MR. W. M. HICKS OK TOROIDAL FUNCTIONS. 
651 
Hence, supposing the tore to be at zero potential and to have no charge, the 
potential of the disturbed fieid is, dashed letters denoting functions of u, 
— _ 4l - - y(C— c)%n (q„,— p 7 - sin nv 
The density at any point of the tore is 
Now at the points where the osculating plane touches the tore z—r and p = Ii, 
whence 
C — c = Ss or c= 1 /C s=S/C 
The greatest density on a sphere similarly influenced is —. The ratio is then 
_4\/2 S~ 4S I 
3 CHCPV CHVJ 
The value of this ratio for the cases already considered are 
for ti — sin 3°, *675 
,, k'= sin 6°, '698 
When the direction of the electric field is perpendicular to the axis, its potential is 
, S cos w 
(p i =[xp cos w— /xa~—— 
Hence clearly the functions for the expansion of this are the tesseral functions, P 1(i 
Q L)! , and the conditions, since the potential holds for space outside the tore, are that 
S cos w , _ 
/x«—+ v (J — c cos ivXA n i J lM cos nv— 0 
when u=u' for all values of v. 
4 p 2 
