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



I proceed now to state the modification which the reasoning 

 in the question under consideration receives by using equation 

 (I.). The equation (A.), cited in my last communication from 

 the Mecanique Analytique, was derived from equation (II.), 

 and from the following three equations, 



a 2 , dp (du\ a*, dp (dv\ a 9 , dp (dw\ 



~P~dir + \di)- ' -JdJ+KTt)-^ ~p~dT + \dtr 



the supposition being at the same time made that {d.<pf) 

 ssudx + vdy + ivdz, <p being a function of z and t only, and^a 

 function of x and y only. The analogous equation obtained 

 by using (I.) in the place of (II.) differs from (A.) in only one, 

 but that a very important particular ; viz. that the variation of 

 the co-ordinates must be from point to point in the direction of 

 motion^ whilst in (A.) x, y, z are independent variables. In- 

 troducing that limitation into (A.), and putting ds for the in- 

 crement of space along the line of motion, the equation becomes 



d M -(*- l d Jl\*\<N± , 9 d^ dy^i 



df* \ \ ds ) ) ds* "\ ds ' dsdt , , A ,* 



d.fp/1 1\ 

 - a --dT\R + R')'~ 0> j 



and at the same time, by analytical geometry, 



(L ±\(Vl + Vl+£ *£\*-£f d Il ,<EL d ll 



\R + R7 W dy 2 + f ' dz 2 / dy 2 ' dx 2 + dx 2 ' dy 2 



; J d& 4f df ^^(df 4?A 



dxdy dx dy <p 2 dz 2 \dx 2 dy 2 / 



+ \<p ' dz 2 f ' dz 2 ) \dx 2 dy 2 )' 



Now since 



u=< 



S *=?■%> «**=/■% 



it follows that, corresponding to the maximum value of/" 

 (which we have supposed to be unity), 



u— 0, v=0, and w= -p-. 

 dz 



The motion is consequently parallel to the axis of z, and for 

 this line of motion the above equations give 



dt 2 V dz 2 )dz^ +Z dzdz4t a, ^VR + R7~°' 

 (I \\ d^__d 2 f cff 



\R + R'/'<pdz~ dx 2 + dy 2 ' 



