OF DEFLECTIVE FORCES. 117 



■J- — ^ — Odfda- -{ ; IWMz, whence, taking the flu- 



da:2+d?/2+dz- 

 ent, and dividing by dt, we have ^ ~ =: y 



(Pdx + Qdj/ + jRdz), and ^-^^l±^l±^:=z2 f Pdx + QCy 



•{-Rdz), Now the first member of this equation is the 



ds 

 square of the velocity -r- or v, and the second may be 

 ci c 



called c + 2(p, supposing the expression to be a possible 



fluxion, or capable of integration, which it must be when 



the forces are in any way dependent on the distance of M 



from their origins, as they generally are in nature: we 



have then r^z=c + 2^ (235). (gj. 



Corollary 2, This supposition of the equality of 

 the variations to the actual evanescent increments of the 

 body's path, is equally applicable to the motion of a body 

 in a given surface : and it follows from the preceding 

 corollary, that the velocity remains unaltered, if no other 

 force be acting on it but the pressure of the surface : as 

 indeed it is obvious that the portion destroyed by the 

 curvature at each step must be infinitely less than that 

 which remains, the hypotenuse of a triangle exceeding its 

 base by a quantity which is only the square of the perpen- 

 dicular, which may be compared with the evanescent 

 sagitta of the curvature. See 286. 



Corollary 3. It is also obvious that the velocity 

 must be the same, by whatever path, or upon whatever 

 surface, the moveable body passes from one given point to 

 another. 



265. Lemma. The variation of the dif- 

 ference or of the fluxion is equal to the dif- 



