EQUATIONS OF MOTION 255 



direction, the equation of motion of the particle, by the second 

 law of motion, will be 



ft W 



S = m-> (90) 





We shall suppose the field of force to be permanent, so that the 

 quantity S may be supposed to depend only on the position occu- 

 pied by the particle on its path, and not on the instant at which 

 it arrives there. In other words, S is a function of s but not of t. 

 Equation (91) is a differential equation connecting s and t\ if we 

 can solve this equation, we shall have a full knowledge of the 

 motion of the particle provided its path is known. 



The equation is a differential equation of the second order, but 

 can easily be transformed into one of the first order. For 



d z s _ dv dv ds _ dv 

 dt 2 dt ds dt ds 



so that the equation can be written 



dv 



S = mv 

 ds 



Since S is a function of s, this equation can be integrated with 

 respect to s, so that we obtain 



(92) 



where C is a constant of integration. 



ds 



Since v is equal to > this equation can be written in the form 

 at 



< 93) 



which is an equation of the first degree. If this can be solved, the 

 solution of the problem is complete. 



