384 PROFESSOR CHALLIS, ON THE DIFFERENTIAL EQUATIONS 



Now if V be the whole velocity and not otherwise, that is, if the line 

 s be drawn in the direction of the motion, 



dx _ v dx dy _ v dy dx _ d% 



di~ d~s' di~ Ts' d~t ~ ds' 



and ( — > = V ( — — — ^ — —) ■ V — 

 dt \ dx ' ds dy ' ds d% ds! ' ds 



Hence by substituting the resulting value of f in the above equation, 



rdV V* 



In consequence of the limitation to which this equation is subject, 

 it cannot be argued, as in the former case, that the motion is rectilinear, 

 and we may therefore conclude that when udx + vdy + wdz is integrable 

 by a factor the motion is curvilinear. 



Another remark may also be made here. By assigning a given value 

 to P — Q, the above equation becomes the equation of a surface for all 

 points of which P — Q has that value at a given instant. Hence dif- 

 ferentiating the equation with respect to space and putting the differential 

 under the form 



dP-dQ= - ~ds - Vd V, 



the variation of P— Q on the left-hand side of the equation will be the 

 same in passing from a point of that surface to a point of another such 

 surface indefinitely near, whatever be the relative position of the points. 

 We may therefore suppose the variation to take place from one to the 

 other of the points of intersection of the line of motion with these two 

 surfaces, to which variation the quantities on the right-hand side are 

 limited. Hence the equation holds good notwithstanding that limitation. 



13. Our next step will be to effect a transformation of equation (1) 

 analogous to that which was made in Art. 4 on the supposition that 

 udx + vdy + wdz is an exact differential. The transformation will now 

 be made by means of equation (13), and therefore on the supposition that 



that quantity is integrable by a factor **>. I 



