OF THE MOTIONS OF FLUIDS. S,SS 



T-./T-v d'w . d'^w ^ d'-io 



E F 1= DZ +-- — DZAt + T— r DXDZAf -|- — — - DyDZAt; 

 dz dzax dzdy 



which only differs from M'N' by quantities evanescent in 



comparison with itself; and in the same manner K'H' 



and L'G'may be shown to be ultimately equal to M'N'. 



Precisely in the same manner, by substituting first y and 



V, and then x and u, for z and w, we obtain 



d 1? d M 



MX'ziDy + — - DvAif,andM'K'=Da;4- -— DxAt; and the 

 dy "^ dx 



opposite sides of the parallelepiped will be found to be 



respectively equal to them, so that the figure still remains 



that of a parallelepiped, although its angles are rendered 



oblique ; but the obliquity produced in the instant At is 



infinitely small, so that, without neglecting the cosine of 



the angles, their sines may still be considered as unity, and 



the volume of the solid will be expressed by the product of 



its three sides WN\WL'.WK\ This product, neglecting 



the terms involving the higher powers of the differences, 



which are comparatively evanescent, becomes vxByvz(i 4- 



(— - -h-— +--—]At): and this is the volume of the element 

 \dx dy dz / 



which, at the end of the time #, was DxDyDz. Now the 



density f being a function of j:, y, z^ and t, it follows that 



when t becomes t-\-Aty and x, ?/, and z are changed to j: + 



uAty y-\-vAty and z-\-wAt, it becomes D-\--^At-\--^uAt + 



dt dx 



~ I'At + —i m;a^ : and if we multiply this density by the 

 dy dz 



corresponding volume, the product will express the mass 



at the end of the time : from which if we subtract ^DarDyDz, 



the initial mass, the remainder will be the variation of 



the mass: and this must vanish. Hence, neglecting the 



