only combination of T, />, A, capable of representing a frequency, 
is T*/T*A-*. 

The above reasoning proceeds upon the assumption of the applica- 
bility of the law of isochronism. In the case of large vibrations, for 
which the law would not be true, we may still obtain a good deal of 
information by the method of dimensions. The shape of the orifice 
being given, let us inquire into the nature of the dependence of X 
upon T, p 9 A, and P, the pressure under which the jet escapes. The 



1879.] the Capillary Phenomena of Jets. 81 

dimensions of P, a force divided by an area, are 1 in mass, —1 in 
length, and — 2 in time. .Assume 

\ oc TyA J P M ; 

then by the method of dimensions we have the following relations 
among the exponents — 

x+y + u=0, —3y-\-2z—u=0 9 —2x—2u=0, 

whence u= — x, 2/=0, z=^(l—%). 



Thus X a T*A*~ i *P""* oc A* 



/TV 

VpaM' 



The exponent x is undetermined; and since any number of terms 
with different values of x may occur together, all that we can infer is 
that \ is of the form 

where / is an arbitrary function, or if we prefer it 

m 



X=T -ipi A ! F 



where P is equally arbitrary. Thus for a given liquid and shape of 
orifice, there is complete dynamical similarity if the pressure be taken 
inversely proportional to the linear dimension, and this whether the 
deviation from the circular form be great or small. 

In the case of water Quincke found T = 81 on the C.G.S. system 
of units. On the same system />=1 ; and thus we get for the frequency 
of the gravest vibration (n=2), 

!_=3-51a- f =8-28A^ .... (3). 

2ir 

Por a sectional area of one square centimetre, there are thus 8*28 
vibrations per second. To obtain the pitch of middle (c'=256) 
we should require a diameter 



2a= (!§) l= - 115 ' 



or rather more than a millimetre. 
Por the general value of n, we have 

i^=l-43a-V(^ 3 -^)=3-38A-V(^ 3 -^) . . (4). 

2tt 

If h be the head of water to which the velocity of the jet is due, 

- </(2gh).A* ( . 

3-38 JW-n) * • • • • W- 

VOL. XXIX. G 



82 Lord Kayleigh on [}^y 15, 

In applying this formula it must be remembered that A is the area 
of the section of the jet, and not the area of the aperture. 
We might indeed deduce the value of A from the area of the 
aperture by introduction of a coefficient of contraction (about '62);' 
but the area of the aperture itself is not very easily measured. 
It is much better to calculate A from an observation of the 
quantity of fluid (V), discharged under a measured head (Ji), com- 
parable in magnitude with that prevailing when X is measured. Thus 
A=V (2# &')"*. ■ In the following calculations the C.Gr.S. system of 
units is employed. 

In the case of the elliptical aperture of Table II, the value of A 
was found in this way to be *0695. Hence at a head of 10*7 the wave- 
length should be 

x _ y(2yxl0-7)x(-0695)L .g.g 7 
3-38 -v/6 

the value of g being taken at 981. The corresponding observed value 
of X is 2*55. 

Again, in the case of the experiments recorded in Table IV, it was 
found that A='0660. Hence for &=29'0 the value of the wave- 
length should be given by 

x _ (2<7x29-0)x(-0660)* ._ 3 . 76 

3-38 x v/6 

The corresponding observed value is 3*95. 

We will next take the triangular orifice of Table V. The value of 
A was found to be '154. Hence for a head of 9 '2 the value of X, 
calculated a priori, is 

,_ y(2,^x9-2)xC15^ __ 1 

3-38 x v/24 

as compared with 2*3 found by direct observation. 

For the square orifice of Table VII, we have A= v 153. Hence, if 
h=16'7, 

^_ y(2^xl67)x(-153)g „ 1 .H A 
• 3-38 x ^60 

as compared with 1'85 by observation. 

It will be remarked that in every case the observed value of X 
somewhat exceeds the calculated value. The discrepancies are to be 
attributed, not so much, I imagine, to errors of observation as to 
excessive amplitude of vibration, involving a departure from the 
frequency proper to infinitely small amplitudes. The closest agree- 
ment is in the case of Table IV, where the amplitude of vibration was 
smallest. It is also possible that the capillary tension actually 



1879.] the Capillary Phenomena of Jets* 83 

operative in these experiments was somewhat less than that deter- 
mined by Quincke for distilled water. 

When the pressures are small, the wave-lengths are no longer con- 
siderable in comparison with the diameter of the jet, and . the vibra- 
tions cannot be supposed to take place sensibly in two dimensions. 
The frequency of vibration then becomes itself a function of the 
wave-length. This .question is investigated mathematically in 
Appendix I. For the case of w=4, it is proved that approximately 



P ~~~WA "30A3"/ 



Hence for the aperture of Table VII, 

\ oc v/ft(l- -088 X" 3 ), 

X being expressed in centimetres. The numbers in the fourth column 
of the table are calculated according to this formula. 

On the other hand at high pressures the frequency becomes a 
function of the pressure. Since frequency is always an even function 
of amplitude, and in the present application, the square of the 
amplitude varies as h, the wave-length is given approximately by an 
expression of the form \/h (M.-\-~NJi), where M and N are constants. 
It appears from experiment, and might, I think, have been expected, 
that frequency diminishes as amplitude increases, so that 1ST is positive. 

When the aperture has the form of an exact circle, and when the 
flow of fluid in its neighbourhood is unimpeded by obstacles, there is 
a perfect balance of lateral motions and pressures, and consequently 
nothing to render the jet in its future course unsymmetrical. Even in 
this case, however, the phenomena are profoundly modified by the 
operation of the capillary force. Far from retaining the cylindrical 
form unimpaired, the jet rapidly resolves itself in a more or less 
regular manner into detached masses. It has, in fact, been shown by 
Plateau,^ both from theory and experiment, that in consequence of 
surface-tension the cylinder is an unstable form of equilibrium, when 
its length exceeds its circumference. 

The circumstances attending the resolution of a cylindrical jet into 
drops have been admirably examined and described by Savart,f and 
for the most part explained with great sagacity by Plateau. There 
are, however, a few points which appear not to have been adequately 
treated hitherto ; and in order to explain myself more effectually 

# " Statique Experimental efc Theorique des Liquides soumis aux seules Forces 
Moleculaires." Paris, 1873. 

f "Memoiresur la Constitution des Veines Liquides lane^es par des Orifices 
Circulaires en mince paroi." Ann. d. Chim., t. liii, 1833. 

g2 



84 Lord Bayleigh on [May 15, 

I propose to pass in review the leading features of Plateau's theory, 
imparting, where I am able, additional precision. 

Let us conceive, then, an infinitely long circular cylinder of liquid, 
at rest,* and inquire under what circumstances it is stable, or unstable, 
for small displacements, symmetrical about the axis of figure. 

Whatever the deformation of the originally straight boundary of 
the axial section may be, it can be resolved by Fourier's theorem into 
deformations of the harmonic type. These component deformations 
are in general infinite in number, of every wave-length, and of 
arbitrary phase ; but in the first stages of the motion, with which 
alone we are at present concerned, each produces its effect indepen- 
dently of every other, and may be considered by itself. Suppose, 
therefore, that the equation of the boundary is 

r=a + a cos 7cz, ..... (6), 

where a is a small quantity, the axis of z being that of symmetry. 
The wave -length of the disturbance may be called X, and is connected 
with h by the equation h=2ir\- 1 . The capillary tension endeavours 
to contract the surface of the fluid ; so that the stability, or instability, 
of the cylindrical form of equilibrium depends upon whether the 
surface (enclosing a given volume) be greater or less respectively 
after the displacement than before. It has been proved by Plateau 
(see also Appendix I) that the surface is greater than before dis- 
placement if Jca>l, that is, if X<2>rra; but less, if ha<\, or \>2tt&. 
Accordingly, the equilibrium is stable, if X be less than the circum- 
ference ; but unstable, if X be greater than the circumference of the 
cylinder. Disturbances of the former kind, like those considered in 
the earlier part of this paper, lead to vibrations of harmonic type, 
whose amplitudes always remain small; but disturbances, whose 
wave-length exceeds the circumference, result in a greater and greater 
departure from the cylindrical figure. The analytical expression for 
the motion in the latter case involves exponential terms, one of which 
(except in case of a particular relation between the initial displace- 
ments and velocities) increases rapidly, being equally multiplied in 
equal times. The coefficient (q) of the time in the exponential term 
(e**) may be considered to measure the degree of dynamical in- 
stability; its reciprocal q~~ 1 is the time in which the disturbance is 
multiplied in the ratio 1 : e. 

The degree of instability, as measured by q 9 is not to be deter- 
mined from statical considerations only ; otherwise there would be no 
limit to the increasing efficiency of the longer wave-lengths. The 
joint operation of superficial tension and inertia in fixing the wave- 

* A motion common to every part of the fluid is necessarily without influence 
upon the stability, and may therefore he left out of account for convenience of con- 
ception and expression. 



1879.] 



the Capillary Phenomena of Jets. 



85 



length of maximum instability was, I believe, first considered in a 
communication to the Mathematical Society,* on the "Instability of 
Jets." It appears that the value of a may be expressed in the form 



2=^(^3) .*W (7), 



where, as before, T is the superficial tension, p the density, and F is 
given by the following table : — 



k 2 a 2 . 


F(fca). 


Jk 2 a 2 , 


F(fca). 


•05 


•1536 


•4 


•3382 


1 


•2108 


•5 


•3432 


•2 


•2794 


•6 


•3344 


•3 


•3182 


•8 


•2701 






•9 


•2015 



The greatest value of F thus corresponds, not to a zero value of 
¥a 2 , but approximately to 7r, 2 & 3 = '4858, or to \=4<'508x2a. Hence 
the maximum instability occurs when the wave-length of disturbance 
is about half as great again as that at which instability first com- 
mences. 

Taking for water, in C.Gr.S. units, T=81, /o=l, we get for the case 
of maximum instability, 



2 



-1. 



a* 



81 x -343 



= '115 d* 



(8) : 



115 



if d be the diameter of the cylinder. Thus, if d=\, g'"~ 1 = -iio; or 
for a diameter of one centimetre the disturbance is multiplied 2*7 
times in about one-ninth of a second. If the disturbance be mul- 
tiplied 1000 fold in time t, qt=S log e 10=6*9, so that t='79dl For 
example, if the diameter be one millimetre, the disturbance is mul- 
tiplied 1000 fold in about one-fortieth of a second. In view of these 
estimates the rapid disintegration of a fine jet of water will not cause 
surprise. 

The relative importance of two harmonic disturbances depends upon 
their initial magnitudes, and upon the rate at which they grow. When 
the initial values are very small, the latter consideration is much the 
more important ; for, if the disturbances be represented by a^i*, a 3 e?3*, 

in which q l exceeds g>, their ratio is -le"^"^ ; and this ratio de- 

"1 
creases without limit with the time, whatever be the initial (finite) 

ratio 03 : a x . If the initial disturbances are small enough, that one is 

ultimately preponderant, for which the measure of instability is 

* " Math. Soc. Proc," November, 1878. See also Appendix I. 



86 Lord Rayleigh on [-May 15, 

greatest. The smaller the causes by which the original equilibrium is 
upset, the more will the cylindrical mass tend to divide itself regularly 
into portions whose length is equal to 4' 5 times the diameter. But a 
disturbance of less favourable wave-length may gain the preponder- 
ance in case its magnitude be sufficient to produce disintegration in a 
less time than that required by the other disturbances present. 

The application of these results to actual jets presents no great 
difficulty. The disturbances, by which equilibrium is upset, are im- 
pressed upon the fluid as it leaves the aperture, and the continuous 
portion of the jet represents the distance travelled during the time 
necessary to produce disintegration. Thus the length of the continuous 
portion necessarily depends upon the character of the disturbances in 
respect of amplitude and wave-length. It may be increased consider- 
ably, as Savart showed, by a suitable isolation of the reservoir from 
tremors, whether due to external sources or to the impact of the jet 
itself in the vessel placed to receive it. Nevertheless it does not appear 
to be possible to carry the prolongation very far. Whether the resi- 
duary disturbances are of external origin, or are due to friction, or to 
some peculiarity of the fluid motion within the reservoir, has not been 
satisfactorily determined. On this point Plateau's explanations are not 
very clear, and he sometimes expresses himself as if the time of dis- 
integration depended only upon the capillary tension, without reference 
to initial disturbances at all. 

Two laws were formulated by Savart with respect to the length of 
the continuous portion of a jet, and have been to a certain extent 
explained by Plateau. For a given fluid and a given orifice the length 
is approximately proportional to the square root of the head. This 
follows at once from theory, if it can be assumed that the disturbances 
remain always of the same character, so that the time of disintegration 
is constant.^ When the head is given, Savart found the length to be 
proportional to the diameter of the orifice. Prom (8) it appears that 
the time in which a disturbance is multiplied in a given ratio varies, 
not as d, but as cU. Again, when the fluid is changed, the time varies 
as p^ T-i But it may be doubted, I think, whether the length of the 
continuous portion obeys any very simple laws, even when external 
disturbances are avoided as far as possible. 

When the circumstances of the experiment are such that the reser- 
voir is influenced by the shocks due to the impact of the jet, the dis- 
integration usually establishes itself with complete regularity, and is 
attended by a musical note (Savart). The impact of the regular 
series of drops which is at any moment striking the sink (or vessel 
receiving the water), determines the rupture into similar drops of the 
portion of the jet at the same moment passing the orifice. The pitch 

* Por the sake of simplicity, I neglect the action of gravity upon the jet when 
formed. The question has been further discussed by Plateau. 



1879.] the Capillary Phenomena of Jets. 87 

of the note, though not absolutely definite, cannot differ much from 
that, which corresponds to the division of the jet into wave-lengths of 
maximum instability ; and, in fact, Savart found thab the frequency 
was directly as the square root of the head, inversely as the diameter 
of the orifice, and independent of the nature of the fluid — laws which 
follow immediately from Plateau's theory. 

From the pitch of the note due to a jet of given diameter, and issuing 
under a given head, the wave-length of the nascent divisions can be at 
once deduced. Reasoning from some observations of Savart, Plateau 
finds in this way 4*38 as the ratio of the length of a division to the 
diameter of the jet. The diameter of the orifice was 3 millims,, from 
which that of the jet is deduced by the introduction of the coeffi- 
cient *8. Now that the length of a division has been estimated a priori, 
it is perhaps preferable to reverse Plateau's calculation, and to exhibit 
the frequency of vibration in terms of the other data of the problem. 
Thus 

frequency =^i|^ (9) . 

But the most certain method of obtaining complete regularity of 
resolution is to bring the reservoir under the influence of an external 
vibrator, whose pitch is approximately the same as that proper to the 
jet. Magnus^ employed a ISTeef 's hammer, attached to the wooden frame 
which supported the reservoir. Perhaps an electrically maintained 
tuning-fork is still better. Magnus showed that the most important 
part of the effect is due to the forced vibration of that side of the 
vessel which contains the orifice, and that but little of it is propagated 
through the air. With respect to the limits of pitch, Savart found 
that the note might be a fifth above, and more than an octave below, 
that proper to the jet. According to theory, there would be no well- 
defined lower limit ; on the other side, the external vibration cannot 
be efficient if it tends to produce divisions whose length is less than 
the circumference of the jet. This would give for the interval defining 
the upper limit w : 4*508, which is very nearly a fifth. In the case of 
Plateau's numbers (w : 4' 38) the discrepancy is a little greater. 

The detached masses into which a jet is resolved do not at once 
assume and retain a spherical form, but execute a series of vibrations, 
being alternately compressed and elongated in the direction of the axis 
of symmetry. When the resolution is effected in a perfectly periodic 
manner, each drop is in the same phase of its vibration as it passes 
through a given point of space ; and thence arises the remarkable 
appearance of alternate swellings and contractions described, by Savart. 
The interval from one swelling to the next is the space described by 

# "Pogg. Ann.," bd. cvi, 1859. 



88 Lord Kayleigh on [May 15, 

the drop during one complete vibration, and is therefore (as Plateau 
shows) proportional cceteris paribus to the square root of the head. 

The time of vibration is of course itself a function of the nature of 
the fluid and of the size of the drop. By the method of dimensions 
alone it may be seen that the time of infinitely small vibrations varies 
directly as the square root of the mass of the sphere and inversely as 
the square root of the capillary tension ; and in Appendix II it is 
proved that its expression is 

T ~^\~w~) ^ 10 ^ 

V being the volume of the vibrating mass. 

In an experiment arranged to determine the time of vibration, a 
stream of 19' 7 cub. centims. per second was broken up under the 
action of a fork making 128 vibrations per second. Neglecting the 
mass of the small spherules (of which more will be said presently) , we 
get for the mass of each sphere 19*7-^-128, or *154 grm. ; and thence 
by (10), taking as before T = 81, 

t='0473 second. 

The distance between the first and second swellings was by measure- 
ment 16'5 centims. The level of the contraction midway between the 
two swellings was 36'8 centims. below the surface of the liquid in the 
reservoir, corresponding to a velocity of 175 centims. per second. These 
data give for the time of vibration, 

T =16-5-f-36-8 = -0612 second. 

The discrepancy between the two values of t, which is greater than 
I had expected, is doubtless due in part to excessive amplitude, 
rendering the vibration slower than that calculated for infinitely small 
amplitudes. 

A rough estimate of the degree of flattening to be expected at the 
first swelling may be arrived at by calculating the eccentricity of the 
oblatum, which has the same volume and surface as those appertaining 
to the portion of fluid in question when forming part of the undis- 
turbed cylinder. In the case of the most natural mode of resolution, 
the volume of a drop is 97ra 3 , and its surface is 18?ra 2 . The eccen- 
tricity of the oblatum which has this volume and this surface is *944, 
corresponding to a ratio of principal axes equal to about 1:3. 

In consequence of the rapidity of the motion some optical device is 
necessary to render apparent the phenomena attending the disintegra- 
tion of a jet. Magnus employed a rotating mirror, and also a rotating 
disk from which a fine slit was cut out. The readiest method of 
obtaining instantaneous illumination is the electric spark, but with 
this Magnus was not successful. " The rounded masses of which the 
swellings consist reflect the light emanating from a point in such a 



1879.] the Capillary Phenomena of Jets, 89 

manner that the eye sees only the single point of each, which is prin- 
cipally illuminated. Hence, when the stream is illuminated by the 
electric spark, the swellings appear like a string of pearls ; bnt their 
form cannot be recognised, because the intensity of the light reflected 
from the remaining portions of the masses is too small to allow this, 
on account of the velocity with which the impression is lost."* The 
electric spark had, however, been used successfully for this purpose 
some years before by Buff,t who observed the shadow of the jet on a 
white screen. Preferable to an opaque screen in my experience is a 
piece of ground glass, which allows the shadow to be examined from the 
further side. I have found also that the jet may be very well ob- 
served directly, if the illumination is properly managed. For this 
purpose it is necessary to place the jet between the source of light 
and the eye. The best effect is obtained when the light of the spark 
is somewhat diffused by being passed (for example) through a piece 
of ground glass. 

The spark may be obtained from the secondary of an induction 
coil, whose terminals are in connexion with the coatings of a Leyden 
jar. By adjustment of the contact breaker the series of sparks may 
be made to fit more or less perfectly with the formation of the drops. 
A still greater improvement may be effected by using an electri- 
cally maintained fork, which performs the double office of controlling 
the resolution of the jet and of interrupting the primary current of 
the induction coil. In this form the experiment is one of remarkable 
beauty. The jet, illuminated only in one phase of transformation, 
appears almost perfectly steady, and may be examined at leisure. 
The fork that I used had a frequency of 128, and communicated its 
vibration to the reservoir through the table on which both were 
placed without any special provision for the purpose. The only weak 
point in the arrangement was the rather feeble character of the 
sparks, depending probably upon the use of an induction coil too 
large for the rate of intermittence. A change in the phase under 
observation could be effected by pressing slightly upon the reservoir, 
whereby the vibration communicated was rendered more or less 
intense. 

The jet issued horizontally from an orifice of about half a centi- 
metre in diameter, and almost immediately assumed a rippled out- 
line. The gradually increasing amplitude of the disturbance, the 
formation of the elongated ligament, and the subsequent trans- 
formation of the ligament into a spherule, could be examined 
with ease. In consequence of the transformation being in a more 
advanced stage at the forward than at the hinder end, the liga- 
ment remains for a moment connected with the mass behind, when 

* " Phil. Mag.," xviii, 1859, p. 172. 
f " Liebig's Ann.," lxxviii, 1851. 



90 Lord Rayleigh on [May 15, 

it has freed itself from, the mass in front, and thus the resulting 
spherule acquires a backwards relative velocity, which of necessity 
leads to a collision. Under ordinary circumstances the spherule 
rebounds, and may be thus reflected backwards and forwards several 
times between the adjacent masses. But if the jet be subject to 
moderate electrical influence, the spherule amalgamates with a larger 
mass at the first opportunity.* Magnus showed that the stream of 
spherules may be diverted into another path by the attraction of a 
powerfully electrified rod, held a little below the place of resolution. 

Very interesting modifications of these phenomena are observed 
when a jet from an orifice in a thin platef is directed obliquely up- 
wards. In this case drops which break away with different velocities 
are carried under the action of gravity into different paths ; and thus 
under ordinary circumstances a jet is apparently resolved into a " sheaf,'' 
or bundle of jets all lying in one vertical plane. Under the action 
of a vibrator of suitable periodic time the resolution is regularised, 
and then each drop, breaking away under like conditions, is projected 
with the same velocity, and therefore follows the same path. The 
apparent gathering together of the sheaf into a fine and well-defined 
stream is an effect of singular beauty. 

In certain cases where the tremor to which the jet is subjected is 
compound, the single path is replaced by two, three, or even more 
paths, which the drops follow in a regular cycle. The explanation has 
been given with remarkable insight by Plateau. If for example 
besides the principal disturbance, which determines the size of the 
drops, there be another of twice the period, it is clear that the alter- 
nate drops break away under different conditions and therefore with 
different velocities. Complete periodicity is only attained after the 
passage of a pair of drops ; and thus the odd series of drops pursues 
one path, and the even series another. All I propose at present is 
to bring forward a few facts connected with the influence of elec- 
tricity, which are not mentioned in my former communication. To 
it, however, I must refer the reader for further explanations. The 
literature of the subject is given very fully in Plateau's second 
volume. 

When the jet is projected upwards at a moderate obliquity, the 
sheaf is (as Savart describes it) confined to a vertical plane. Under 
these circumstances, there are few or no collisions, as the drops have 
room to clear one another, and moderate electrical influence is without 
effect. At a higher obliquity the drops begin to be scattered out of 
the vertical plane, which is a sign that collisions are taking place. 
Moderate electrical influence will now reduce the scattering again to 

# " Proc. Roy. Soc," March 13, 1879. On the Influence of Electricity on 
Colliding Water Drops. 

f Tyndall has shown that a pinhole gas burner may also be used with advantage. 



1879.] the Capillary Phenomena of Jets. 91 

the vertical plane, by causing the coalescence of drops which come into 
contact. When the projection is nearly vertical, the whole scattering 
is due to collisions, and is destroyed by electricity. If the resolution 
into drops is regularised by vibrations of suitable frequency, the 
principal drops follow the same path, and unless the projection is 
nearly vertical, there are no collisions, as explained in my former 
paper. It sometimes happens that the spherules are projected laterally 
in. a distinct stream, making a considerable angle with the main stream. 
This is the result of collisions between the spherules and the principal 
drops. I believe that the former are often reflected backwards and 
forwards several times, until at last they escape laterally. Occasionally 
the principal drops themselves collide in a regular manner, and ulti- 
mately escape in a double stream. In all cases the behaviour under 
electrical influence is a criterion of the occurrence of collisions. The 
principal phenomena are easily observed directly, with the aid of 
instantaneous illumination. 



Appendix I. 

The subject of this appendix is the mathematical investigation of 
the motion of frictionless fluid under the action of capillary force, the 
configuration of the fluid differing infinitely little from that of equi- 
librium in the form of an infinite circular cylinder. 

Taking the axis of the cylinder as axis of z, and polar co-ordinate rfi 
in the perpendicular plane, we may express the form of the surface at 
any time t by the equation 

r=a o +f(0,z) (11), 

in which / (0,z) is always a small quantity. By Fouriers' theorem, 
the arbitrary function / may be expanded in a series of terms of the 
type oc n cos nO cos fez ; and, as we shall see in the course of the in- 
vestigation, each of these terms may be considered independently of 
the others. The summation extends to all positive values of k, and to 
all positive integral values of n, zero included. 

During the motion the quantity a does not remain absolutely con- 
stant, and must be determined by the condition that the inclosed 
volume is invariable. Now for the surface. 

r=a o + a n cosn0 cos kz (12), 

we find 

volume=-| ff r%cl6dz=f (jra^ 1 -\-\iroLy? co^kz)dz~z(7ra ^ + l7ra, n 2 ) ; 

so that, if a denote the radius of the section of the undisturbed 
cylinder, 

7TO, 3 = 7Ttf q 3 + ^7Ta M 3 , 



92 Lord Rayleigli on [May 15, 

whence approximately 



a Q =a\ 1 



8a 
For the case w = 0, (13) is replaced by 



w (13) " 



a =a 



( ~~A O ! .♦•••• \J"'*/' 

4<xv 



"We have now to calculate the area of the surface of (22), on which 
the potential energy of displacement depends. We have 

Surface = Jj{l + (J)V(^) 3 }W,^ 

I 2 Vfe / 2r 2 W/ J 
==//"{ 1 + |-& 2 ^ 2 eos%0 sin 3 /^ +^i 2 a» 2 a~" 3 sm%# cos 3 /^}r $0 cZ^ 

= 3 {2wa + \irWa,^a + iTT^a^a -1 } ; 

so that, if a denote the surface corresponding on the average to the 
unit of length, 

cr=27ra-fj7T<X"" 1 (^ 3 a 3 4-% 2 -— 1)«» 2 . . . (15), 

the value of a Q being substituted from (13). 

The potential energy P, estimated per unit length, is therefore ex- 
pressed by 

P=|7r^iT(^ 2 a 2 + ^-l)^ 2 .... (16), 

T being the superficial tension. 

For the case « = 0, (16) is replaced by 

P=^a-iT(^a8-l)^ (17). 

From (16) it appears that, when n is unity or any greater integer, 
the value of P is positive, showing that, for all displacements of 
these kinds, the original equilibrium is stable. For the case of dis- 
placements symmetrical about the axis, we see from (17) that the 
equilibrium is stable or unstable according as ka is greater or less than 
unity, i.e., according as the wave-length (2,7rkr x ) is less or greater 
than the circumference of the cylinder. 

If the expression for r in (12) involve a number of terms with 
various values of n and h, the corresponding expression for P is found 
by simple addition of the expressions relating to the component 
terms, and contains only the squares (and not the products) of the 
quantities x. 



1879.] the Capillary Phenomena of Jets, 93 

The velocity potential (0) of the motion of the fluid satisfies the 
equation 

g0 1 j0 1 d?ct> d?<t> _ Q . 
^ r 2 r <ir r 2 c?0 2 <&: 2 

or, if in order to correspond with (12) we assume that the variable 
part is proportional to cos n6 cos hz, 



A6 + l^_/^ +jfc2 \ _ (1 

dr z r dr \r A J 



The solution of (18) nnder the condition that there is no introduc- 
tion or abstraction of fluid along the axis of symmetry is — 

0=j3 w J w (^r)cosn#cos lez .... (19), 

in which i= V( — 1), and J n is the symbol of the Bessel's function of 
the nth. order, so that 

(ler) n / - & 2 r 2 &V> 






i+ „ ; „ + 



2«r(w + l) L 2.2^ + 2 2.4.2^ + 2.2^ + 4 

+ ^r 6 + \ ^o) 

2.4.6.2^ + 2.2^ + 4,2^+6 J '' 

The constant /3» is to be found from the condition that the radial 
velocity when r=a coincides with that implied in (12). Thus 

ik^ n J n f (iha)=:^L (21). 

di 



The kinetic energy of the motion is, by Green's theorem, 

ad0dz~\wpz . ilea . J n (iJca) J n '(ika) . /3 n % ; 

so that, by (21), if K denote the kinetic energy per unit length, 



\p 



" ,dd> 
dr 



r=a 



^Trpcfi — (^). ( ^ | . . . (22). 
ilea . J n '(ilea)\ dt J 



ilea . J n (ilea) \ 
When w=0, we must take, instead of (22), 

ilea , J (ilea) \dt / 

The most general value of K is to be found by simple summation, 

with respect to n and le, from the particular values expressed in (22) 

and (23). Since the expressions for P and K involve only the squares, 

dot. 
and not the products, of the quantities cc, --, it follows that the 



94 Lord Kayleigh on [May 15, 

motions represented by (12) take place in perfect independence of one 
another. 

For the free motion we get by Lagrange's method from (16), (22), 

^ + ii^W( R s + W-lK=0 . . (24), 
dt' z pa 6 *j n (ika) 

which applies without change to the case % = 0. Thus, 

if a n OC COS (pt—e), 

/=1"^)( % HW-1) . . . (25), 
pa 6 J n (ika) 

giving the frequency of vibration in the cases of stability. If n = 0, 
and ha< 1, the solution changes its form. If we suppose that a n cce^K 

a== TiW|^ (1-/^2) m , m ( 26). 

From this the table in the text was calculated. 

When n is greater than unity, the values of p 2 in (25) are usually 
in practical cases nearly the same as if ha were zero, or the motion 
took place in two dimensions. We may therefore advantageously 
introduce into (25) the supposition that ha is small. In this way we 



get 



_p 3 = fiQn? — 1 -f & 3 & 3 ) _ n 

po? 

or, if ha be neglected altogether, 



n . 2??< + 2 



(27), 



i» s =(» 8 -»)— (28), 

pa 6 

which agree with the formulae used in the text. When w=l, there is 
no force of restitution for the case of a displacement in two 
dimensions. 

Combining' in the usual way two stationary vibrations, whose 
phases differ by a quarter of a period, we find as the expression of a 
progressive wave, 

r=a + ^ n cos nO cos hz cosjp£ + 7» cos nO sin hz &mpt 

= a + 7*t cos nO cos (pt— hz) . (29). 

For the application to a jet the progressive wave must be reduced 
to steady motion by the superposition of a common velocity (v) equal 
and opposite to that of the wave's propagation. The solution then 
becomes 

r=a + 7« cos nO cos hz .... (30), 



1879.] the Capillary Phenomena of Jets. 95 

in which ^ n is an absolute constant. The corresponding velocity- 
potential is 

0=- w + ^^$ sin kz cos nO. . . (31). 

It is instructive to verify these results by the formulas applicable to 
steady motion. The resultant velocity q at any point is approxi- 



mately equal to — ; and 

dz 

~-=— -V+ 1 * n n \ s cos kz cos nO. 
dz i/cJn (ilea) 

At the surface we have approximately r=a, aud 

1^2— 1^2 _p fa v r (n »y — I cog i iZ cos ^ 

ihJ\ika) 

Thus by the hydrodynamical equation of pressure, with use of (25), 
since v=:ph~ 1 , 

Pressure = const, -f- 7»a"~ 3 T (% 3 — 1 + 7tf% 3 ) cos kz cos w# . (32). 

The pressure due to superficial tension is T (Rfi^-R^"" 1 ), if R 1? R 2 , 
are the radii of curvature in planes parallel and perpendicular to the 
axis ; and from (30) 



■ R 9 1 = LV = — W*i n cos nO cos 

R x " x = r"" 1 + - = ft -1 + 7»a~ 3 ( -ft- a — 1 ) cos %0 cos ib ; 

so that 

Pressure^ const. + 7^~ 3 (^ 3 — 1 +7f&& 3 ) cos %0 cos 7^. 

Thus the pressure due to velocity is exactly balanced by the capillary 
force, and the surface condition of equilibrium is satisfied. 



Appendix II. 

We will now investigate in the same manner the vibrations of a 
liquid mass about a spherical figure, confining ourselves for brevity to 
modes of vibrations symmetrical about an axis, which is sufficient for 
the application in the text. These modes require for their expression 
only Legendre's functions P n ; the more general problem, involving 
Laplace's functions, may be treated in the same way, and leads to 
the same results. 

The radius r may be expanded at any time t in the series 

r=a + a 1 P- 1 (fi)+ . . . + aJ?«0*)H- . . (33), 



96 



Lord Ray lei gh on 



[May 15, 



where a Y a 2 are small quantities relatively to a , and ja (according to 
the usual notation) represents the cosine of the colatitude (0). 
For the volume included within the surface (33) we have 

f+i 

=l 7r I K 3 +W{ a i p i(^)+ • • +fl«P»C"-)+ . . } 

+ 3a {o 1 P 1 (/*)+ . . +a w P w (/i)+ . . } 3 + . . ytfju 



=f^o 3 



'+i 



-l 



[l + 3a Q ~^a n ZFn*M¥/* 



=%7ra \l + 3a -2S(2^. + 1) " W, 

approximately. If a be the radius of the sphere of equilibrium, 

a*=a ( ?[l + 3ar^(2n+l)- 1 a n z'] . . . (34), 
We have now to calculate the area of the surface S. 



S = 2tt [ r sin J |^ + (— Y j ^=2w[ {r2 + i(— Yjsi 
For the first part 



sin 0d#. 



+i 



-l 



*r*d/i=2a * + 2S(2w + 1)~W. 



For the second part 



JL 
2 



r/ ~Ysin6»^=i 



-fl 



(1-^) 



2 a. 



ftr«. 



w 



n- 



djUL __ 



cZ/a. 



The value of the quantity on the right hand side may be found with 
the aid of the formula* 



*+i 



.dP m dF 



m "'-»• n 



-l a/a dfjb 



dju=n(n-{-l) 



'+i 



-1 



■*- »&-*- «^/*» 



Thus 



JL 

2 



+ Y^Ysin0cZ0=i 



"+1 



+1 /dV \2 

-1 \djLL J 



=^2n(n + l)a n A P n ^=2 »(w + l)(2w+l)-W. 

Accordingly S = 47ra 3 + 27r2(2?i+l)~ 1 (ri 3 + ^+2)^ 3 ; 

or, since by (34) 

a 3 =a 3 -22(2^+l)~V, 

S=47ra 3 + 27rS(^-l)(^+2)(2^ + l)-% M 3 . . (35). 



# Todliimter's " Laplace's Functions," § 62. 



1879.] the Capillary Phenomena of Jets. 97 

If T be the cohesive tension, the potential energy is 

P=2«rT2(w-l)(^ + 2)(2w + l)-W • • (36). 

We have now to calculate the kinetic energy of the motion. The 
velocity-potential may be expanded in the series 

0=/3 o + /VPiOO+ • • +P»r w PnW+ • (37); 
and thus for K we o:et 



i 



K=i P 



^td&=\p.2na* [ +l ^dfi=i P . W . 2 (2^ + l)-%a^-i^ 
<#r ~ J-i cZr 

But by comparison of the value of — from (37) with (33), we find 

dr 

n — ]n CKX'ii 

lb Co M«, \ 

H dt 

and thus 

K=27rpa^(2n+l)-hi-^(^^J . . . (38). 

Since the products of the quantities a n and — — do not occur in the 

expressions for P and K, the motions represented by the various terms 
occur independently of one another. The equation for a n is by 
Lagrange's method 

^ + w(»-l)(» + 2)-5-a n =0 . . (39); 

at* pa 6 

so that, if On oc cos (p£ + e), 

^=7z(rc-l)( w + 2)-L ". . . . . (40). 

The periodic time t given in the text (equation (10) ) follows from 
(40) by putting T=27rp-~ 1 , w=2, V=|- tt^ 8 . 

To find the radius of the sphere of water which vibrates seconds, 
put jp = 2?r, T = 81, /o=1, w=2. Thus a=:2*54 centims., or one inch 
almost exactly. 



The Society adjourned over Ascension Day to Thursday, May 29. 



VOL. XXIX. H