284 Mr. Challis on the Theory of the 



(7) f^—m\ sin (^ .X — at + cj + m'\' sin {~,.a: - at + c'\ + &c. 



— 7n,X, sin (^ .x + at +c,j — W2X2 sin (^.x + at +c^ + &c. 



and in consequence, 



(S) aS = m\ sin (~ .x - at + A +m'X' . sin (~ . x — al+c'J + &c. 



+ OT,Xi sin (— .X + at -b cA + W2X2 sin (^ .x + at + C2 j + &c. 



Let s, »', &c. Si, So, &c. be the condensations arising from the 

 respective disturbances. 



Then r= a (* + s' + &c. - s, - S2 — &c.) = a (o- - o-'), 

 a .^ = a (s + s' + &c. + Si + S2 + &c.) = a{(T+ a'). 

 Whenever <T = a', V = o, and S=2<r; and whenever 0-'= -(7, S = o, 



and F= 2 Act. 



The equations (7) and (^) apjily to the most general motion 

 that can possibly take place, because the principle of the co- 

 existence of small vibrations, coupled with the reasoning of 

 Art. 3, shews that the vibrations of the particles must always 



primarily be such as the equation y = m\ sin — indicates, and there- 

 fore can in no case be other than those which arise from a 

 composition of motions of this kind. As the terms of these equa- 

 tions are to be taken between certain limits, depending on the 

 durations of the respective disturbances, which it is allowable to 

 do, because when the terms are so taken, the expressions for 



V and as .satisfy the equation -i—; = a' r?, the line which defines 



the resulting velocities and condensations at a given instant, is not 

 continuous, unless the disturbances be all supposed to continue 



