269 



and that for all other values of x the functions ^^^Q^nd ^/'^ o 

 vanish: which is equivalent to supposing that at the origin 

 of i, and for a large number i of wave-lengths (each zz 7i) be- 

 hind the origin of ^, the displacements and velocities of the 

 particles are such as to agree with the following law of un- 

 dulatory vibration, 



y ,:=i 1] vers [2x ^at sin - ), (3)' 



*' ^ \ n nJ 



but that all the other particles are, at that moment, at rest : 

 it is required to determine the motion which will ensue, as 

 a consequence of these initial conditions. The solution is 

 expressed by the following formula, which is a rigorous de- 

 duction from the equation in mixed differences (1) : 



r? ( . ttN^C %\x\in% cos(2j:^0 + m0— 2a/sin0) ,^ ,.. 



II , zi - ( sm - \ -r-^ — ^ ■ dd; (4) 



•^*.^ ttV nJ J. smO cos6/~cos^ 



n 



an expression which tends indefinitely to become 



t V /^ TT ^ , . 7r\ 



y zz - vers \^ x 2 a^ sin -) 



•^*' 2 \ n n) 



^ A ( sin ^y { sin(o.a-o.,sin0 ) ^^^ ^^^, 



27rV nJ J« siu0^cos0-cos^) 



as the number i increases without limit. The approximate 

 values are discussed, which these rigorous integrals acquire, 

 when the value oit is large. It is found that a vibration, of 

 which the phase and the amplitude agree with the law (3)', 

 is propagated forward, but not backward, so as to agitate 

 successively new and more distant particles, (and to leave 

 successively others at rest, if i be finite,) with a velocity of 



progress which is expressed by a cos -, and which is there- 

 fore less, by a finite though small amount, than the velocity 

 of passage a ~ sin - of any given phase, from one vibrating 



2 A 



