288 CELESTIAL MECHANICS. I. vii. 52. 



^ da do dc 



d'i ^ d'z ^, d'z ^ 



By substituting these values in the equation (F) (363), 

 we may obtain three separate equations of the coefficients 

 of Sa, S6, and Sc, considered as vanishing separately ; these 

 equations expressing the relations of the partial fluxions 

 of the coordinates x, y, and z, the primitive ordinates a, 

 6, and c, and the time t. 



We must next investigate the conditions required for 

 the continuity of the fluid. For this purpose we may con- 

 sider the elementary portion of the fluid, at the beginning 

 of the motion, as a rectangular parallelepiped, of which 

 the sides are Da, d6, and dc, and the mass (f) DauhDc, 

 We may call this parallelepiped (A) : and it is easy to see 

 that after the time t it will be changed into an oblique pa- 

 rallelepiped ; for all the molecules at first situated in any 

 face of the parallelepiped {A) will still be in the same plane, 

 at least if we neglect the infinitely small eff"ect of curvature 

 on the infinitely small faces; and all the particles situated 

 in the parallel edges of (J.) will be found in elementary 

 right lines equal and parallel to each other. We may call 

 this new parallelepiped (^B), and we may conceive two 

 planes, parallel to that of x and y, to pass through the ex- 

 tremities of its edge formed by the particles which in (A) 

 occupied the edge Dc. Then if all the edges of (B) be 

 prolonged, until they meet these two planes, they will form 

 a new parallelepiped (C), equal to (B); for it is clear that 

 as much as one of these planes cuts off from the parallele- 

 piped (B), so much is added to it by the other. The 



