o 
LORD RAYLEIGH ON THE CIRCULATION 
the places of maximum vibration, and falling back to it at the nodes. In a vacuum 
the phenomena observed by Savart do not take place, all kinds of powder collecting 
at the nodes. In the investigation of this, as of the other problems, the motion is 
supposed to take place in two dimensions. 
It is probable that the colour phenomena observed by Sedley Taylor* on liquid 
films under the action of sonorous vibrations are to be referred to the operation of the 
aerial vortices here investigated. In a memoir on the colours of the soap-bubble,t 
Brewster has described the peculiar arrangements of colour accompanied by whirling 
motions, caused by the impact of a gentle current of air. In Mr. Taylor’s experiments 
the film probably divides itself into vibrating sections, associated with which will be 
aerial vortices reacting laterally upon the film. 
The third problem relates to the air currents observed by Dvorak in a Kijndt’s 
tube, to which is apparently due the formation of the dust figures. In this case we 
are obliged to take into account the compressibility of the fluid. 
[My best thanks are due to Mr. W. M. Hicks, who has been good enough to 
examine the mathematical work of the paper. The results are thus put forward with 
greater confidence than I could otherwise have felt.] 
§ 1. In the usual notation the equations of motion in two dimensions are 
1 dp 
P d d. 
1 dp 
p d v 
du 
dt 
dv 
dt 
du du ] 
-\-V V M— U — — V -- 
dx du 
[ 
0 dv dv I 
+ W~ v — u - v — 
dx d y j 
( 1 ), 
and since the fluid is incompressible, 
In virtue of (2) we may write 
du . dv 
t & + ( fe =0 
dAr d\P 
U ~dy’ V= ~~dA . 
Eliminating p between equations (l), we get 
0 fdu dv\ d (du dv\ d ( du , du \ d ( dv , dv \ 
v ^\^y -t: I=i7.1 u a:. j- T Ju—+v— ). 
Now 
dy dx) dt \dy dx J dy \ dx dy j dx \ dx dy 
du du , d(u 2 + v 3 ) /du dv 
U ~dx^ V dy = *~~dx \dy ~Av 
dv dv , d(u~ + v~) (du dv 
U dy = ? dy U \dy ~dx)’ 
( 2 ). 
( 3 ). 
* Proc. Rot. Soc., 1878. 
f Edinburgh Transactions, 1866-67. 
