594 Proceedings of the Royal Society 
7. We may now easily find the value of 
taken over the whole spherical surface. For 
and 
/( ) dcr =f /( ')d<pdfx: 
o +1 
27r 
f dp cos. (sp + a s ) cos. (s'p + a/) 
vanishes unless s and s' he equal, in which case its value is tt. 
Hence, attending to § 4, and to (14), 
f S S — 0 , 
and 
/ 
q 2 7 —7 r . 2 
Oj Cl C — 9 £ j_ 
& + s 
• (15). 
I — s 
8. Another curious expression for is given by (4)* For 
that equation gives 
(1 " ^ s ~c = - (*'(*'+!) - <« - 1)) yh - /A 1 ^ 
= + {«■(»■+D-«(«-l)J{i(i+l)-(*-l)(»-2)} JR-V?)' ''fpW 
, K + s r 
=(-) s =y^^ • 
(5) 
Hence 
= c - y jf= a-/**>”* 
) Qj 
(16). 
(17). 
10. Let 
y 1 -|- ‘IjjJi -j- A" — 1 -f- 7zy . . 
(19), 
where y is a function of 7i and /x, never beyond the limits +1 and - 1. 
Then 
h 
= h 
__ _ Jy 
J l + 2yh + h 2 "dp 
But the first equation gives, at sight, 
7 1 -y 2 
y — [i + h o 
( 20 ), 
