282 WILSON AND MOORE. 



Multiply by Crt and sum over t. The double sum S,t on the left reduces 

 to the single term ark by (12), and we have 



ark = ^)3t^ikCisCrt(o.ts)- 



Now j = k alone contributes something. Hence, finally, 



If we compare this with (10), we see that the transformation of the a's 

 is contravariant, and the theorem is proved. We shall therefore in 

 conformity with our general notation use upper indices and indeed 

 write 



an = a(''^ 



for the cofactor of an in (8) divided by the determinant \aij\. 



If X° = Z(i), Z(2),. . ., A"(") be the notation for a contravariant 

 system, the results of this article may be written 



(X°) = M-i.X°, X° = M-(X°), (X°Y°) = M-iM-i:X°Y°, etc. 



If A be the matrix Ilaji||, and I the idemf actor we have further 



A-A-i = 1, A = (A):M-iM-\ [(A):M-iM-i]- A-i = I. 



Now if [C:MN]-D = I, then C[NM:D] = I. For 



[C:MN]-D = Mc-C-N-D, C-[NM:D] - C-N-D-Mc, 



Mc-C-N-D = I, C-N-D=Mc-S C-N^D-Mc = I 



Hence [(A):M-iM-i]- A-^ = (A)-[M-iM-i:A-i] = I 



or (A-i) = M-^M-i:A-\ A-^ = MM:(A-0. 



This analysis could, of course, be carried out without the conversion 

 of the double products into simple products ; the conversion has been 

 used because it may seem simpler to those familiar with matrices 

 (products with a single opening only). 



