378 PROFESSOR CHALLIS, ON THE DIFFERENTIAL EQUATIONS 



If now equation (6) be differentiated with respect to space, the re- 

 sult is, 



dP-dQ +d.^ + VdV=0; 



and as this equation is subject to no limitation with respect to the direc- 

 tion of variation of co-ordinates, let us suppose the variation to be from 

 one point to another of a surface of displacement. Then from what 

 has been just shewn, dV = 0. Also 



, dd> d 2 <b j d'cb , d 2 d> , du dv t dw . 



d 'dt = ttt dx + dftt d * + Mt d% = Tt d * + Tt d * + -3i dz 



d.{udx + vdy + wdz) _ 



= di ' 



because for a surface of displacement udx + vdy + wdz = 0. Conse- 

 quently dP — dQ = 0. It follows from this that when udx + vdy + wdz 

 is an exact differential, the surface of displacement coincides with a sur- 

 face for all points of which P - Q has the same value. 



If Q = 0, that is, if there be no impressed forces, the surface of dis- 

 placement evidently coincides with a surface of equal pressure, and the 

 motion of each fluid particle must consequently be rectilinear. In this 

 case only equation (5) is subject to no limitation in regard to the 

 direction of the variation of the co-ordinates. 



The motion is rectilinear also when Q is not equal to nothing. For 

 though in this case the pressure varies along a surface of displacement, 

 the effect of this variation is just counterbalanced by the impressed forces, 

 as may be thus shewn. Let da be the increment of any line drawn 

 on the surface of displacement. Then 



dP dQ _ a?dp ^.dx v dy ydz _ d z a 

 d<r da pda da da da df 



the effective accelerative force in the direction of a. Since therefore, 



dP dQ = Q 



da da 

 the effective accelerative force in any direction along the surface of dis- 

 placement is nothing; and the velocity being the same at all points of 

 this surface, it follows that the motion is rectilinear. 



