320 



LIPKA. 



a(ax,//0 fdQidf, 



dqi \dUrdUk 

 = 0, (r,^-= 1,2, ...,n- 1). 



Making the further hnear transformation 

 (98) j; = S ax, in 



equations (97) take the simple form 



(99) 2 ^(^^^-^^Ws^l^^^-^^^ =0, 

 /M dxi \durduk dukdurj lyi dqi \durduk dukdurj 



ir,k= 1,2, ...,n- 1). 



Here the form of J' is to be determined if the /'s are arbitrary 

 functions, the only relations between the /'s and the Q's being the 

 identities 



(89) 

 (90) 



2Q^f^ =0, (r= 1,2, ...,n-l); 



u OUr 



- (j^f" - ^-'f") = °' (-■,*■= 1,2,. ..,«-l). 



fi \dUkdUr OUr OUr J 



We note that equations (99) and their accompanying identities (89) 

 and (90) have the same form as equations (41) and their accompanying 

 identities (33) and (34), with i replaced by n, F by J', and P by Q. 

 The form of F is determined by (80) and (85), and we may therefore 

 immediately write down the form of J', as 



(100) 



J 1 — 01 _ J2 — 02 

 91 92 



J i — <i>i 



Qn 



where the 0's are partial derivatives of a single function L(xi, X2,... Xn), 

 or 



dL 



(101) 



0i = 



dXi 



(i = 1, 2, . . .,n). 



