38, 39] DEFINITE INTEGRALS. 75 



The total gain through these two sides is, accordingly, the 

 difference 



-- -^ - dxdydz. 

 Similarly through the sides normal to the F-axis the gain is 



and through the sides normal to the 



9(p&) , 

 -- ^ - ax ay dz. 



oz 



The total increase of the mass in the parallelepiped is therefore 



,, 



8dm = - f + --^ + -* 



a dy dz } 



and this being taken for an element of our integral, the total 

 increase of mass, or variation of the integral, is 



f/Y (d(p8x) d(pfy) d(pSz)] , 

 cm = - \ -^ '- + \ v ; + ; r '\ dxdydz. 

 JJJr( fa dy dz j 



We may obtain this result in a more rigid manner by the use of 

 Green's Theorem. Through each element of surface dS of the 

 boundary of the space in question there moves inwards an in- 

 finitesimal prism of matter whose volume is 



dS [x cos (nx) + &y cos (ny) + Sz cos (nz)}. 

 The mass of this is 



{pSx cos (nx) + py cos (ny) + pSz cos (nz)} dS, 

 so that the total gain of mass in the space r is 



8m = 1 1 {p x cos (nx) + p Sy cos (ny) + pSz cos (nz)} dS. 

 But by Green's Theorem this is equal to 



This result will be of frequent use. 



39. Reciprocal Distance. Gauss's Theorem. Consider 

 the scalar point-function, V = - , where r is the distance from a 



