OF THE MOTIONS OF FLUIDS. 283 



dp d'u d'w d'w d'tt 



fdr "" d^ do; dy dz 



d'o -^ d'v d'v d'u d'v 



fdy d* ax ay dz 



d'o ^ d w 6!w d'w d'w 



fdjz d^ djr Ay dz 



*' Each of the elements, into which the fluid is supposed 

 to be divided, will change its form during- the instant ^ty 

 and it may also change its volume, if the fluid is compres- 

 sible : but since the mass must always remain constant, it 

 follows that if we find its volume and its density at the 

 end of the time f + Af, their product must be the same as 

 at the end of the time t : and by making the variation of 

 this product vanish, we shall obtain a new equation for the 

 motion, 



" In order to form this equation, we may consider the 

 rectangular parallelepiped, of which the volume was ex- 

 pressed by BxDyDz at the end of the time t, and examine 

 the form which it will assume at the end of the time t + At, 

 supposing M to be the summit of the parallelepiped which 

 corresponds to the coordinates x, y, z, and MN, ML, MK, 

 the three sides or edges which meet in it, and which are 

 parallel to the axes 02 0y and 0a7 respectively, so that we 

 have MNzzDZ, ML=DY, and MK=DX: supposing 

 also E,F,G, and H, to be the four other angles of the pa- 

 rallelepiped ; and the points M,N,L,K,E,F,G,H, to be 

 removed, during the instant Af, to M',N',L',K', K,F,G', 

 H'. The new soUd will still be a parallelepiped, as may 

 be thus demonstrated. 



