340 Prof. Challis an some Theorems in Hydrodynamics. 



If the rate of propagation be supposed equal to a constant a 1 , 

 we obtain by integration, 



Vp = of(p-l) + 



<t>M 



Since m varies as the product R R', this equation may be written 



V,=^- !) + £$• 



We have thus arrived at a general relation between V and p on 

 the hypothesis of uniform propagation of the kind above enun- 

 ciated. It will be seen that if /o = l, the expression for the 

 velocity coincides with that obtained for an incompressible fluid, 

 in which no propagation can take place. Propagation implies a 

 change of density. 



We are now prepared to obtain the general equations which 

 arc the object of this research. By differentiating the last 

 equation, 



d-Vp _j dp 4>(t)( 1 , 1 \ 

 «!Il rfR RR'\R + R7 



.^ + (^-i)-v,)(i + i). 



Hence by substitution in the general equation of Prop. VIII. 

 given above, we have 



Introducing now the condition that the motion is rectilinear, 

 this condition having been obtained in a general manner, and 

 taking z any abscissa along the line of motion, we shall have 

 dz=dR=dW. The above equation now admits of being exactly 

 integrated, giving 



p RR' ' 



Hence 



T~ RR' ^RR'' 



As these equations have been arrived at anterior to any sup- 

 posed case of motion, it may be concluded that they are express- 

 ive of laws of the mutual action of the parts of a compressible 

 fluid. They depend, it is true, on the supposition that the rate 



of propagation ^ is a constant quantity. But this supposition 



is justified by leadrag through exact integration to definite ex- 



