618 
MR, W. M. HICKS OK TOROIDAL FUNCTIONS. 
has to some extent considered spherical harmonics with imaginary argument, but he 
has not developed them, at least for fractional indices, in a form suitable for appli¬ 
cation here. Consequently they will be here considered independently and with 
especial reference to physical applications. Hereafter, C, S will be used in general to 
represent cosh u, sinh u, respectively. 
We have then in general 
f— v/C—cos v 2(A„P„+B„Q„) cos (nv-\-a) 
where P„, Q lt are two independent integrals of equation (9). We first discuss the 
integral already obtained 
iW 
P.= 
2 n +1 
(C — S COS 0) 2 
• ( 10 ) 
It is well known that this integral is the same as 
( (C—S cos ff) 2 clO . .(11) 
2 0 
the second solution obtained above ; this may be easily shown by the transformation 
(C —S cos 0)( C —S cos 0') — 1 or by means of the sequence equation (14) below. 
6. Discussion of P„. 
We have 
Whence 
Similarly from 
cU\ 
du 
2n+i\r s-c cos e 
2)i + 3 
(C —S cos 6) 2 
2S d P„ 
2n +1 du 
=P„ +1 -CP„ 
dP H _2y—] [* tn 
die 2 j 0 
2S d? 
l)i—l du 
I (C — S cos d) 2 (S —Ceos 6)b0 
J o 
“=CP„—P„ 
n x u —1 
( 12 ) 
(13) 
Combining (12) and (13) we have 
(2 n-\- l)P« +1 — 4«CP„+(27i— ^P^j—0 
• • • (14) 
This sequence equation may also be deduced at once from (10) or (11). 
