through Mediums, and of its Reflection at their Surfaces. 163 



the velocity varies in a manner independent of the arbitrary 

 distui-bance, and also point out the laws of this variation. 

 But the equations (A) and (B) do not immediately apply to 

 points at any fnite distance from the disturbance, because the 

 motion at such points is not strictly arbitrary, but depends on 

 the laws of the propagation of motion through a continuous 

 mass. Hence we have need of an independent theorem re- 

 specting propagation of motion, of the same nature as the 



theorem, vel. = -A^ — , regarding motion itself. Such a one 



time ° ° 



will be found in the paper referred to at the beginning of this 

 communication, in which it is shown generally and indepen- 

 dently, that the velocity of uniform propagation 

 the velocity of the fluid 

 ~ Nap. log of the density ' 

 Now in the case before us, we infer from the equations (A) 

 and (B), that, 



,, , F {r-a t) 

 V = a. Nap. log q ^—^ . 



We may therefore conceive v, as soon as it is impressed 

 on the fluid, to resolve itself into two parts, v' and 5y", of which 



•d = a . Nap. log g, and u" = ^^ — ^ — . The theorem 



above informs us that x) begins to be propagated with the 

 velocity a. The propagation will go on with this velocity 

 continued uniform, because by the incipient propagation, the 

 particles next in succession are brought into the same state, as 

 to the relation of velocity and density, as those immediately 

 disturbed ; and so on successively through all the particles of 

 the fluid. For a like reason the varying of the velocity in- 

 versely as the distance from the centre of the sphere, indepen- 

 dently of the form of F, holds true for any distance : for this 

 law is true with respect to an indefinitely thin spherical shell 

 of fluid immediately contiguous to the disturbing surface; 

 and so for every similar portion in succession. Hence at a 



distance R the velocity is -^ after a time from the 



instant that v is impressed. The other part of the velocity u", 

 not being accompanied with change of density, is transmitted 

 instantaneously, as if the fluid were incompressible, but varies 

 at diflerent distances inversely as the square of tiie distance; 

 for this law obtains at the disturbed points, and, as may 

 readily be seen, iluist obtain in incompressible fluids. Hence 

 Y 2 the 



