Fundamental Equations for determi?mig Motion of Fluids. 429 



Let the fluid be incompressible and the motion be parallel 

 to the plane containing the rectangular axes of or and?/; then 



by (1.), -z — h -r- =0. Supposing udx-\-vdy to be an exact 



^2 A d/^ A 



differential {d^), equation (1.) becomes ^—^ + -y-|= 0, which 

 by integration gives 



v= \/ — \{Y{x+y\/-\)-f{x-yV-\)]. 



In the same case, as is known, the integral of equation (2.) is 



rf4> u^ + v^ . ^ ,. .. 



p=2<p{t) — jj- — . As no reason appears for limitmg 



the arbitrariness of the functions in the above values of « and 

 V, let us assume that 



and 



, ^ m , 



Then u = mXf »=— wj/, and -^ = 0. 



in 

 Hence p =: <p (^t) — {x^+y^). 



It follows from this result, by putting ^ = 0, that the boun- 

 dary of the fluid may at all times be a cylindrical surface ; but 

 it is impossible this can be true, because the particles at the 

 surface are all moving with the same velocity in directions 

 making different angles with the surface. What then is the 

 reason that we have been brought to an absurd result by a 

 process all the steps of which are legitimate if two funda- 

 mental equations are sufficient? The answer simply is, that 

 the third equation cannot be dispensed with, as will be made 

 apparent from the argument that follows. 



When the two recognised general equations are alone made 

 use of, it is not possible to treat any instance of fluid motion 

 for which udx -^-vdy ■^ixidz\% not an exact differential. I know 

 of no writer who has attempted to indicate generally the pro- 

 cess to be followed when that condition is not fulfilled. Nei- 

 ther has any general rule been given to distinguish the cases 

 in which it is allowable to assume udx + vdy+wdz to be an 

 exact differential. Lagrange asserts that the assumption may 

 be made when the motion is small, and when it commences 

 from rest. These rules I have proved to be without foundation 

 (see Phil. Mag., S. 3. vol. xxi. pp. 106 and 426, and Number 



