92 Prof. Challis's Researches in the 



state his reasons for objecting to equations (8.)) (B.) and(C.). 

 The subject, being entirely new, offers a fair field for discus- 

 sion ; and I am quite prepared to find that I have been mis- 

 taken on some points. In particular, I have recently disco- 

 vered that the equations just named are not generally true 

 beyond the first order of approximation, as I shall presently 

 show. From the nature of the problem the processes applied 

 to it must be in a great measure tentative, and can only be 

 tested by the results. The equation (A.), for instance (Phil. 

 Mag., vol. xxxiii. p. 99)? as far as I know, is the first example 

 of an application of analysis to a question in physics, which 

 presents for solution a partial differential equation containing 

 two principal variables and two independent sets of variables 

 mixed up with each other. I have proceeded on the prin- 

 ciple that if particular and consistent values of one set of va- 

 riables be substituted in the equation, the resulting equation 

 will be true ^oy general, if not the most general, values of the 

 other set. But the application of this principle is restricted 

 by any limitation to which the supposition which conducted 

 to the equation (A.) is subject, viz. the supposition by which 

 udx + vdy-\-'iSodz was made an exact differential. Now it is 

 well known that for small vibrations, the equations 

 du _ dv du _ dtso dv _ dw 

 dy dx* dz dx dz dy ' 

 must be at least approximately verified. This will be the 

 case for approximate values of w, v, w, if the complete values 

 make udx + vdy + ii^dz an exact differential. On this account, 

 for the purpose of verifying those three equations approxi- 

 mately, it was assumed (Phil. Mag., vol. xxxiii. p. 99) that 

 udx + vdy + wdz was integrable for the complete values of u, 

 V, and w. But it is equally possible that the same three equa- 

 tions may be exactly verified by approximate values of u, v, 

 and w, in which case the condition that tidx + vdy-j-ivdz be 

 integrable, may be only satisfied by approximate values of 

 u, V, w. It does not seem possible to decide which of these 

 is the true state of the case but by trial. I supposed the 

 former to be true, and obtained equations applicable to ray- 

 vibrations on this presumption. It turns out on further in- 

 vestigation that the latter is the true theorem, when the inte- 

 grability depends on the supposition that {d.f(p) = udx-\-vdy 

 + isodz, f being a function of s and t, andy a function of .r and y. 

 For this reason those equations require certain modifications 

 which I now proceed to develope. 



The equation (B.) in page 99 of the Phil. Mag. for last 

 August, viz. 



o_ o^ + a.^^^ ^^2 U% dzdt dz^ dz^ ' ^^'^ 



