174 
MR. W. M. HICKS OH THE STEADY MOTION AND 
cos nd 
ijrA„R'„= 4 a 2 VS' 2 f - 77 ^ 
j o — C 1 
(O' — cos df 
d6 
but 
Hence 
a convergent series. 
= -a s vs 7- T 1 = de 
du J 0 \/ C— cos 0 
= — a 2 V \/ 2 S' =« 2 V ^/2T' n 
du 
7rA 0 R/ 0 =orV 2T' 0 
^=^S){T'o|+2Vr,|cos OT } 
The circulation of this is by (4) 
ZuN'l — 
la ^\ R ' 0 ^R', 
Let the stream function for this be 
2^q\/2 [ rp, Ro _ ^' n 
7rv/(C- C ) j °R ' 0 2,1 4 w 2 — 1 Rh 
cos nv 
Then 
If then 
[ T 0 
00 1 T' 1 / T' co T' 
. ,-5'° +22 U-^e^ = 2aV -^- 22 ^ 
A = 
T' 00 T' 
_ 0 _ n 
p/ -“^1 TTT 
JX n -K n 
T' 00 1 T' 
_4lO_i_9'V - 1 - x « 
R'o ^4»»-lR'. 
The stream function for translation alone is 
aV 2 Y 
r v/(C-c) 
Rn 
A \ rpy R')i 
(! X)T °R' 0 + 2S 1 \ l +±n*-lJ T * Rh C0S llV \ ‘ ' ' 
The principal term here is the second, in R i . To the value of A is 
A=1-4(L-2)F-(2L 2 -| 1 L + K^H.(17) 
The value of xp along the tore is 
iV(/-Xa 8 ) 
The stream lines will of course in general be closed curves, having their extremities 
on the surface of the tore; one set going through the aperture, and the other outside. 
