PEOFESSOE CATLET ON THE THEOET OF MATEICES. 



29 



and hence, from the foregoing values of (T— 0)(T+n)~' and (T+fl)(Q— T)~', we 

 find 



n=-o-'(T-nxT+n)-'a= -i+2d, 2c, 



2h, -l+2g. 



21, 

 2p, 



—25, -2a ) 

 -2/, -2e 

 2k, —l—2j, —2i 

 2q, ^2n, -l-2m 



and 



II-'=-Q-'(T+Q)(T-0)-'Q=( -1— 2m, -2i, 



-2n, -l-2j. 



2e, 2a ) 



2/ 2b 



-2o, -2k, -l+2g, 2c 



-2p, -2Z, +2A, -l+2d 



which shows that the matrix 11 for the automorphie transformation of the function 

 j[:w'-\-yz'—zi/ — wa/ is such that writing 



II=( A, B, C, D ) wehaTOn-'=( P, L, -H, -D ) 



E, F, G, H 

 I , J , K, L 



M, N, O, P 



O, K, -G, -C 







B 

 A 



which is the theorem in question. 



12- I remark in reference to the foregoing proof that writing 



T=( a, h, g, I ) 

 h, b , f, m 



l^ m, n, f 



tlien the actual value of 



(T+Q)-S =( a 

 h 



h , g , l-l )-« 



_ b , /—I, m 



g , /+1, c , n 



l+l, m , n , d 



IS 



1( 



"A 



lI-\-n-v , B+c , F-f+K, M-g+ff 

 G— m+f*, F-/-X, C+b , N+A+r 

 li—l—g , M—g—ff,. N+^— r, D+a 



