Pressure in Fluids. 



119 



w 



where </> is arbitrary; whence it follows, putting </> = «, where a 

 is any constant, that (2) is capable of being derived from an 

 equation of the form 



<**Hh«> « 



where a is an arbitrary constant independent of the value of any 

 constant appearing in F. 

 Differentiating (3), we get 



0=F W+ F(,)| +P (|)^ + F@)J.j 



By hypothesis, (2) is derivable from these last two equations 

 and from (3) ; and since « appears in (3) and does not appear 

 in (2) or in either of the equations (4), it is clear that (2) must 

 be derivable from equations (4) alone, without taking account 

 of (3). 



Hence the right side of (2) must be identical with the right 

 side of the following equation, viz. 



+ B{r W+ F W j + r(g.at«'.(4)S> 



dv dv 

 where A, B are functions of txy -j- -j,' Therefore, comparing the 



coefficients of corresponding terms, we must have 



S= BF '©+ AF '(S> 



-R 



dx 



0= B {P(<) + ¥'(¥) % } + A {l»« + P(y) g}. 



whence we have, eliminating A and B, 



