130 THE LAWS OF 



mits of an analogous expansion in entire powers and products of 

 x, y, z. Moreover, as the density p^ retains the same numerical 

 value, and merely changes its sign when we pass from the ele- 

 ment dv to a point diametrically opposite, where the co-ordi- 

 nates #', y', z f are replaced by x, y ', z': it is easy to see 

 that the function F 1? depending upon p lt possesses a similar 

 property, and merely changes its sign when x, y, z, the co-or- 

 dinates of p, are changed into x, y, z. Hence the nature 

 of the function V t is such that it can contain none but the 

 odd powers of r, when we substitute for the rectangular co- 

 ordinates x, y, z, their values in the polar co-ordinates r, 0, TV. 



Having premised these . remarks, let us now suppose V 1 is 

 divided into two parts, one F/ due to the sphere B which passes 

 through the particle p, and the other V" due to the exterior 

 shell 8. Then it is evident by proceeding, as in the case where 

 p= (1 - 7* /2 )0/(r 2 ), that F; will be of the form 



F; = r*- n {A + Br* + O 4 + &c.} ; 



the coefficients A, JB, (7, &c. being quantities independent of the 

 variable r. 



In like manner we have also 



F/' = fr'Wdffdw sin & (1 - / 2 )0. /^ (V 2 ) f* ; 



the integrals being taken from r =r to r=l, from & = to 

 0' = TT, and from -&' = to vr f = 2?r. 



By substituting now the second expansion of # 1-n before used 

 (Art. 1), the last expression will become 



F 1 "=r + r 1 +2 7 2 +r 3 +& c . 



of which series the general term is 



T a =fdffdrf sin ff Q 8 fr'*-dr' (1 - / /. 



Moreover, the general term of the function ty (r' 2 ) being repre- 

 sented by A t r*\ the. portion of T 8 due to this term will be 



sn 



the limits of the integrals being the same as before. 



