Prof. Challis on the Vibrations of an Elastic Fluid. 363 



For the right-hand side of the second, equation we have 



6 2 

 already substituted & so that the first equation becomes 



** (a* ^VVo^ d ** +We-0 (B>) 



This is the general equation for determining <p. It is iden- 

 tical, as might have been anticipated, with equation (B.). 



By integrating (B'.) to the first order of approximation 

 (see Phil. Mag. for April, p. 279), I found the only value of 

 <p expressible in finite terms to be 



<p = m cos q(z — a't + c), 



the constant a! being such that 



b 2 



a!*=a*+ % 



Setting out with this first approximation, and including 

 terms which contain the third power of m, I find the following 

 expressions for <p and a' 2 : 



<p — m cos q(z — a't + c) 



vfiua 



(1-) 



aW + p +MY (^ + !). • • • (*•) 



The equation (1.) serves to define the motion parallel to the 

 axis of z, and the equation (2.) determines the rate of propa- 

 gation (a 1 ), which, it thus appears, is not altogether indepen- 

 dent of m. I have before stated, that the different terms of 

 the expression for <p account for harmonics. 



Let us now investigate the equation which defines the mo- 

 tion transverse to the axis of z. For this purpose, on account 

 of the independence of the variables in <p andj^ suppose p = 0. 

 Then u = and v=0. Hence the motion at each point of the 

 plane corresponding to <p = is parallel to the axis of z. For 

 any point of this plane R and R' each become infinitely great. 



But ( p" + t>7 ) ' ~ ma y nave a finite value. Hence the general 

 expression for p- + 777 gives for this particular case, 



\R "*■ R'/<p ' dz ~ dx* ^ df f W 2 df/' 

 2 B 2 



