of the Laws of transmitted Motion, 271 



considered constant, when it is required to ascertain whether 

 the function F varies with the time. 



In applying the formula to find v from the equation 



y = Y {x-^at) -\-f(x-\-at), 



a value is obtained which is dependent on the arbitrary func- 

 tions, and consequently leads to no general law of transmis- 

 sion. There is, however, an analytical circumstance to be 

 now attended to which has an important meaning with re- 

 spect to the motion. The given differential equation is satis- 

 fied by each of the equations z/j = F {oc—a t), y<^ ■=. f{^x-\-at\ 

 and also by the equation y^ + j/g = F [x—a f) + f{^x + a t)^ 

 which shows that the motion corresponding tft the last equa- 

 tion is compounded of the motions corresponding to the other 



oL u 

 two. Now, from^i = F (x—at)^ - y = — a F' {x—at)^ 



and -T^= ¥' {x—a t). Hence v =^ a. So from 3/2 ■=-f{x-\-at\ 

 a X 



V =^ — a. These results, it must be observed, are obtained 

 without reference to any particular derangement, and inde- 

 pendently of the time ; and ^the general inference from tliem 

 is, that whatever be the initial derangement, the motion which 

 is taking place at each point at any time results either from a 

 propagation in a single direction, or from two propagations in 

 opposite directions, and that the law of propagation is such 

 that the ordinates existing at any instant are transferred 

 through space undiminished with the uniform velocity a. It 

 results from this law that the functions F andy do not change 

 with the time. Had we obtained a variable rate of propaga- 

 tion, these functions would not have applied to the motion at 

 two epochs separated by ^.finite interval, without first ascer- 

 taining from the law of propagation the change they undergo 

 during that interval. 



This being premised, with the help of what Lagrange has 

 proved respecting the discontinuity of the motion, we are 

 prepared to go strictly through the reasoning by which it was 

 before shown thatj/ = ^ <^\x—at)-\-^ <^ {x + at) -^ and it is 

 to be observed that this equation will give the value of y at 

 a7zy distance from the origin and at any time, i^hoXh x-\- at 

 and x—at be subject to the condition of lying between — / 

 and +/, and either function be supposed evanescent when 

 this condition is not fulfilled : also / may be of any magnitude. 



The rest of the reasoning by which it is commonly shown 

 that the velocity of propagation is a, is open to the objection 

 of making the determination of this velocity depend on the 



