MR. W. M. HICKS ON TOROIDAL FUNCTIONS. 
G39 
In this write i/>= \/uG when 
cPCt 1 dG 3 p rn~ __ 
7 ~ i-;- n Gx— — Lt — l) 
cm - 4 w cn< Vj~ 
the equation of cylindric harmonics (Bessel’s functions) with imaginary argument. 
Let G.H be the two particular integrals corresponding to the cylindric function J.Y 
of the first and second kind. Then 
G m (nu)=J m (nui) 
U ill (nu)=Y m (nui) 
And the potential function can be expressed in the form 
cf)= y/u 2 -j- v 2 t%(A m G m (nu) -f B m H m (nu) ) cos (nv-\-oi) cos (»ue+/3) 
Many of the properties of these functions can be at once written down from the 
analogous properties of J.Y. Thus 
So also 
and 
G m (nu) = - 
( nu) % 
2.4 .. . 2m 
/d 
n*U' 
2.2m+ 2 1 2.4. (2m+ 2) (2m+ 4) 
= — cos (mi sin 6—m6)d6 
(pm) 
1.3.5 
&c. 
2 r T . t 
--- . cos ( nz cos 6) sin' 2 " 
2m —1 7rJn 
ode 
d'G ul /- < 
U ^Y (/ =VlG ul — UGx m+1 
— llGm^ — rnGm [> 
G m+x — G,„ fi- G w _]_—0 
which equations the H also satisfy. 
The sequence equation has been solved by Lommell," so as to fully express G m 
and H m in terms of G () , G 1? H 0 , H 1 . But in any particular case where the values are 
required it is best to calculate successively by means of the sequence equation direct. 
In the space within a tore u can become infinite, viz. : at the origin, and is never 
zero ; this is evident from the equation 
ap 
u=- 
O I o 
P~ + 3“ 
* ‘ Studien liber die Bessel’schen Functionen,’ p. 4. 
