I 
5G 
PROFESSOR W. M. HICKS ON VORTEX MOTION. 
(P'L , ^ f/Z , „ 
-r- + tan v-r mA = 0 i 
(iv civ I 
clX ^ ^ 
—^ — cotn u- -wiX = 0 I 
cm J 
As will be seen later m must be of the form n{n — 1 ), n being any integer. 
TT 
Writing for a moment v = ~ — 6, the equation in Z becomes 
whence 
Therefore 
- cot 6 ' + n (ir - I) Z = 0, 
cW 
Z„ = — sin 6 
d0 
Z. 
sin“ 6 = cos^ V 
6 sin“ 0 — ^ 2 - sin^ 0 
6 cos“ V — A- cos^ V 
cos^v 
4 7 _2_ 7 
Zj, 1 0 ^^4 
- 5-^2 
To determine X, we proceed by the same analogy to put 
dV 
x = s 
dll 
Then 
dll 
— — coth u-r — n (?^ — 1) X = S — 1 + coth u- - nin — 1) P k 
hd clu ' ' cki ydu- v d^l '■ ' J 
If then P denote a zonal harmonic with imaginaiy argument and of order n 
the right hand of the above vanishes, and the value of X is a solution. That is 
- 1, 
X.„ = S 
clu. 
Now we have 
_ 1 . 3 ...( 2 ?^ - 1 ) r _ n {11 - 1 ) 
2(271-1)^ 
nl 
Hence 
Xo = SI Xi = 6 S“ + Sb S^ = lA X 4 - I X. 2 . 
This set oT solutions gives values finite and continuous at all points inside a given 
ellipse of the family, but infinitely large at an infinite distance. Let Y denote the 
second integral of the equation. Then, in the usual way, it may be shown that 
Y 
= xf|*. 
