360 Prof. Challis's Determination of the Velocity of Sound 



compatible with fluid motion. The same objection might be 

 raised against the results of the hypothesis now under consi- 

 deration, ifj on carrying the investigation to higher degrees of 

 approximation, any similar incompatibility appeared. To 

 determine whether this be the case, it is now required to in- 

 tegrate equation (B.), taking account of the two last terms. 



This may be done by successive approximations, beginning 

 with the value of ^ already obtained. The result to three 

 terms, as given in my communication to the Philosophical 

 Magazine for November 1848 (p. 363), is 



(p = w cos q{z— alt + c) 



"■"3k~^^"^2'(^~'^''^ + ^^ 



and 



4^2 V W 



It: 



=«'+r2+^V\^-^2 



q being substituted for 



So far as this result indicates, <p is a function ofz—a't, and 

 the velocity of propagation at all points of a wave is the con- 

 stant a'. To ascertain whether constant propagation, the same 

 for all points of the same wave, accords exactly with equa- 

 tion (B.), let us introduce into this equation the condition 

 (p = F(z — a^t). For the sake of brevity write v for z — a^^t, and 

 F for F(v). Then, supposing «j constant while z and t vary, 



^_ ^ (l^ ^_^ _^__ ^ 



dt"" ~^i • 'd^' dz~ rfv' dzdt ~ '^^ dv^' 



Consequently 



d^F^ o o ^ dF dF^\ ,01. ^ , V 



-^(«,^-a^-2«,-^4--^)+6^F=0. . . («.) 



It may be here observed, that if the term involving b^ be 

 omitted, the resulting equation, which applies exactly to the 

 case of plane-waves, is satisfied by either of the two equations, 

 d^F ^ ( dF\^ 2 ^ 



Neither of these equations gives a form of F compatible with 

 vibratory motion : whence we may infer that, on the hypo- 

 thesis of plane- waves, all the parts of a wave are not propa- 

 gated with the same velocity. This result leads to the incon- 

 sistency already spoken of. 



