123] POISSON'S EQUATION. 301 



The area of the face is 17 , so that this face contributes to the 

 integral / / Pco$(Pn)dS the amount |i^g. 



o i y- 



At the opposite face, since o is continuous, we have for its value 



terms of hiher order in 



and therefore, the normal being directed the other way, this side. 

 contributes to the integral the amount 



and the two together 



d*V 

 ^y^Wx* ~^~ ^ erms ^ high er order. 



Similarly the faces perpendicular to Y-axis contribute o~ 



d*V 

 and the others %>n^ i [V 



Thus the surface integral is 



and by Gauss's theorem this is equal to 



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

 parallelepiped infinitely small, and dividing by |^g, we get 



An important application of Poisson's equation has been made 

 to the attraction of the earth. The acceleration g is made up of the 

 resultant of the attraction of the earth and of the centrifugal accel- 

 eration. Since the latter has the components RPx, &?y along axes 

 perpendicular to the axis of rotation ( 104), it has the potential 



function (x 2 -f i/ 2 ) , so that if y denote the positive value of the 



gravitation constant, and n the inward normal to an equipotential 

 surface, we have, putting 



where 



