120 PROBLEMS 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 #, y = r sin 9 cos &&amp;gt;, z = r sin 6 sin &&amp;gt;. 



The simplest way of effecting the transformation is to apply 

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

 r$0 . r sin #&&amp;gt; . 8r. Thus the difference of flux across the two 

 faces perpendicular to r is 



-.. 



dr \dr 



Similarly for the two faces perpendicular to the meridian (&&amp;gt;= const.) 

 we find 



d d() ... 



and for the two faces perpendicular to a parallel of latitude 

 (6 = const.) 



sin 6da) 

 Hence, by addition, 



. a d ! ^d6\ d ( . ad&amp;lt;f)\ , 1 &&amp;lt;j&amp;gt; , t v 



sin -y- (r 2 ^- + -77; I sm - + -. 7, -^ = . . .(1). 

 dr \ dr) dB \ dOJ sm da? 



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 &amp;lt;f&amp;gt; is homogeneous, of degree ?i, and put 



we obtain 



1 d / . ~dS n \ 1 d~S n 

 d d0 



n \ 1 d~S n 

 } + sin&quot;* ^ 



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 -n-i n 



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



* &quot; Th^orie de 1 attraction des sph6roides et de la figure des planetes,&quot; Mem. 

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

 Mecanique Celeste, Livre 2 mc , c. ii. 



