198 Mr CHALLIS's RESEARCHES IN THE THEORY 



15. If r be indefinitely great in equation (C), the motion is in 



parallel lines, and putting r = c + s, j=j- Let -^ = w, and 



suppose no force to act ; the equation for this case becomes 



d''(p 2w (Pep 1 d'(p _ 



~d? ~ '^^' ■ dsdt "^ o^^T^ ' dF~^' 



This equation combined with a* N. 1. p = — -^ — — , gives as a particular 



integral, u] = al:iA. p =/"{«- {a + w)t\. By varying a little the mode of 



_ ^( as. 



integrating, I found also w — a^A. p =/( ■ atj, {Camb. Phil. 



Trans. Vol. III. Part III. p. 399), and endeavoured to shew the way 

 in which each integral ought to be applied. But this enquiry was 

 unnecessary ; for the integral may present itself under an unlimited 

 number of different forms. The equations 



w = a^.\.p=f{.^-{a + io)t + ^{w)], or «, = «N.]. jo=/(^^^^%i^l , 



will equally satisfy the differential equations, being, in fact, only 



different forms of the first-mentioned integral. The principle according 



to which it now appears to me, an integral of this nature should be 



employed, is to apply it immediately only to the parts of the fluid 



immediately acted upon by the arbitrary disturbance, in order to 



determine the law according to which the initial velocity is transmitted 



to the contiguous parts ; then to determine the law of transmission 



from these to the next; and so on in succession. In the present 



instance by making * and t vary so that w and p remain the same, 



ds 

 we shall find a + w for -j~ the velocity of transmission, under whatever 



form the integral may appear. The second term m of this quantity 

 is due to the transmission of velocity through space by the motion of 

 the particles themselves ; the other a is the velocity of propagation 

 along the particles. In this example, as the velocity and density are 

 propagated uniformly and undiminished, it is easy to determine at any 



