PROFESSOE W. M. HICKS ON VORTEX MOTION. 
79 
where 
b = (Six — sin X) xjj. 
TT/xa ' ' ^ 
In deterraininof the vortex filaments we will take 9 to be measured in the same 
O 
direction as rj : that is, to the right of meridians as looked at from the polar axis. 
In this case 
j x’l 
J sin-^? cb. 
or 
sin 6 (-T — xrj') 2 sin 6 cos 0 
5 = ^+ yJ'f' 
(1.) Case of X small. 
j dx 
b)} 
z" dx 
, v/(J - (J - - b) ■ 
J - = i W \ - 1) 
also h is of order Xk Put 
Hence 
1 
■ ' Xi 
10 
- X dx 
Xi, Xo are the roots of the denominator equated to 0, viz., of 
A first approximation is 
1 ± v/l -^c 
X- 
— a;- + c = ^ (t® — x“). 
= i\ or Vo (say), where ?‘i denotes the smaller root. 
Let for a second approximation 
x~ = Ti 4- where ^ is of order X". 
+ c = Xi + — Vi — ^ c = (2r| 
x^ = r\ 4- S/’i^ — Xi — 
l)e 
X 
Therefore, 
X- 
