120 PKOBLEMS IN THREE DIMENSIONS. [CHAP. V 



84. In the original investigation of Laplace*, the equation 

 V 2 < = is first expressed in terms of spherical polar coordinates 

 r, 6, a), where 



x = r cos 0, y = r sin 6 cos &>, z = r sin 6 sin &>. 



The simplest way of effecting the transformation is to apply 

 the theorem of Art. 37 (2) to the surface of a volume-element 

 r6 . r sin OBco . r. Thus the difference of flux across the two 

 faces perpendicular to r is 



dr \dr 



Similarly for the two faces perpendicular to the meridian (&>= const.) 

 we find 



d fdcb . a , \ I- 

 r* sin uoo} . or bu 



) ' 



and for the two faces perpendicular to a parallel of latitude 

 (6 = const.) 



sin 

 Hence, by addition, 



9 d 



-T-lr*- +- sm- 



sin c&> 



This might of course have been derived from Art. 82 (1) by the 

 usual method of change of independent variables. 



If we now assume that </> is homogeneous, of degree n, and put 



<t> = r-S n , 

 we obtain 



which is the general differential equation of spherical surface- 

 harmonics. Since the product n (n + 1) is unchanged in value 

 when we write n 1 for n, it appears that 



+-r-*-*8n 



will also be a solution of (1), as already stated. 



* " Th^orie de 1'attraction des sph6roides et de la figure des planetes," Mem. 

 de VAcad. roy. des Sciences, 1782; Oeuvres Completes, Paris, 1878..., t. x., p. 341 ; 

 Mecanique Celeste, Livre 2 me , c. ii. 



