OF THE M0TI0N5? OF FLUIDS. 



289 



parallelepiped(C) 

 will have its two 

 bases parallel to 

 the plane of.T and 

 y: its height be- 

 tween the bases 

 will evidently be 

 equal to the ele- 

 ment of 2 ; and y ^' 

 since in this point "^vt^ 



of view jr, y, and t may all be considered as constant, and 

 the same values only of a and t enter into the determina- 



dz 

 tion, the element will be merely — ^ Dc [, which must be 



dc 



d'w 



equivalent to the Dz 4- ^^-r— ^^^^ ^^ Poisson]. 



The base of 



the parallelepiped (C) will be found by observing that it is 



equal to the section of (B) by a plane parallel to that of x 



and y; and we may call this section (e) : with respect to 



the particles situated in it, the value of z will be the same 



„ „ , t 1, 1 /^ d'z d'z , d'z 



for all, and we shall have Dz-=0=:-r- oa-h --7- Tio-\--— dc. 



da do dc 



Now if up' and nq be two contiguous sides of the section 

 (e), the first derived from the face answering to d6dc of 

 {A)f the second from DaDc : if through the extremities of 

 the side Dp' we imagine two right lines to be drawn paral- 

 lel to X, and the side opposite to Dp to be produced so as 

 to meet these lines, they will intercept a new parallelogram 

 (x) equal to (f), having its base parallel to .r. The side Dp' 

 is formed by some of the particles belonging to the face 

 D&Dc, that is, by those particles with regard to which the 

 value of z is invariable, and it is easy to see that the height 

 of the parallelogram {>) is the element of y, taken on the 



u 



