OF THE MOTIONS OF FLUIDS. 291 



dVd;z_d>d;z 



Dc= — rr-'^a:^ consequently do; =-- .Da + 



d y dz d y d z j ^ tj^ 



d6 *dc dc db 



d'x d'y d'z d'x d y d'z d'a: d'y d'z d'x d'y d^ 



d6*dc*da d6*da*dc dc'dadft dc .d6 'da ^ ^j 



rr— p -TT—p Da ; and 



dy dz dy dz 



db dc dc db 



Dx=d'yd'z d'yd'2; which is the base of the parallelo- 

 d6 'dc dc *d& 



gpapft 

 gram (a); and its height being Dy, its area is = dz : 



dF 

 which is also the area of the parallelogram (e), and which, 



d'z 

 multiplied by — . dc, will become CoaD^Dc, for the vo- 

 lume of the parallelepipeds (C) and {B) : and f being the 

 density after the time t, the mass must be ^ CoaD^Dc, which 

 being equal to (f) DaD^Dc, we shall have f^=:(f) for the 

 equation implying the continuity of the fluid. 



§ 33. Transformation of these equations : shown to be 

 integrdble provided that the density be any function of the 

 pressure, and that the sum of the velocities parallel to three 

 orthogonal coordinates, each being multiplied by the ele- 

 ment of its direction, make an exact variation. This con- 

 dition fulfilled at every instant if it is at a single one, 

 P. 94. 



366. Theorem. If ez, v, and w be the 

 velocities of a particle in the directions of j, 



t/, and z, we have S V— ^=^^(^"+ u^+ ^ \^ 



u 2 



