048 
MR. W. M. HICKS OK TOROIDAL FUNCTION'S". 
_os' 
- r 2 Q'r 
■1 4/ 3 — 1 PC 
_ r I - Q i " 
Yim —! 4 ? o_- 
■1 PC 
and where PC. Qh stand for 
d P,-Q 0 ) ^Q/Pti) 
dw 0 ’ dw 0 
We may now find the energy of the fluid motion. This is. the density of the fluid 
being unity, 
m I-* 7 deb 
1 = — 2lTp-r-clv . (h- 
J 0 r d o r dn 
and 
dn a 
dv C — c 
#_y£_y_§L_ 
dn r C — c 
«S 
P=n 
C—c 
.T = — 27t« 2 YS 5 
(C-c) 
*<f>dv 
But 
„ 5TT7O , 0 . . „ f^sm« sm nv 7 
-8 v /2a s V 3 S 2 SA„P„ 
J o U~— C J 
16\/2 ,T T .irt^ I -r. d /I (f \ f 2,r cos (n— l)v— COS (7l + l)i; 7 
:-3- a s V-SSA„P„-f g --=- dv 
a/C-; 
f> cos nv 7 
l„ 7nw*= 3 ' /2 Q” 
But 
• T— — — r/ 3 Y 3 S2A P -CO' —O' ) 
3 ” j (/«S Wh k ' !+1 ' 
—Q'»+i= — 2 hSQ w 
‘•T= -;-a s V 2 S2*nA i ,P„Q', 
(44) 
which is more convergent than the series for <£. 
In a similar manner may he found the velocity potential for any motion of trans¬ 
lation, or the magnetism induced in a uniform field of force. 
■ o 
