594 Proceedings of the Royal Society 
7. We may now easily find the value of 
fS&fdtr 
taken over the whole spherical surface. For 
and 
2 TT — 1 
/( )^<r=//( ')d<pd(x: 
o +i 
f d<p cos. (sp + a,) cos. (s'<p + <v) 
vanishes unless s and s' be equal, in which case its value is 7 r. 
Hence, attending to § 4, and to (14), 
and 
f S S da~ = 0 , 
/ q2 , 2 tt *■ .2 lh 
S^cr _ 2i+ i 2 0 A s 
i + s 
• (15)- 
8. Another curious expression for ©£ s) is given by (4)- For 
that equation gives 
= - (*'(‘+l) -«(»-!)) f-!’*) 1 
=+{«'+ i)-.(*-i)}{»(«+i)-(*=i)(^2)} 
(S) 
= ( -yifi/w .... (i6). 
Hence 
=(-) ! {S( i -/^( 7^*)* Qi • (17) - 
10. let 
JT+^xh + hF = 1 + hy . . . (19), 
where y is a function of h and (a, never beyond the limits -f 1 and - 1. 
Then 
h n dy 
J l + Zfxh + l? = h df' 
Hut the first equation gives, at sight, 
( 20 ), 
