acted upon by the Undulations of an Elastic Fluid. 345 



tion in order to ascertain how far the law of the coexistence of vi- 

 brations holds good when terms of the second order are included. 

 For the present purpose it is not necessary to carry the inquiry 

 further. 



Let it be supposed that udx-\-vdy-\-wdz is an exact differen- 

 tial (d'yjr) for vibrations relative to an axis; and, regard being 

 had to the series for the velocity along the axis previously ob- 

 tained on that supposition, let us assume that for any distance 

 from the axis we have 



mf m*kg . v m 3 Bh y 



n/r = - -± cos 55 - -£-*- sin 2q£+ ~- cos 3g£ 



2tj- 

 q being put for -— -, %£ovz—a't + c, A for 



A, 



2* 



and B for 



3(7/e 2 -f-l) * . . 3 «(* 2 - 1 ) , 



3 g — ^W and /, ^, A being functions of r and constants. 



These values of A and B are extracted from a formula given near 

 the beginning of Part L, where also will be found the following 

 relation between a 1 , the rate of propagation, and a : — 



■ + 5 5 4 5+ l2«P 



5* 2 + 3 



Since a! and m are constants, -5-3 is a numerical constant, being 



q'a* 



in fact, as has been already shown, the quantity k 2 — 1. Hence, 



«' 2 M -, ^ 



— - = k! z ~ 1 + -5-, we nave 



«r q l 



if-2=4e, 



AT 



and 



k' 2 = k 2 + 



wr 



12a 2 



e'= e + 7-— >• 



5/c 2 + 3 

 5* 2 + 3 



12« 2 (* 2 -l) 2 



On the supposition that (dty) =udx -\- vdy + wdz, we have, as 

 is known, the exact differential equation 



U — " * V //*£ "•" ^,,2 T J <2 / ,7,2 



\ flfa? 



^ 2 



<fy 2 ' ^ 2 / 

 dty d^ d^r d*yfr 



dec dxdt dy dy dt 

 d^d^r_d^ dNf_d^ dhjr 

 dx 2 dx* dy* dy* dz 2 dz* 



9 dyfr d*yjr 

 dz dz dt 



>(A) 



djr dyjr d*yjr dty d^ d*ty » #; df d*ty 



dx dy dxdy dx dz dx dz dy dz dy dz' 



