47-49] Equations of Laplace and Poisson 41 



with a continuous distribution of electricity, the volume-density of electrifi- 

 cation in the neighbourhood of the small element under consideration being 

 p. The whole charge contained by the element of volume is accordingly 

 pdxdydz, so that Gauss' Theorem assumes the form 



(16). 



The surface integral is the sum of six contributions, one from each face of 

 the parallelipiped. The contribution from that face which lies in the plane 

 x % ^dx is equal to dydz, the area of the face, multiplied by the mean 

 value of N over this face. To a sufficient approximation, this may be 

 supposed to be the value of N at the centre of the face, i.e. at the point 

 \dx, 7j< f, and this again may be written 



/8F\ 



fcW.; 



so that the contribution to llNdS from this face is 



dydz 



Similarly the contribution from the opposite face is 



'8F\ 



the sign being different because the outward normal is now the positive axis 

 of x, whereas formerly it was the negative axis. The sum of the contributions 

 from the two faces perpendicular to the axis of x is therefore 



(17)- 



8F 

 The expression inside curled brackets is the increment in the function = 



8 /8F> 



when x undergoes a small increment dx. This we know is dx (^ ),so 



das \ or / 



that expression (17) can be put in the form 



8 2 F , 

 - JT dxdydz. 



The whole value of llNdS is accordingly 



9 2 F 



, , 



w+^r y ' 



and equation (16) now assumes the form 



