58 PROCEEDINGS OF THE AMERICAN ACADEMY. 



where the dx's and Bx's are two independent sets of differentials, will 

 be an absolute covariant. 



Proof. Consider an expression ^? CijCLxfixj; and let it go over by 



i, i 

 our change of variables into ^ Cijd$iHj- Then for the c's we readily 



i,i 

 calculate the formula, 



= 2 <* 



J V1 



dxi dxj 



Now the coefficients, — — , in (49) above are transformed co- 



dxj dxi 



grediently with the c's. For we have, from (48), 



■^ d 2 Xi . ^ dxi dxj dli 

 ^7 dip d£ q l ^ dip d% q day' 



-^ d 2 xi . ^ dxi dxj dlj 

 i* di p di q ij^p dS q dxi 



dk dlj \ dxi dxj 



dip diq 



f*. \dXj dxi) 

 as asserted. Hence it follows that just as we have 



^ Cijdifiij = ^ Cijdxi&Xj, 

 i, i i, i 



so also we have 



2(g-t)^ = 2(|-g>M. q.e. d . 



Now the covariant (49) is identical with the expression (43) which 

 we wish to prove a covariant. To establish this identity we need 

 merely to obtain the explicit form of (49). From formulas (47), (45) 

 we get 





