638 
MR. W. M. HICKS ON TOROIDAL FUNCTIONS. 
A] so 
— \/ 2L». „ (say) 
Since the distance between two points is 
{CC' — cos (v—v') — SS' cos (iv—w')} 1 
^/(C-oXCW) 
It follows that the potential for a unit point at (u'.v'.w') is 
for points outside the tore u'; whilst for points inside, it is 
y • • (35) 
where, when m— 0 or n= 0, half the above value for L mM must be taken. 
When u —0 P //; ,,= 0 except when m— 0 when P 0 . ;V =7r, which agrees with the 
value found in section II. 
Also P ynji behaves in a similar way to P 0 „j for increasing n, whilst P / „ +] ., l >P m ., l when 
m is large, as is clear at once from the integral expression for P, iWi . 
Also since 
it is clear that when n=cc Q rn r = 0 for all values of m.n. Also Q m ,„ behaves as Q 0 , ; 
for increasing n. 
IV. 
TORES WITH NO CENTRAL OPENING. 
15. In the case where the hole of a tore vanishes the functions hitherto considered 
become nugatory. In this case we must have recourse to the co-ordinates already 
referred to in (8). It is not here intended to develop the theory with the fulness of 
the general case. The functional differential equation has been shown to be 
