326 Mr. Challis's General Considerations respecting 



and V becomes v + -r— u t + -— t, and the relative velocity 



a X at 



, d . (i>''^ — i'^) d . (v — v) 



^-^+ \d. "+ -V-" 



But as t/ = u + -^ 8 .r, the two last terms of the preceding 



expression will involve small quantities of an order which may 

 be neglected. Hence the relative velocitywill be t/— v during 

 the small time t. Now by propagated motion it is to be un- 

 derstood that the state of the masses y, |3, passes into that of 

 /3, a, supposing the propagation to be directedy)o??i the origin 

 of.r. Hence the distance between the centres of gravity of 

 y and /3, will become that between the centres of )3 and a. If 

 this take place at the end of the time t, {^-d — v)r = the dif- 

 ference of these distances. Hence 



(d-v)T = —^ — = 2 • 



But z" = s' -H ^Ix, and c = 2' - 4^ Sj:. Therefore, 



dx dx 



x' — z = 2" — z', and consequently (t/ — r) t = z" — sf. If 

 g' = the density of /3, g of y, g' z' = g 2", and z" — z' = z' . ?^. 



Hence, u' — u = -^ . -^^^. 



Or, since t/— t; and q'— g are the variations of 7; and g at a 

 given instant from one point to a contiguous point, 



dv z d f 



d X T f dx 



Here — is the rate at which a given small mass passes into 

 the state of the contiguous mass, abstraction being made of 

 the velocity common to the two. Hence -"- is exactly the 

 velocity of propagation. If it be constant and equal to b, 



77 = ^ 77x ' '''"'^ ^y integrating, 



v = bh.l.g + (Pit) (A) 



It is to be observed, that the proposition here proved is inde- 

 pendent of all consideration of the constitution of the medium, 

 and the influence of caloric and extraneous forces. 



When the medium is a fluid, and the relation between the 

 density and pressure is, y? = a'^ g, the equations relating to the 

 motion are, 



c/i (p 2v d- (p 1__ d- (p Pv _ 



drat a- — I'- ' dl' a^ — v"^ 



