628 
MR. W. M. HICKS OK TOROIDAL FUNCTIONS. 
Further if u >u (P', Q', here standing for P (it), Q (u)) 
Hence the series 
V P O' < 
and is therefore convergent, 
convergent. 
Again if u'<u 
Much more then is the series 
P«+iQ 
»+i ■ 
L ± S .p' o 
CHS 
£P»Q'» COS u(v + a) 
and as before, the series SP „Q„ cos n(y-\-a) is always convergent. 
11. Both the functions P ;j .Q„, except along the critical circle and axis respectively, 
make <f> finite, continuous, and single valued when n is integral. The first statement 
has already been proved, the second follows from the way in which y-, — are expres- 
sible in terms of two successive P„ or Q„, and the third is seen to be at once true by 
integrating ~ round a circuit lying on any tore u— constant, when is seen to 
vanish. Now the space is a cyclic one. Hence the above functions are not suitable 
for expressing any general conditions in the space without a tore, though they are 
suitable for any given surface conditions whatever. 
Still keeping to physical analogies in order to obtain solutions suitable to this case, 
we will consider the potential due to a vortex ring or electric current along the critical 
circle. This would give cyclic functions, but also certain surface conditions. In any 
particular case then it will be necessary to take account of these surface conditions 
by means of the P„ or Q„. This potential is measured by the solid angle subtended 
by the ring. 
The (solid angle) X/x can be expressed in the form c.s. denoting cos v, sin v ,— 
or 
where 
/— . / 7 s- [ n C + c— S cos <4 
2/xtt— V sm vVC-c\ 8 , 
dcf) 
! + S 2 sin 2 </> y'C’ — S cos (p 
n Jcsinv f v / C + c+^/C-c TT/ 7X 
2^-^ 7 ==- 5 -n(» 1 ./') 
I ^/0 + C — y/ C —C , 7N 
+ “7c i =5+s -n(%J) 
«1 = 
2S_ 
—c 3 + S 
