96 Lord Rayleigh on [May 15, 



where a Y a% are small quantities relatively to a , and (according to 

 the usual notation) represents the cosine of the colatitude (0). 

 For the volume included within the surface (33) we have 



=fTp[a s +3a S{a 1 P 1 W+ . . +«»P»W+ . . } 

 + 3a {a 1 P 1 0<,) + • • +a; i P«W + . . } 2 + • • W 

 =%™ ^ + \l + 3a -22a/P^( y a)]^ 



=f*V[l + 3a - 3 S(2w. + 1) ~ V, 

 approximately. If a be the radius of the sphere of equilibrium, 

 a 3 =a 3 [l + 3a- 2 2(2^4-l)- 1 ^ 2 ] . . . (34). 

 We have now to calculate the area of the surface S. 



S=&r j r sin 6 </ j r 2 ] d0 =2^ | r 2 j sin 



For the first part 



j + V 2 d/*=2a 2 + 22 (2n + 1)"W. 



For the second part 



4 |(|)\in^=^;(l^)[ S ^]V 



The value of the quantity on the right hand side may be found with 

 the aid of the formula* 



f +1 (l _^2)^ ^ d^=„( w + 1) f +1 p„P„^. 

 J -l a/* a /u, J-i 



Thus 



43l) 3sin ^ fe} f-! (1 - /)2a » 3 @ v 



r+i 



=l2rc(w + l> a 3 J P„ 2 ^=2 n(n + l)(2^ + l)" 1 a w 2 . 



Accordingly S = 47^a 2 + 27^S(2^ + l)- 1 (n 2 + 7^ + 2)c^, 2 ; 

 or, since by (34) 



a 2 - a 2 _ 22 (2n + 1 )~ W, 

 S=47ra 2 H-27r2(^-l)(^ + 2)(2n + l)- 1 a iJ 2 . . (35). 



* Todlmnter's tt Laplace's Functions," § 62. 



