382 PROFESSOR CHALLIS, ON THE DIFFERENTIAL EQUATIONS 



Hence, if we substitute \js for N~cp, the preceding equation will become, 



^(^ ) + p_Q T _ + ^. ^4.^ + ^J _0, 



which is the equation obtained when udx + vdy + wd% is an exact 

 differential, as it manifestly ought to be. The general expression for 

 cf> inclusive of all the cases in which that differential is exact, would be 

 obtained by integrating the equation, 



dt + V U* 2 + dtf + dtf) ' 



dtf 



on the supposition that N is a function of t only. By multiplying this 

 equation by N, it becomes N -~ + V 1 = ; whence it appears that since 



-^ and V are constant for a given surface of displacement at a given 



time when udx + vdy + wdz is an exact differential (see Art. 6), the 

 factor N is constant under the same circumstances. With this limi- 

 tation, therefore, as to the value of N, the equation (13), holds good 

 at the same time that udx + vdy + wd% is integrable of itself. 



12. Resuming the equation obtained in Art. 10, we have in general, 



by Art. 9, if the variation with respect to space be from one point to 

 another in the line of motion. Also, 



„cfo d.jN{d<p) 

 ~ dt - dt 



