286 CELESTIAL MECHANICS. I. vii. 32. 



terms which contain the square of A^, and dividing by 

 vxDT/Bz^t, we obtain 



d'c . d'p dV , dV /d'w , d'v , d'w\ ^ , ^ 



d^ da: dy dz ^ ^dx dy dz / 



amounts to the same, -^ + — ^^— ■ + — f — + -^^L-^zz 0. It is 1 

 dt dx' dy dz M 



unnecessary to pursue Mr. Poisson's investigation any fur- ^ 

 ther, since it is only introduced as an illustration of some of 

 the less perspicuous parts of Laplace's mode of consider- 

 ing the subject, to which we are now to return.] 



363. Theorem. The motions of fluids in 

 general may be deduced from the equation 



^V heing= P^j^-hQ^y+R^Zy p the pressure, f 

 the density, and P, Q, R the external forces 

 acting in the directions of the coordinates a;, 

 Tjy and z. 



It will be convenient to deduce the laws of the motions 

 of fluids from those of th^ir equilibrium, in the same man- 

 ner as, in Chapter V, the laws of the motions of a system 

 ofsohd bodies have been deduced from those of the equi- 

 librium of the system. For this purpose we may resume 

 the equation ^p=:^ {P^x -\- Q^y + R^z) from the demonstra- 

 tion of article 316. 



Now when the fluid is in motion, the forces unemployed 



in generating motion are F — -, Q r-f , and JR— --— , 



d^^ d.t^ dt^ 



which must hold each other in equilibrium : we must there- 

 fore substitute these forces for the P, Q, and R of the 



