MR. W. M. HICKS ON TOROIDAL FUNCTIONS. 
621 
Un—a^u i— n- 
u x , u 0 may be expressed as elliptic integrals, viz. : 
where 
or 
and 
Hence 
1 
r n cie /j-a*- 
' f ° J o \/C—S cos 6 V sin 2 0 
iq =1 \/C—8 cos 6d6= - 7 = [ = a/I — & 3 sin 3 6<16= —t=E j 
Jo v k Jo V h J 
¥= 
2 S 
C + S 
IFF 
1 
(C + S) 2 
jt® =1 _ e -2« yl' /3 =e-2» 
x=2C=F+} 7 
(17) 
„(2»-2)(2w-4) . . . 2/ i /T7 / r \ 
p *- 2 ~>-i)... 3 Itp w - iF 1 • 
(18) 
where we may suppose the numerical factor dropped if we are dealing with the 
differential equation, but not if we are dealing with the sequence equation. 
The value of P ;t when u= 0 is 7 r 
,, ,, u= co is co 
These statements are at once seen to be true. Since u becomes infinite alone: the 
critical circle it follows that the P„ are not the suitable functions to use by which 
to express functions which are finite in spaces containing the critical circle, i. e ., within 
any tore. But it is finite and continuous for all space outside any tore. 
7. If we put for P„ in equation (9) IfiQk we find in the usual way 
P;/Q n —BP„ + AP„ 
u (hi 
oS~lV’ 
Begarded as an analytical solution of the equation this is complete, but in this form 
it is altogether useless for application. Now Heine”' has shown that the spherical 
harmonic of the second kind is expressible in the form 
Q« 
_ cW _ 
(x— \/x 2 — 1 cosli d) n 
* ‘ Krigelfunctionen,’ Ivap. iii. 
