593 
of Edinburgh, Session 1871 - 72 . 
From equation (10) we may prove, as usual, by multiplying by 
Si and integrating over the unit sphere, that 
i(i + nArS& = j(j+ U/ArSiS,, 
the expression for either being symmetrical in i and/, so that the 
integral vanishes unless i—y. 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, 
Sj - 2 0 As cos. (s<p + a s ) ® C i } . . . (11). 
where A 5 , 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) ) 
®(s) 
i 
(i - Q s . 
(13), 
for with this substitution it becomes, by a process the same as that 
of section 2 above, 
(*x*+1) - <«+1)) a - ix*) s 6s + dfX (o r *) + f 
= 0. 
But, by (4), putting s +1 for s, 
(y | -i\ c , iA q 2\s d s Qi d / s +ieT'+iQA 
Comparing these equations, and remembering that all the permis¬ 
sible arbitrary constants have already been introduced into the 
solution of (10), we have 
d s Qi 
6< 
Hence, finally, 
dp* * 
Si = X As cos. (s<p + a 5 )(l - p 2 ) 
l 
d[x s 
(14.) 
