SIMILAR TO THOSE OF HYDRODYNAMICS. 



373 



of Xt , X2 . . . . x n , t, denote the Jacobian of these by J, i. e. 



d9_ d0_ 



dx 1 dx 2 



dA dA 



dx r dx 2 



' dx H 

 ' dx n 



dx„ 



= J. 



(43) 



&-JC-\ &3C2 



The minors corresponding to elements in the first row may be denoted by 



1 



■"11 ■> -"12 J • • • • 1 -"!»> 



and in general for the minors to the elements in the i th row 



-"il > -"i2 >••••) -"iw 



Of course we have now 



<»,-. 



d»i-i 



<»,-i 



J — A * d Xl ' + -^a «&/ + ■••• + 4» &;„ ' 



The minors ^4 satisfy identically the equations 



<*4i . rf-4,2 , 1 dA ln __ 



^r7 + ^r7+ + Hx7 t —°> etC ' 



If we substitute A u A n .... J. lB for n 1} w 2 • • • • w„ in equations (38) we 



at once 



*=■■ DA, 



dU=%^dx } 



or 



dET= 2 ~w dx J +22 A lk ^r dxj. 



} = 1 k — l j = l k 



Introducing the quantity 2 q which is now 



— : A2 _|_ J 2 I J. J! 



(47) becomes 



(44) 



(45) 

 obtain 



(46) 

 (47) 



_ j ^dA v 



d{U—q) = ^-^ dxj + 2 2 A ik 



j=l k—lj=l 



dX; 



For brevity write 



fc = rc 



Kj = 2 A u 



k = l 



d_ d_ 

 dx k dxj 



Au;Aij 



(48) 



(49) 



Assume a set of functions n — 1 in number 



vol. x. 



<J>!,<J> 2 , * K _l 



48 



