282 Mr. Challis on the Theory of the 



cease to be developed, and the particle will remain at rest. 

 This is a particular case of that discontinuity of the motion, 

 which was proved by Lagrange to obtain. In general, it may 

 be concluded that any motion is possible, which is indicated 



by a portion of the curve y = m sin -— , included between any 



two jjoints at which it cuts the axis; but in every case, the nature 

 of the disturbance must determine between what two jjoints the 

 portion is to be taken ; as will appear more plainly by examples 

 hereafter to be adduced. 



5. Let us now proceed to the application of the integral 

 of the differential equation of the motion to particular cases. 

 First, conceive a series of waves of the primary form to 

 be generated at a given point, and to be propagated in the 

 positive direction. Let us fix upon origins of < and x, and suppose 

 the given point to be at a distance .t;' from the origin of x, and 

 the commencement of the disturbance to be separated from the 

 origin of t by an interval t. Then the motion at any time t 

 and distance x, will be given by the equations 



V = «s = m\ sin - .{x — x' — a .t — t) ; 



for they satisfy the differential equation of the motion, and 

 V and as each = o, when x = x , and < = t. It is not allowable to 

 take X greater than x' + at, because at the end of t the propaga- 

 tion has not proceeded beyond this distance. If the propagation 

 had been towards the origin, we should have had 



f = — fls = — m\ sin - .{x — x -^ a A ~ t), 



A 



and X must not be taken less than x —at. 

 As x' and r are constant and arbitrary, 



TT 



V •=. ± as — ± m\ sin - . (a: + a< + c), 



