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V. Further Discussion of the Analytical Principles of Hydro- 

 dynamics, in Reply to Mr. Moon. By Professor Challis, 

 M.A., LL.D., F.R.S., F.R.A.S* 



IN vol. xxxvi. of the Philosophical Magazine (p. 117), and in 

 the Number for September 1873 (p. 247), Mr. Moon has 

 adduced an argument which, as expressed in the later publica- 

 tion, is as follows : — " For motion in one direction it is axiomatic 

 that we must have in every fluid, in every case of motion, 



P=f\ fo Oj P=/a(*j *)> v =fs(x> 0; 

 whence it follows that, for every value of as and t for which each 

 of the foregoing equations represents a substantive relation be- 

 tween the variables, we shall have^?= funct. (p, v)." 



I have admitted the truth and the comprehensiveness of this 

 conclusion, considering it to be a legitimate inference from the 

 axiom that all physical relations expressible by functions of space 

 and time are comprehended by the processes of abstract calcu- 

 lation (see Phil. Mag. for August 1873, pp. 160 & 165). 



Also in my reply in the October Number (p. 310), I have ex- 

 tended the argument to space of three dimensions, and obtained 

 the general equation ¥(p, p, u, v, w) =0. 



I have, besides, ascertained that the values of p, p, v obtained 

 by Mr. Moon in vol. xxxvi. (p. 124), and reproduced in the Num- 

 ber for December 1873 (p. 448), satisfy the differential equation 



^ + -^=o m 



df*+Ddx ' ' ' * ' {L) 



Having made these statements, I am prepared to admit that 

 Mr. Moon has rightly urged (in the December Number, p. 447), 

 in opposition to an assertion I had made, that the assigning of 

 a relation between p and v in the equation p=f(p, v) does not 

 simply define the fluid, but imposes also conditions on the mo- 

 tion existing in it, and that an analogous remark applies to the 

 equation F(jo, p, u, v, w) = 0. Let us, however, inquire what in- 

 ferences relative to this point are deducible from Mr. Moon's 

 equations. 



By a solution of the equation (1) he obtains for determining 

 p the equation 



a being an arbitrary constant, and the form of the function % 

 being arbitrary. Putting on the left-hand side of this equation 



f(p, v) for p, it will be seen, since %(v -f -J is arbitrary in form 



* Communicated by the Author. 



