On the Received Principles of Hydrodynamics, 447 



special relation p = a 2 p obtains has the slightest claim on our 

 attention. 



Waiving this consideration, however, I join issue with Pro- 

 fessor Challis on the point of our being able to assign to the 

 relation ¥(p, p, u, v, id) = a definite form which is independent 

 of the particular problem. By assigning a definite form to the 

 arbitrary function Y(p, p } u,v,w)=^0, it is not true, as assumed 

 by Professor Challis, that we simply define the fluid. We do a 

 great deal more. We define the fluid, and at the same time 

 impose conditions on the motion existing in it. 



In proof of this I must again direct attention to the analytical 

 argument of my last paper, which Professor Challis does not 

 appear to have examined. 



Confining ourselves to motion in one direction — suppose that 

 we have a cylindrical tube filled with any particular fluid (say 

 air of the mean density of the atmosphere) ; and suppose that at 

 a given time / we have a disturbance extending over a limited 

 portion of the column. Let the velocity and density throughout 

 the disturbance at the time t follow any law whatsoever consist- 

 ent with continuity; then I have shown in the paper in ques- 

 tion that the pressure prevailing throughout the disturbance at 

 the same time may follow any law whatever consistent with con- 

 tinuity*; from which it is evident that Professor Challis's argu- 

 ment, in proof " that it is allowable to make for fluid in motion 

 the hypothesis that the pressure varies as the density always and 

 at all points/' must be fallacious. 



I observe that Professor Challis mistakes the points on which 

 I mainly rely for my " objections to the received principles of hy- 

 drodynamics." 



My main argument is this : — Taking, in the case of motion in 

 one direction, 



p=f ] (zt), p=M%t) } v=f 3 {xt). 

 I hence derive the equation 



jt?=funct. (p, v); 

 whence it follows that the equation of motion 

 _ dhj 1 dp 



will be satisfied by the three relations 



* It must be borne in mind that the condition of continuity requires 

 that at the limits of the disturbance the velocity shall be zero, the density 

 = D, the atmospheric mean density, and the pressure = a a D. 



