640 
HR. W. M. HICKS OK TOROIDAL FUNCTIONS. 
Without, it may become zero along the axis but infinite nowhere, for as it approaches 
the origin u must approach a finite limit which depends on the circle along which it 
moves. Now when u is infinite, G is infinite. Hence the G functions belong to space 
outside a tore. We are led to conclude that the H functions belong to space within. 
This may be proved as follows : Amongst many integral expressions known for Y 0 one 
is given by Heine,'" viz. : 
f e ™ cosh e d0 
H 0 (A)=a[ e-' lC0she cW 
Jo 
This is easily verified, for substituting in the differential equation it has to be 
shown that 
(si nk 3 6 — - cosh $)e~" coshe cl0= 0 
which, on integrating the first term by parts follows at once. From this form we 
gather that 
r 
when u=0 H (J = cW= co 
Jo 
„ U — OD H (J =0 
whence H (l is the proper function for space within a tore. From the sequence 
equation this is seen to apply also to the H,« in general. 
V. 
EXAMPLES AND APPLICATIONS. 
In this section I propose to give a few examples of the application of the foregoing 
theory, to the solution of physical problems. 
16. Potential of a ring whose axis is the same as the critical circle. 
Let z be its distance from the plane of the critical circle, b its radius, u', v its 
dipolar co-ordinates. 
Then the potential is 
0 T f _ cW _ 
1/X J o ^/(z-z')* + p i + ¥-2bfPP~6 
This expanded takes the form 
VC— cAA /( P /t cos (ra?+a„) 
for points outside the tore u . 
This suggests 
* ‘ Kugelfuncfcionen,’p. 191. 
