Mr EARNSHAW, ON FLUID MOTION. 229 



differential coefficients of with regard to A, B, C... we may change 

 the equation 



into another not containing x, y, %. It is to be observed that A, 

 B, C ... are dependent only on / and quantities which are absolutely 

 constant; B, C, ... are therefore expressible in terms of A, and con- 

 sequently there is only one independent parameter A. Hence the 

 process above pointed out will give us an equation between f/,'^. 

 d_c(p, d,,(p and A: that is, (j> will be a function of A and / only. 



30. As one of the simplest examples of this process, let us sup- 

 pose the form of the waves to be spherical. Then ;( = is 



x" + y- + z^ = r", 



r being the parameter A. Now (p being a function of / and r 

 only, and x, y, z being functions of r only, we have 



di<P = '<-d,'<P+[l-'^)d.<l>, 



.-. d,'<j> = a' (rf,> + - . d,^) = - . d; (rep) ; 



.-. d;'(r(}>) = a:d?{r(p). 



This equation being integrated gives 



r<p = F(r- at) +f{r + at), 



which expresses the state of the fluid when the motion is in spherical 

 waves: the former term F {r ~ at) shews that spherical waves may 

 diverge from a fixed centre; and the latter y(r + «0 that they may 

 converge towards a fixed centre. The velocity of either wave is a. 

 The motion of each particle is directed towards or from the fixed 

 centre Art. 3, and its velocity = rf,^ 



G G 2 



