MR. W. M. HICKS OK TOROIDAL FUNCTIONS. 
023 
In these change 6 into 20, then 
v u =2 
cie 
0 C — S + 2S cosh 2 0 
_ ^ f" dd 
1 "J o (C — S + 2S cosh 2 6)1 
Again, write cosh 0 —sec <f>, then sinh d=tan </>, d0 =sec ^d<f>, and when 0— 0 or co, 
Mor^ 
Hence 
d<j> 
Also 
Now 
and 
v n = 
— 9 
o\/C + S — (C — S) sin 3 (f> 
= 2y/k'Y . 
cos 3 <jjd<J> 
J 0 (C + S-(C-S) sill 3 0}* 
2 N A/ 3 —/V 3 sin 2 0 
v/FJ o (1 -/■•'" sin 2 '<f>yrV 
2F' 2 Id N dtf> 
~ x/k'~ v/FJ o (1 -/A sin 3 # 
fr // 2 sill 2 
C did J 0 (1 —7/ 3 sin 2 0)t 
_/V -#-„ 
.1 0 (1 —7a' 3 sin 3 </>)« 
/w 1 — 1 F'—F' 
did id 
( 20 ) 
and finally 
^=7F( f - e ') 
(2 K -2) . 
(2rc-l) . 
2 /^%(F-E')-K-iv^F' 
3 1 \/i' 
( 21 ) 
( 22 ) 
The value of Q /( for u — 0 is oo, and for u— oo is zero. Hence Q„ is suitable for 
space within a tore, and not for space including the axis. 
8. The foregoing value of Q ;i has been obtained from analogy with that for V n ; but 
