162 Rev. J. Challis's Theory of the Transmission of Light 



For the object in view it will be necessary to solve the fol- 

 lowing problem. A disturbance is made in an elastic fluid 

 like air, by means of a spherical ball, which is supposed to 

 contract and expand in a given niannei', its form remaining 

 spherical and its centre fixed : — what will be the consequent 

 motion of the fluid at any distance from the centre of the ball 

 at any time? The velocity of the surface of the ball must be 

 conceived to be small compared with the velocity of propa- 

 gation in the fluid. The following equations, applicable to this 

 case, ai'e well known : 



a- Nap. log? + -^ = 0; (3) 



In which r is the distance from the fixed centre, t any time, 

 V the velocity of the fluid at the distance r, § the density, 

 a" xq = the pressure, and 4> is a function of r and t, deter- 

 mined by the equation (1). 



The general integral of ( 1 ) is, 



rip = F (r—at) +f{r + at); 



and r <$ = F(r — «i5) is a particular integral. Let us see what 

 may be inferred from this latter. By differentiating we obtain, 



d(t> F'(r-at) F''lr-at) ,., 



— --!-, or V = — 5^ :^^ — s— ^ (A) 



dr r r- 



d(p oxT 1 Y'(r-at) 

 -jj, or -a- Nap. log g =-a —^ ', 



whence a Nap. log g = — 5^ (B) 



We have now to consider what the equations (A) and (B) 

 inform us. There is a difficulty hei'e in interpreting the lan- 

 guage of analysis, and ascertaining the physical facts it ex- 

 pounds, especially as they are of such a nature that they can- 

 not be objects of sight. Having met with nothing on this point 

 that appeared altogether satisfactory, I offer the following- 

 reasoning for the consideration of mathematicians. The pre- 

 ceding values of v and §, because they contain arbitrary func- 

 tions, will apply to that part of the motion v/hich is arbitrary ; 

 viz. to the motion of the particles which are immediately dis- 

 turbed; and this is true if the disturbance continues during 

 any given time. The denominators r, r^, not being involved 

 in the arbitrary functions, show that in passing at a given in- 

 stant from the disturbed particles to those indefinitely near, 



the 



