Prof. Challis on the Principles of Hydrodtjnamics. 239 



value of X will be found to satisfy equation (3,), supposing the 

 arbitraiy quantity pj;(^) to be zero. 



Example III. Let the motion be parallel to the plane of xy, 

 anA, as in the example of page 37, let u = mx and v=. —my. 



These values of u and v were obtained (p. 37) by integrating 

 the equation 



du <^v _ 

 dx dy ~ ' 

 on the supposition that udx + vdy is an exact differential, and 

 then giving particular forms to the arbitrary functions contained 

 in the integral. This is equivalent to assuming that the motion 

 is of a certain kind consistent with the principle of constancy of 

 mass ; and the problem for solution is, to determine under what 

 circumstances such motion can be produced. It may be readily 

 shown that the surfaces of displacement in this case are cylin- 

 drical surfaces, of which the general equation is x^—y'^ = a^; and 

 that their orthogonal trajectories are rectangular hyperbolas, of 

 which the general equation is xy — c^. 



Assuming, for the reasons given in the preceding example, 

 that cfi is a certain function (T) of the time depending on the 

 position of a given element, we have 



d±__dl df d^ 



dt ~ dt dx ""^^ % - "^^^ 



and since u^mx=X-£, \= ^. Hence the equation (3.) be- 

 comes 



--^+2>w(^'+y')+xW=0. . . . (11.) 

 The equation which gives the pressui'e is 



Hence by combining these two equations, 



This result gets rid of the contradiction arrived at in p. 38. It 

 , , . (/T . 



also shows, smce -y- is given for each position of a given element, 



that the pressure at each position depends on arbitraiy quantities. 



Again, since this is a case of steady motion, c^ will be constant 



for a given element in the equation xy = c^. Hence the values 



of a? and y deduced from the combination of this equation with 



