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INDIANA UNIVERSITY 



2 



Whence, dividing by --i?^, dropping subscripts and the terms in 7/, 



■') 



we have in the limit 



OX Oy Oz 



For an outside point (2) gives us Laplace's equation, for in 

 that case, 



Y - ^'o = -Yi - Fi 0. 

 The method that has been used above for deriving Poisson's 

 equation for the Ne"\^i;onian potential can be applied to the case 

 of the logarithmic potential by developing over a circle instead 

 of a sphere. 



After Poisson's equation has been derived as indicated, Gauss's 

 theorem of the integral of the normal component of the force 

 follows immediately on the application of G-reen's theorem. 



In developing the Newtonian potential it is customary to derive 

 as one of the first examples the potential due to a homogenous 

 spherical shell both at an exterior and interior point. The inte- 

 grals involved in this are of precisely the same form as those 

 referred to above in deriving Gauss's theorem of the mean. The 

 latter theorem is therefore really shown at an early stage in the 

 subject. The above method then gives Poisson 's equation, and with 

 the aid of Green's formula, Gauss's theorem of the normal com- 

 ponent of the force. This order of developing the subject is the 

 reverse of the order generally given in American texts. 



