OF THE MOTIONS OF FLUIDS. 293 



1+Af.— -— , so that the first term of the value of C becomes 



QZ 



equal to the product of these three quantities, and the five 



other terms vanish, since --=:0, -— =0, -J: =0,-^ =0, 



do dc da dc 



d'z dz 



— =^> -n =^«] The equation {G) becomes therefore 



da Qo 



^ ,d'u , d'u , d'w\ . V /^ 1 .p , 



fAt ^j- +— +-T— j+f — (f)=:0: and if f be considered 



as a function of x, y, z, and /, we have 



(f)zzf— A^ --/ — uAt -j^—vAt -r — wdtrr\ so that the pre- 

 d^ dx dy dz 



ceding equation becomes 



d> _^ d\^u) ^ d\^v) ^ dXpw) _. ^ ^ 



d^ dj7 dy dz 



It is easy to see that this equation is the fluxion of the 

 equation (O) (365) taken with regard to the time < [ : for it 

 has been deduced from G by taking the difference of its 

 terms with regard to the evanescent element of time A^]. 



368. Theorem, liu^x + v^y + w^z— 5^, ^ 

 being any function of the pressure p^ we shall 



and, for homogeneous fluids, tt~+"tt"'^ 



dz^ "• 



When u^x-^v^y-^-w^z is an exact variation of x, y, and z 

 (313) and f is also a function of the pressure, the equation 

 (£r) is susceptible of integration, for it becomes SF — 



^»S*»'U-S)"+(^)^(-a'r)"Sc^ 



since 



