269 

 and that for all other values of x the functions y . and 1/ , 



*S •*» u «/ .1*, u 



vanish: which is equivalent to supposing- that at the origin 

 of I, and for a large number i of wave-lengths (each = n) be- 

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

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

 dulatory vibration, 



V« t = 1 vers ( 2 x \~ 2 ^ sin ^), (3)' 



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 iu mixed differences (1) : 



1 f ■ 7r \ 2 C sininO cos (2x6 -\-in9 — 2at sin Q) .„ ... 



y t — -\ sm - ) \ T-Q- (19; (4) 



■>*>* ttV n J J. sm0 C os0-cos^ 



n 



an expression which tends indefinitely to become 



— - vers 

 2 



I 2 x 2 at sin - ) 



\ n n) 



2ttV «/ J <> sin0(cos0-cos^) 



as the number i increases without limit. The approximate 

 values are discussed, which these rigorous integrals acquire, 

 when the value of t 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 



7T 



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 



