SIMILAR TO THOSE OF HYDRODYNAMICS. 377 



If we multiply now equations (58) by A n , A n , . . . ., A ln respectively we have by 

 virtue of this relation between the two operators 



d(U-q) _ dW 1 dq 



dd »~ d$ >~Jdt ( G1 ) 



and consequently 



W =U-g = fj§dd + V {6A ■■■■ «L-iO (G2) 



^ being a perfectly arbitrary functional symbol. This might be expressed differently ; 

 since 6 is the only variable function we must have 



dO, ffl dO 



17, **■ + C K ** + •••• + ^ d*« = dd 



cle . ^ , , de, . 



*, dx * + d* 2 ** + •■•• + SE ^ = ° 



from which follows 



W.l 2 ,r 



Substituting the value of f?# thus obtained in the above equation we are enabled to 

 express the integral in terms of a^, .r 2 , . . . ., .r re . In order however to obtain the 

 value of the integral in (62) it will be necessary to express x^ .... x n in terms of 

 6, 6 X , 2 , • • • •, 0,i_i- To this end consider the Jacobian 



_ dfflfl. fl„_i IF) 



^ ~ d (.; r r 2 .... ;•„) ( G3 ) 



Tbe minors of A corresponding to -7- , -pr- . . . . are the same as the minors of Jcorre- 



M d9 _. . ' '-' ,. d6, . 



sponding to ^r, ^r . . . . Ihe minor corresponding to -^r is 



*f 41 + s; 41 + ..•■ + 5^. (64) 



which is again a Jacobian. Denote the principal minors of A corresponding to ele- 

 ments in the first row by A u . A 12 , . . . ., A ln and by A n , A, 2 , . . . ., A*, corresponding 

 to elements in the i th row. Now as we know 



d\ tl dA ia dA,„ 



^ + A^ + - •••+ dx n -°; ( 6 -->> 



