SH6 Prof. Cliallis's Determination of the Velocity of Sound 

 and 



udx + vdy + 'wdz=<^ [£ dx+ dy^yj^^'d'z'^''' 



It hence follows that the left-hand side of the last equality is an 

 exact differential. We have now to trace the kind of motion 

 resulting from this general hypothesis. 



For this purpose I shall assume, for the present, that the 

 supposition by which uda; + vdy-\-'wdz was made integrable, 

 holds good for exact values of w, v and w, although the proof 

 of the integrability of that quantity extended only to terms of 

 the first order of approximation. Thus we shall have the 

 following seven exact equations : — 



»=^l <'•> 



^-4 ^'-^ 



»=/| («•) 



:^ ^. ^ + iifl + '^'P'^ =0 . . . (4.) 



dt dx dy dz 



,., + ©=" («•) 



"^ + (7:)-' (^•' 



with an eighth deducible from the first three and the last three 

 by integration, viz. 



By means of equations (1.), (2.), (3.) and (8.), u, v, w, 



—4-, -4-, — T-» and —i-. may be eliminated from (4.), and 

 pdt' pdx* pdy' pdz* ^ ^ ' 



the resulting equation is 



