524 
DR. E. W. HOBSON ON A TYPE OF SPHERICAL HARMONICS 
The functions P„_A (cosh cr), Q„_f" (cosh c r), where m and n are positive integers, are 
consequently called toroidal functions. Various expressions for these functions mav 
be found, as particular cases of the various definite integral expressions which have 
been given above for P(p), Q n m (p). 
We find from (113) 
,<r COSh 71(f) 
o y /(2 cosh cr — 2 cosh cf>) 
Also from (81), (82), 
2 
P„_.l (COSh cr) = 
7T J 
P a _j M (cosh cr) = 
From (68), (69) 
1 IT (n + m — |' 
7T n (n — J) , 0 
( - i)™ n (w, - 
(cosh cr + sinh cr cos <f >)" * cos m<f> d(j>. 
7T 
n (ii 
- j-) r 
— m — i) J 
COS 771(f) 
J) J 0 (cosh cr + sinh <x cos <£) 
i-i ^(f)- 
P„_j ,H (cosh cr) 
II (?i + m — J) 
1 
II (n — m — J) 2"'1I ( — ^) II (??i — |) 
n + m — i 
sinh m cr 
(cosh cr+ sinh cr cos <f >)' 1 m * sin 2 " l (f> dtp, 
-Sinh' 1 cr 
II (n — m — |) 2" i II ( — f) fl (to — J) J 0 (cosh a + sinh <r cos (f>y i+m+ij 
f- 
Jo (cc 
sin 2 "' (f> 
i d<f). 
Again from (92), we find 
Q a _i m (cosh cr) = (— 1 ) m — yj ~ -- 1 - ■— ~ ^ f° scoth v(cosh cr — sinh cr cosh cosh miv div, 
II (n — f) Jq 
and from ( 122 ), 
„ , . v . ,. 2 m TL (m — A) II (— 4-) . , f 7r ■ cos n<f> 7 , 
Q„_f (cosh cr) — 1 -~-— smlY" cr —-.-y-—7 —tt d<f). 
^ 2 x / V / tj. J 0 (2 cosh a — 2 cos ^ 
In the case in which the real part of n — m + ^ is positive, w r e find from (90) 
and (91), 
Q*-j* (COSh cr) 
— (— —77 sinh’" cr I ' coth 2 (cosh cr— sinh cr cosh sinh ~ m w dw, 
' • II fn on A \ 11/ or? ) I ^ ' ' 
IT — 771 — 4) Il(?n —J) 
n [71 — 4) r cosh viw 
I) J 0 (cosh cr + sinh cr cosh w) ,l+ 
= (_!). °(—» r_ 
n (?i — — 4 ) J 0 (cosh 1 
r dw. 
57. From (125), (126), we find, on writing u — ^ for n, the relations 
