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LVIII. Reply to Professor Challis's further Remarks* " On the 

 Received Principles of Hydrodynamics." By Robert Moon, 

 M.A., Honorary Fellow of Queen's College, Cambridge']'. 



IN his recent paper Professor Challis reiterates, in a slightly 

 different form, his argument derived from the equation 

 p = f\mct. (p, v), without taking any notice of my refutation 

 of it. 



Putting u, v, w for the resolved parts of the velocity, and 

 x, y, z for the coordinates of a particle at the time /, he assumes 

 that the motion in three dimensions of any fluid may be repre- 

 sented by the five following equations; viz. 



P=f\(z,y>z>t)> P=Mx>y,z,t), u=f 3 (x,y,z,t), 1 



v =*f4 ( & y> *>*)■> w =f& (%> y> 2 > X • J " 



Eliminating x, y, z, t from equations (a), Professor Challis obtains 

 the equation F(/?, p, u, v, w) = 0, which he assumes to be capable 

 of being "satisfied by an arbitrary relation between the quantities 

 — that is, by one which is independent of the particular problem." 

 "The arbitrary condition," says Professor Challis, "has the 

 effect of defining the fluid; and evidently the number of differ- 

 ent kinds of fluids is unlimited." 



I must here point out the very extraordinary character of the 

 admission made by Professor Challis when he asserts that the 

 number of fluids is unlimited, each fluid being defined by the 

 relation between p, p,u,v,w prevailing in it — his sole object, so 

 far as I understand it, being to prove that one only of this 

 unlimited number of fluids, and that a fluid in which the highly 



* See Phi]. Mag. for Oetober last. 



t Communicated by the Author. 



X From this view I must express my dissent, believing that the equa- 

 tions (cc) are insufficient to determine the motion in the case in question. 

 Not wishing upon the present occasion, however, to enter upon a fresh 

 field of controversy, I shall content myself with observing that this assump- 

 tion of Professor Challis contradicts his subsequent assumption that the 

 resultant of the elimination of x, y, z, t from equations («) will be a per- 

 fectly arbitrary function of p, p, u, v, w. For in the case of equilibrium the 

 five equations («) reduce to the pair 



the elimination between which of z gives 



p= funct. (p,x,y)', 



which is irreconcilable with Boyle's law in the case of equilibrium, unless 

 we suppose that x and y both disappear along with z in the elimination. 

 But this would require the functions on the right-hand sides of equations 

 (j3) to be of a special form ; from which it follows that the functions f,f 2 

 must be of a special form, and therefore the result of elimination from them 

 of x, y, t cannot be perfectly arbitrary. 



