28i.] POTENTIAL. l - Jl 



done by the attraction on a particle of mass I if it were brought 

 up to the point PJ along any path from infinity. 



The relations of Art. 278 can be written in the form 



dV m 



= --cos</>; 



ds r* 



i.e. the derivative of the potential with respect to any displacement 

 is equal to the component of the attraction in the direction of the 

 displacement. 



280. When there are given several masses m, m\ m n , ..., con- 

 centrated at points Q, Q f , Q", ..., their potential at any point P 

 is defined as the sum 



jr_m m' m n _^m 



v I -f H 77 ~r"* Z > 

 r r r" r 



when the given masses are continuous, the sign of summation 

 must be replaced by an integral, and we have 



The fundamental properties proved in Art. 279 remain the 

 same. 



281. Let there be given a continuous mass m, referred to 

 a rectangular system of co-ordinates. The attraction at any 

 point P (x, y, z) due to this mass has three components X, F, Z, 

 which can be found as follows. The attraction produced at P 

 by an element dm at a point Q (x' t y' , z') of the mass is dm/r 2 , 

 where r=PQ, and its direction cosines are (x' x)/r, (y* y)/r, 

 {z' z)/r; hence its components are 



Integrating, we find the components of the total attraction 

 a.tP: 



\z'-_z)dm 

 r" 



= C 



*J 



