56 PROCEEDINGS OF THE AMERICAN ACADEMY. 



/— ^ d_ T ^ an djj d£,\ du~\ 

 /7 ^ _9_ f^ _Oki_d$i /^ 6u dg/X"" 



r /I X A T^ _aki_&U du~ 

 ~ JVA i<dt i lf l JVAdz k dx l ] 



t*j~a ^ T— — f afeZ — M 



^ r du ^ f d% dxA~] 



ttl akl dxi^\dx i dx k di i )j 



The first half of this expression 



,— ^ d f aw du \ 1 ^ du dJ 

 ^dx k \ */A dxi) J £{ kl dxi dx k 



Now 



1 ^ du dJ 

 J ~f dxi dx k 



dJ_ = y d% j__ 



dx k ~~ i?j dXjdXk %1 ' 



if J a be the cofactor, in J, of — - . Further 



dxj 



J% > ~ J a& ' 



so that we have finally 



A 2 M = A 2 w — "> a^r— ^ — :r~ z^ + the same quadruple sum; 

 i.tt.l dxi dxjdx k d$i 



that is, 



A 2 w = A 2 w. Q. E. D. 



