of Fluids as investigated by Prof. Challis. 299 



In any possible case of fluid motion, the motion, which 

 would result by supposing the whole mass of fluid so in motion 

 to be besides moving forward in space with a uniform velo- 

 city, would also be possible. But if the components u, v, w 

 of the velocity in the first case be such that udx + vdy + ivdz 

 is an exact differential, it will be easily seen that the com- 

 ponents u'j ?/, id of the velocity in the second case will also 

 be such that u' dx + v 1 dy + Do' dz will be an exact differential. 

 But if the lines of motion in the first case be right lines, they 

 will not be so in the second, unless the velocity at each point 



a x 



of the same line be the same. If, for instance, u — —* 5 . 



x l +y l 



v = Q • y -- a ) 10 = 0, and if we now suppose the whole mass of 



x + y 

 fluid to be moving besides with a uniform velocity parallel to x, 

 the lines of motion and of direction will both be right lines in 

 the first case, but neither of them will be right lines in the 

 second. 



Professor Challis objects to the case of motion to which he 

 alludes, where u = a x, v = — ay, w = 0, by saying that 

 the arbitrary quantities introduced in the process of inte- 

 gration cannot be satisfied, unless the fluid be in confined 

 spaces or narrow canals ; that is in indefinitely narrow canals, 

 as his reasoning which follows shows to be the meaning. It 

 will appear however from the following reasoning that the 

 canals need not be narrow. 



Conceive a mass of incompressible fluid to be at rest,bounded 

 by material parallel planes, and by cylindrical surfaces whose 

 bases are part of a branch of a rectangular hyperbola, its 

 asymptotes (which I shall take for the axes of x and y), and 

 two lines perpendicular to them. Of the two planes whose 

 bases are the two latter lines, conceive one, whose equation 

 is y = y v to be made to move with a velocity — f{t)y x par- 

 allel to y, and the other, whose equation is x m x v with a 

 velocity f(t) x y parallel to x, and conceive the planes to con- 

 tract or expand, so as always to reach from the hyperbola to 

 an asymptote. Then the motion is determined by the equa- 

 tions of motion, the equation of continuity, and the condition 

 that the particles in contact with a surface, whether fixed or 

 moveable, neither penetrate into, nor separate from it. Since 

 the motion is determinate, and these are the only conditions 

 to be satisfied, any values of w, v, w and p which satisfy them, 

 will be the true values. Such values will be found to be 



u=f{t)x, v=-f{t)y, wsO, £ =v|/(/)-M" 2 + ^ 2 ). 



