of the Laws of transmitted Motion. 269 



let ^ (:r) be the function that represents the initial values of y. 

 Then at the beginning of the motion, 



<^(^)= F(^)+/(x) 

 = - a F (^) + af {x). 



Hence F (^) =.f'(x)\ and as <^' (jt) = F (.r) +/'(^), it 

 follows that 4)' (x) = 2 F (x), and <^ {x) = 2 F (.r). So also 

 <^ {x) — 2f(x), Hence Y {x—at) — ^<p{x—at), and 

 f(x j-at) = ^ ^ {x-\-at). Thus it is inferred, from the equa- 

 tion^ = F (x — at) -\-f(x-\-at), that 



■ ^ 1/ — ^<p(x—at) + ^<p(x-\-at). 



As in the first instance the values of 4) (x) are restricted to 

 those corresponding to values of x from —I to +/, no other 

 values of this function are in any case to be taken. Hence as 

 soon as x + at becomes greater than /, the function ^ (x-\-at) 

 must be considered evanescent, and the function ^ (x—ai) 

 alone applies. The motion at any point commences when x—at 



X ^~ I X I Z 



= /, or t = , and ends when x—a t = — I, or t = • 



a a 



As x—l, the distance between the point in question and the 

 extreme point of the initial derangement, is equal to a ty it 

 follows that the velocity of transmission in the positive direc- 

 tion is a. Analogous reasoning may be employed to show 

 that there is a transmission of equal velocity in the negative 

 direction, the function <p (x-\-at) being alone applicable when 

 x—a t is a negative quantity greater than l. 



It will be seen that in the above process, after supposing 

 (which is permitted,) the initial values of 3/ to be expressed by 

 F (or) +y(jr), it is assumed that its values at any subsequent 

 period are expressed by F {x—a t) + f(x+at), the forms of 

 the functions F and f being the same in both cases : in other 

 words, it is assumed that the forms of these functions do not 

 change with the time. But this is a consequence of the uniform 

 transmission, and cannot, therefore, be assumed in the proof 

 of it. Whenever the velocity of propagation happens to be 

 uniform this method leads to no error, bec&use it rests on a 

 supposition which implies a uniformity of propagation. It 

 could not, however, be applied without error to an instance 

 of propagation like that of waves at the surface of water, 

 where the forms of the propagated waves, though dependent 

 on the initial state of the fluid, are continually changing with 

 the time. It is with reference to this point that, in the Re- 

 port, I have suggested for consideration, whether the arbi- 

 trary functions obtained by the integration of the diflerential 

 equation can be immediately applied to any but the initial 



