Differential Equations applicable to the Motion of Fluids. 87 



The elimination of N between this equation and equation (4.) 

 presents no difficulty. 



I proceed now to draw some conclusions from the equation 

 (2.), after substituting in it the expressions for w, u, and "w. 

 By the substitution the equation becomes 



(d.^p-\ /rf.N''n (d.si^x 



Hence by integrating, 



-J{di+^{d^+^+^dr^ •" ^'-^ 



The equation (4.) makes the last term of this equation dis- 

 appear. 



First, it may be remarked that if N be a function of ^, the 

 equation becomes 



/W-P = <1^H- 



W_ /d^ d^ d^\ 



and if N (^ = i|r, by differentiating with respect to space, 

 N rf <$ = t/ i/r ; so that we have 



'^}^>~^ -Tt '^ T-ydx^'^d^'^d^)* 



which is the equation that is obtained when icdx + vdy + iad z 

 is an exact differential. 



Next let the motion be supposed very small. We may then 



substitute -.— for (-77-) in the expression tor — dVy and 



neglecting the last term as being of the second order, we have 



dt d 

 Hence by integrating. 



/P \^ 1 d<P (l^ J y, d.d'p , dN . d.NdA 

 dt dt dt dt dt 



/W-P='^^ (6.) 



At the same time equation (1.) to the same degree of approxi- 



tion becomes, 



I dP (In dv dw 

 -7-. 77- + -7— + -,— + -,— =0. ... (7.) 

 Ic at dx dy dz ^ ' 



