230 CELESTIAL MECHANICS. 1. Vl. 24. 



The preceding equations of the motion of a system of 

 bodies have been derived from this law: but the same mode 

 of investig-ation may be easily extended to all other rela- 

 tions between force and velocity which are mathematically 

 possible, and the principles of motion may thus be exhi- 

 bited in a new point of view. 



For this purpose we may suppose F the force and v the 



dF 

 velocity, putting F=<p(v)^ and let ^'(^)i=-^* [or, more 



simply, let (p denote <p(v) or F, and (p'dv, dtp ]. If this 

 force be reduced to the direction of x, it will become 



<p.y- . and in the next instant it will be 9'-r--T-A(<P'-^\ or 

 ds » ^ ds^ ^^ ds) 



^ dx /& dx\ . , , 

 ^. -7- + ^(-'-r-)> smce dszivdf. 

 ds \vdt^ 



Now if P, Q, and jR be the forces, acting on the body m, 



in directions parallel to the respective coordinates, the 



system would remain in equilibrium in virtue of the action 



of all such forces combined with the elementary differences 



A (-J-* — ) considered as negative, since these differences 



are the effects of the results of the forces, and the fluxions 

 are their measures: we shall therefore have, instead of the 

 equation (P) (317), 



0=2^5 §x(d(^.-^)-PdO + gy(d(^.-J)- Qdt)^ gz(d 



which only differs from it by the substitution of 

 ---for — or unity. This alteration would render its gene- 



V V 



ral application to mechanical problems very diflScult: we 

 may however derive from it some principles, analogous to 



