410 Mr. Challis on the general Equations 



Hence w- a hyp. loff. — ^ — = o ; 

 from which, neglecting (jj , &c., 



just as in the above-mentioned example. The foregoing remark 

 may also be verified by the integral of the equation, 



d'.r< p _ , dr. r<p 

 ~d¥'~^ ~d?~' 



which is derived from (n) Art. 4, by taking only the two first 

 terms of this equation, and supposing ^ to be a function of r 

 and t. If /t>=i+s, this integral gives, when s is small, 



as = -^\F'{r-at)-f'{r + at)\, 



u>=\\F'{r-at)+f'{r+at)]-^,{F{r-at)+f{r+at)\*. 



Hence when w is small and r very small, 



, F{r - at) +f{r + at) 

 as = 0, and a. = ^^ ' j^ '' '- , 



nearly ; and the arbitrary functions may be so assumed that w 

 shall be equal to ^^-^. 



• Id Art. 14 of the communication I made to the Society, on the Small Vibrations 



of Elastic Fluids, the term involving -^ in the value of <u is by mistake written % , and 



the reason assigned for neglecting this term is inaccurate. It tan only be neglected when r is 

 great, compared to the dislnnce to which the effect of the disturbance in the first instant 

 extends, or is great compared to A the breadth of an undulation. Hence also, the manner 



in 



