172 STATICS. [282. 



Differentiating the relation 



r*= (x 1 x)' i -\-(y 1 y)*' + (z'z) t * t 

 partially with respect to x, y, and z, we have 



dr _ _ , , _ . dr_ _/ /_ \ dr _ , . 



Substituting these values in the above integrals, we find 



v C I dr j C & fl\ j d Cdm 

 JL = I am= I - }am = I > 



J r*dx J dx\rj dx^ r 



and similarly 



Y= C^ m Z= C^ m 



dyJ r dz*s r 



As J is the potential Fof the given mass, we have 



X= > Y= > Z 



dx By 



i.e. the components of the attraction at any point are the deriv- 

 atives of the potential at that point in the direction of these com- 

 ponents. This may be regarded as a special case of the last 

 proposition of Art. 279. 



282. It is to be noticed that the proof given in the preceding 

 article can easily be extended to the case of forces of the form 

 (2), Art. 260. In other words, even in the case of forces not fol- 

 lowing the Newtonian law of the inverse square, but expressed 

 by any function f(r) of the distance, there exists a function 

 corresponding to the potential of Newtonian attractions ; it is 

 called \he force function. 



We have, just as in Art. 281, 



hence -= 



_, - 

 dx r dy r dz 



