364< Prof. Challis on the Vibrations of an Elastic Fluid. 

 at the same time the equation (A'.) becomes 



J 'dP \ J ' dz*) J 'dz* + J *«fe dswfc 

 Hence substituting the above value of ^-+ vr-,, we have 



f ** ( a * ,**£}/ ** l2f *< 1 *, d *<? 



J '<pdF \ J 'dz*J J '<pdz 2 + J 'dz fdzdt 



~ ' \A& T df ) f ' W? + rfz/V* 



In this equation must be substituted for the factors involving 

 <p, the several values which they have when <p = 0. These may 

 be obtained from equation (I.), which clearly shows that when 

 <P = 0, 



q(z-a't-Yc)= |. 



This process having been gone through, the following is the 

 resulting equation for determining/: — 



~ ?• V^^^-lW 2 - 1 ))=° j 



It is now clear that where /^O, we have also -— = and 

 j- —0. The equations are consequently consistent with non- 

 divergence of the vibrations. 



The equations (1.), (2.) and (3.), define the motion and 

 propagation of the motion of the vibrating particles. To com- 

 plete the theory of non-divergent vibrations, an expression for 

 the density remains to be obtained. This I reserve for a future 

 opportunity. 



Cambridge Observatory, 

 Oct. 21, 1848. 



