276 Mr. Challis on the Theory of the 



posed, the velocity is always proportional to the condensation. 

 Also if the curve be traced which gives the velocities and con- 

 densations at any instant, by supposing t constant and x variable, 

 it is plain that the same curve is obtained, whatever be t; but 

 it will be at different distances from the origin of x, according 

 to different hypotheses made on the value of t. Also if t be 

 supposed variable and x constant, the series of values which v 

 and as take whilst t varies from < to f + t, are the same as are 

 obtained, when t is supposed constant, by making x vary from 

 X to X ~ ar. Hence, the velocities and condensations which the 

 particles at a given point undergo during the time t, are the 

 same as those which the particles in a space ar, measured from 

 the given point towards the origin of a, are undergoing at the 

 instant t commences. The motion will therefore be understood, 

 by conceiving the curve which gives the velocities and conden- 

 sations at every point, to move without undergoing alteration, 

 along the axis and from the origin of x. The velocity of its 

 motion will be found by making x and t vary at the same 

 time, in such a manner that F{x — at) does not alter. Hence, 



d.{x- at) = 0, -^ =a. 



This is the velocity of propagation ; which is thus shewn to be con- 

 stant, and in all cases the same. Considering now the function y by 

 itself, we shall have v= — as =f{x + at). A discussion similar to that 

 applied to the function F, will prove that these equations indicate 

 a series of velocities and condensations, which may be repre- 

 sented by a curve of invariable form, moving toivards the origin 

 of X with the uniform velocity a. For making d {x + at) = o, 



dx 



-rr = —a, which shews that the direction of propagation is opposite 



to what it was in the former case. Also v has changed sign, 



