118, 119, 120] POTENTIAL FUNCTION. 353 



where r 19 r 2 , . . . r n are tlie distances from the repelling points, is the 

 force -function for all the forces acting upon the particle m s . If we 

 put the mass m s equal to unity, at a point P whose coordinates are 

 x, y, 2 the function 



49) F=^ + ^ + . .+ ^=V- 



n ' r n ^r 



is called the potential function at the point P of the field of force 

 due to the actions of the particles m lt m 2 , ... m n , and y times its 

 negative vector parameter, 



50) x = - r d J, Y-- r g, z--&, 



f cx r dy * dz 



is the strength of the field, that is, the force experienced hy unit 

 mass concentrated at the point x y y, 0. 1 ) 



Since any term - possesses the same properties as the func- 



i Tr 



tion -9 118, we have for every term, for points where r is not 



equal to zero, /4 \-\ = 0, and consequently 



12O. Potential of Continuous Distribution. Suppose now 

 that the attracting masses, instead of being in discrete points, form 

 a continuously extended body K. 



Let the limit of the ratio of the mass to the volume of any 

 infinitely small part be o = lim > which is called the density. Let 



^r=0 4* 



the coordinates of a point in the attracting body be a, &, c. 



1) It is more usual among writers on attracting forces to write the force 

 as the positive parameter of the potential. The convention above adopted in 49) 

 amounts to defining the potential as the work necessary to remove the attracted 

 particle of unit mass from the given point to infinity against the attracting 

 forces, thus keeping the potential function positive, instead of negative as in 

 28 (end). It is the usual practice to adopt such units that y is equal to 

 unity. In order to preserve consistency with the units previously employed and 

 at the same time not to be obliged to introduce y throughout all the equations 

 of this chapter, we shall define potential as above 49) and introduce the factor y 

 into those equations which involve the relationship of the force to the potential. 

 If the force is attractive, y will be negative, and putting y = 1, we get the 

 usual formulae. Putting y = -|- 1, our notation agrees with that customary for 

 electricity and magnetism , for example in the author's Theory of Electricity and 

 Magnetism. 



WEBSTER, Dynamics. 23 



