762 
PROFESSOR W. M. HICKS OX THE THEORY OF VORTEX RIXGS. 
Also 
rfU 0 
dk 
ah 
2?r a cos nv + 7 sin nv 
n C — c 
dv 
7 dTJ 0 [^ « 2 er nu 
ak - I a cos" 5 nv-- , 
dk J o ^ 
2iraka.c~ nu dU 0 
S dk 
= 4z7rCLak n+z 
cK Jo 
dft 
and can therefore be neglected except in the case of pulsations. 
Hence A+2AJEt„=0 our order of approximations. The terms in e can therefore 
be neglected except in pulsations, i.e., n= 0. 
15. The next step is to determine cf >. Now <f) is the flow along any line up to the 
point in question. Choose this line to be the two portions 
(1) straight line v=tt from u=u 1 to u—u 
(2) circle u—u from v—v to v—tt 
The part of c/j depending on \jj 0 depends on the path of integration, but is constant, 
and therefore will disappear in <£. Consequently the part depending on y is the only 
portion needed. Hence 
Now 
Hence 
=f 
1 dy dv dn1 
p dv dn' dn 
1 d X 
p dv 
du-\- 
Ji)=7T 
du-\- 
1 dy du dn' 
_p du dn dv 
1 d X 
dv 
IP 
du 
dv 
= <t>i + <f>- 2 (say) 
X= 
- (L cos nv +M sin nv) (say) 
v (C— c) 
^n±>Mdu 
J t/j 
=(-)-£ V(|)(CE, + DT.)^ 
_2)i+l In— 1 
Whence, as the lowest terms only are required, and as R„cc h and T„oc k 2 
& = (-)" 
V(2£) 
a 
(CJR.-DT,) 
w 
"l 
