PROFESSOR W. M. HICKS ON VORTEX MOTION. 
89 
in which a are small, and in the third integral, x and 6 are nearly 1, 0 respectively. 
Let cc = 1 — sin d ~ t]. Then near the pole the rectangular co-ordinates of a 
point referred to the pole are connected by (if f= J — x“J') 
_/sin^ 6 = small constant = (say). 
so that when ^ = rj each = y. 
The third integral is 
J sin^ d v/ (df + dv-) 
2 S 
- If = y 
dx 
2X f — 
J r ?i 
sin 9 {{dfjdrf sin^ 0 + 4 siid 6 cos- 6 (//r)^}* 
+ df) 
= 2X 
I (1 - ^) i (d//dx)- sin^ 0 + 4 cos’- ^ ^ ^ 
2XJ' 
dfjdx 
The curve is given by 
Therefore 
]:v 
1 + (d-nld^f' 
~VTW 
+ I. V 
1 + (d^ldyf 
f + 4P 
dr] 
___ _ \ fl _ _ I 
i^T? 
7 
3/2 
i / 
7 
Therefore 
Integral = ^ ^V 
+ 47'V 
+ Afjn'' 
-+w;y 
4^3 ^ 
4f + yf ^ 
i 
-IV 
7y> + 4y 
/ 
'’o drj~ 
V _ 
2XJ' 
dfjdx 
The second integral is 
ilog| + log^^ 
XJ' , 
- lOP’ - 
dfldx ^ 7?1 
2\S' d- dd _ 2\,y 
dfjdx J 0 sin 0 dfjdx 
XV 
dfjdcc 
and 6 is nearly = 0. Therefore 
log tan 
, 1 + cos 0 
I - COS0 “ 
A,J' , (1 + COS 6V 
log 
dfjdx 
sin^ 0 
Second Integral = log Ar , 
^ dfjdx ^ vl 
and the second and third together 
XJ' , 4 
dfjdx 7 I 1 
VOL, cxcn.—A, 
N 
