87, 88] NEWTONIAN POTENTIAL FUNCTION. 173 



Transforming the other two integrals in like manner, 



,dV , v VdF 



+ h 2 cos (n 2 n s ) + h s ^ cos (n 3 n s ) j- ct>S 



L (A. a.^ + 9 1 ^ |Z) + ^. (A|Z)1 ^ 5l 



Now this is equal to 1 1 1 A Fc?r. 

 But cZr = 



Multiplying and dividing in the last integral of (4) by kjiji 3 , 

 we find that, since the integrals are equal for any volume, the 

 integrands must be equal, or 





( \ 



This result was given by Lame, by means of a laborious direct 

 transformation. The method here used is similar to one used by 

 Jacobi, and is given by Somoff *. 



88. Laplace's Equation in Spherical and Cylindrical 

 Coordinates. Applying this to spherical coordinates 



h r = 1, h0 = - , 



/,9F\ 8/1 9F\) 



'w).+ 



= 28F 1^8^F 1 8F 1 9 2 F 



8r 2 + r 8r + r 2 8(9 2 + r 2 CC 8^ + r 2 sin 2 6> 8< 2 ' 



We may apply this equation to determine the attraction of a 

 sphere. For external points AF=0, and since by symmetry Fis 

 independent of 6 and 0, 



* Lam6, Journal de VEcole Polyte.chnique, Cahier 23, p. 215, 1833 ; Lemons sur 

 les Coordonnees curvilignes, n. Jacobi, "Ueber eine particulars Losung der par- 

 tiellen Differentialgleichung AF=0," Crelle's Journal, Bd. 36, p. 113. Somoff, 

 Theoretische Mechanik, n. Theil, 51 2. 



