1883.] Mr M. J. M. Hill, On some General Equations. 23 



(4) On some General Equations which include the Equations 

 of Hydrodynamics. By M. J. M. Hill, M. A. 



[Abstract.] 



1. If u x u 2 . . .u n p be (n + 1) functions of x x x r . .x n t which satisfy 



(a) the n equations which may be obtained by changing r 

 successively into 1, 2, S...n in 



du 



du r 



du 



du r _ d / fdp 



dt 1 dx, 2 dx„ ' ' n dx„ 



dx r \J p /' 

 and (/3) the equation 



dp d , v . a , v d . N _ 



dt dx 

 it is proved that 



dx n 



dx„ 



u 1 dx 1 + u 2 dx 2 + ... + u n dx n 



= dK+f 1 dP 1 +f 2 dP,+ ...+f n dP n , 

 where K satisfies the equation 



_f#_ F , < + <*+•-. +u * 

 J P 2 



fd d d d \ Tr 



= {dt +U ^ 1 +U ^ 2 + ''^^dx n ) K ' 



and where P t P 2 ... P n are n independent integrals of the equation 



#\„, *f 



df 



if 



= 0: 



(JUL (Auk* \JjOC~ iX/OCr 



and/^ •••/ 3 are arbitrary functions of P 1 P Z ... P„. 



2 ' If &• = i^~ai and if ^ 2 "" ^» be Square r °° tS ° f the 



s r 



coefficients of % n i; 22 ... % nn in the determinant 



SlM5» 



