1856.] 



and on Partial Differential Equations, 



309 



where R and V are fiuictions of p and v only, we |liave for our equations 

 of motion the following, viz. 



'dt'^Bdi li)di' 

 Q_6/y _j_D dp 

 dx ^ dt 



of which the following pair of equations constitute the solution, viz. 



fM=1> 2D J ' 



where /^(p,u) = const., and//|0,w) = const, are the respective integrals of 

 the ordinary differential equations, 



^^p = 0, - 



^^„y+VvH:4iv 



2p' 



which involve the variahles v and p only. But this result is dependent on 

 the fact of the following conditions heing satisfied, viz., that we have 



dp dv 



' dp ' dv 



where 



K,=V- Vv- + 4Rp^ 



K, =y+ VVH4Rp^ 

 which conditions cannot be satisfied if the law of pressure depends upon 

 the density only (in which case V=0 and R contains p only), as may easily 

 be shown. 



With regard to the theory of Partial Differential Equations, I conceive 

 that the methods indicated in the Paper will serve to elicit every solution 

 of a partial differential equation of the second order and fi.rst degree, save 

 one, viz., an integral solution consisting of a simple relation between the 

 variables x, y, z, free from arbitrary functions, and which is not derivable 

 from a solution containing arbitrary functions, by assigning particular 

 values to the latter. 



If the equation 



0=R^-f-S-^ + T— ;-v 

 dx^ dxdy d^/ 



have a first integral consisting of the pair of equations 



¥{xyzpq) = {/{xyziHi)} , 



Y^ryziKi) = xl, {/(xyzjiq)}, 



