PROFESSOR W. M. HICKS ON VORTEX MOTION. 
97 
! cl d cl d_ 
7, + 7 "i" cr -{■ IV \ T 
dt dr r cW dz j r 
(dd d d 
\Tt + ’■ dr +^ ^\b 
_L — 
cr + — = 
d d d cl 
fZ , , da . d , . 
('") ++ *<’■») = 0- 
dL [HL 
dr \ p 
d IP 
rcW \ p 
d fp 
+ V 
H-V 
4- V 
ch 
It is desired to find a solution in which all the quantities are independent of 6. 
Therefore 
dr 
da 
cie cie 
^ 0 - + V 
cie ’ cie\p^ 
0 . 
The last gives 
d , d , d \ ^ 
+ ^ V: + VT ) = 0- 
dt 
dr 
clz 
If therefore t// be the equation of a surface always containing the same particles of 
fluid, it is possible to take 
= o:: J W- 
TT 
Also 
ch 
Let K be the current function (which I distinguish throughout from \Ij). 
Therefore 
Substituting in 
it follows that 
1 (I^ 
27rr clz ’ 
IV = 
1 dK 
2'7rr dr 
d , d cl 
* +“■ * 
d'yj/' I cl/c clyjr 1 die cl^lr 
dt 2nTr dz dr 27rr dr clz 
Now make the further supposition that the surfaces i// = const, move without 
alteration parallel to the axis of 2 ; with velocity Z. 
Therefore 
Therefore 
cly\r diy 
•“ Aa ~ *• 
dt 
clz 
d^ d . • d^jr d , 
.h s V - ’Ti'-Z) - - {k 
77/'“ Z) = 0. 
VOL. cxcir.- A, 
Q 
