77, 78] NEWTONIAN POTENTIAL FUNCTION. 153 



and hence, the normal being directed the other way, this side con- 

 tributes to the integral the amount 



dV 3 idV\ . 



and the two together 



* 



+ terms of higher order. 



3 2 F 

 Similarly the faces perpendicular to F-axis contribute 77 f ~ , 



and the others tf?-;rr . 



d^" 



Hence the surface integral is 



and by Gauss's theorem this is equal to 

 4?rm = 



where p is the mean density in the parallelepiped. Now making 

 the parallelepiped infinitely small, and dividing by 77 f, we get 



AF=-47T/!>. 



78. Abbreviations for Operators. If p is any point 

 function, the potential function at any point due to a distribu- 

 tion through all space of matter whose density at any point is p 

 has been denoted by Gibbs and Heaviside by the abbreviation 

 Pot p } standing for the definite integral 



**,-/// 



P -dr. 



(The suffix oo denotes integration through all space.) 

 We may thus abbreviate Poisson's equation 

 - A Pot p = V 2 Pot p = 4-777) , 



so that the operation Pot followed by the operation A = V 2 , 

 performed on any scalar function, has the effect only of multiply- 

 ing it by 47T, or the operations A and -.- are the inverses of 

 each other. 



