MR, W. M. HICKS ON TOROIDAL FUNCTIONS. 
f> 35 
AP„=pr 
• n Jo 
Here A = 0 unless n— 0. Hence 
cos mio cos nv 
o V {C—S cos id] 
r ckudv 
AP* i0 =47t[ 
n cos md 
q y/ C—S cos 6 
do 
We have already found that 
P 
die 
2n. + l 
(C—S cos 6}~*— 
we are therefore led to expect that in general 
fi 77 cos m6dd 
P M.H X 
2)1 + 1 
{c—s cos e } 2 
which can easily be shown to be the case. 
By taking the fixed point at the origin we have 
BQ m.n^ 
f "■ p T cos miu cos nvdiodv 
Jo Jo \/C—cosv 
Here B = 0 unless m— 0, and then 
Q 
cos nd 
oc 
f" COS' 
J n a/ 0 — 
\/c— cos e 
dd 
an expression which has been already found. 
These expressions as single definite integrals are already known to be solutions of 
the differential equations, and are given by Heine in his ‘ Kugelfunctionen.’ They 
may easily be proved directly, and connected with the values found already by the 
sequence equations, and the values for P 00 , P 01 , &c. Thus writing 
cos mddO 
2n + l 
(C—S cos 6) 2 
the integral is easily shown to satisfy equations (32), and the only further condition 
requisite is that A shall be chosen so as to make it agree with V 0 H , P,,.,^. 
Now 
dd 
(C —S cos 
4 N 2 
