402 Mr. Challis on the general Equations 



Again, as <« may be considered a function of c and t, or of 

 Cj and t, we may also have, 



2au,=f\s-at-f^j,(c,t)dt\+F\s + at-/i,,{c„t)dt\, 

 2a- hyp. log. p=f{s-at-f^(c,t)dt]-F\s + af-fiy,{c„t)dt]; 



from which, by reasoning as before, may be obtained the two 

 particular integrals, 



2aw= 2 fl^ hyp. log. |o = f{s-at-wt) (3), 



2aw= —2a- Uyp.\og.p=-F{s + at — (ot) (4). 



These give the same velocity of propagation as the integrals 

 (1) and (2). 



The integrals (3) and (4) were first obtained by M. Poisson, 

 {Journal de I'Ecole Poly technique, Cah. xiv.) and he infers from 

 the consideration of discontinuous functions, that one of them 

 relates to motion on the positive side of the origin of s, the 

 other to motion on the negative. But this inference is made by 

 employing the arbitrary functions in a manner, which their nature 

 does not admit of The existence of arbitrary functions in the 

 integral is the proper proof of the discontinuity of the motion; 

 for were the motion necessarily continuous, the value of w would 

 be given by a determinate function. The arbitrary character 

 of the functions has reference to the mode of disturbance, 

 which may be any we please, either obeying or not the law 

 of continuity. Those properties of the motion which are in- 

 dependent of the manner of disturbance, must be ascertained 

 by reasoning independent of the arbitrary nature of the func- 

 tions. By reasoning in this way, we have shewn that one of 

 the preceding solutions belongs to propagation in the positive 

 direction, the other to jiropagation in the contrary direction, 

 whether on the positive or negative side of the origin of s. 



