593 
of Edinburgh, Session 1871-72. 
From equation (10) we may prove, as usual, by multiplying by 
S j and integrating over the unit sphere, that 
i(i + 1 )/ dtrSfij = j(J+ lXArSiS,- , 
the expression for either being symmetrical in i and/, so that the 
integral vanishes unless i —j : or, if negative values be admitted, 
unless i + j + 1 = 0. 
6. We must now express S* in terms of <p and Q ». Let, then, 
S< = 3 0 As cos. (s<p -f ai)®® . . . (11). 
where A s , a s are virtually 2 i+ 1 arbitrary constants. Substituting 
this value in (10), and supposing all the coefficients A to vanish 
except A s , we have 
This equation is materially simplified by assuming (as is suggested 
by (6) and (9) ) 
®f=( l-ff 6 *. ■ ■ ■ (13), 
for with this substitution it becomes, by a process the same as ''that 
of section 2 above, 
(^+l)- S ( S +]))( 1 -^ + |((l-^) S+1 f) = 0. 
But, by (4), putting s + 1 for s, 
+ «) 0 -<■•> T + 1 (o -*"**$) - »• 
Comparing these equations, and remembering that all the permis- 
sible arbitrary constants have already been introduced into the 
solution of (10), we have 
Hence, finally, 
Si = 2j A, cos. (s<p + a,)(l - ^ 7 - . 
(U) 
