346 NEWTON S PRINCIPIA. 



IfLe -7 - d .LL 



pdz ~ dt* 



These three equations are not, however,, sufficient to 

 determine the motion. For we have four quantities x,y&amp;gt;z&amp;gt;p 

 to determine in terms of t. A fourth equation is necessary. 

 This D Alembert supplied from the condition that any 

 portion of the fluid, in passing from one place to another, 

 preserves the same volume if incompressed, or dilates, ac 

 cording to a given law, if the fluid be elastic, in such a 

 manner that the mass is unchanged. It is usual at present 

 to derive this equation from a principle that in reality is 

 only the above in another form. If u, v, w be the velocities 

 of the fluid at the point xyz in the directions of the axes ; 

 then the result arrived at is 



dp u dp v d pw dp _ 

 - -- (- _ -- 1 - -- 1 \j, 



ax ay a z at 



It will be observed that these equations do not make any 

 assumption as to the molecular constitution of the fluid. 

 All that is required is that the pressure, no matter how 

 transmitted, shall be equal in all directions. 



These equations are so complicated that hardly anything 

 can be done with them. But there is one general case in 

 which the equations are greatly simplified. This is when 

 X, Y, Z, u 9 v, w, are such that 



dy + Zdz - - (A) 



udx -f vdy + wdz - --^ - (B) 



are perfect differentials, upon the supposition that the time 

 is constant and the density either also constant or a func 

 tion of the pressure. &quot; It becomes then of the utmost 

 consequence to inquire in what cases this supposition may 

 be made. Now Lagrange enunciated two theorems by 

 virtue of which, supposing them true, the supposition may 

 be made in a great number of important cases ; in fact, in 



