52 PROCEEDINGS OP THE AMERICAN ACADEMY 



Therefore, if e l5 e 2 j etc, severally denote the matrices formed from 

 7] by erasing successively all its constituents except, first, those in 

 the square rji, second, those in the square -q 2 , etc., then 



V — ci + c 2 + > 



and 



4 = «i2i? [r, r] + ftiSiV 1 [r, r-+ 1] + c^i"*" 2 [r, r + 2] 



+ 4AV [r, r + 3] + ei 2iV 4 [r, r + 4] + • • • • 



€ 2 = a 2 Si" O + r, m + r] + JjSw 1 [m + r, m + r + 1] 



+ c 2 ^i"7 2 [«» + r, m + r + 2] + d 2 Si" 7 3 [m + r, m-+ r + 3] 

 + e 2 Si"7 4 [m + r, m + r + 4] + • ■ • ■ , 

 etc., etc. 



But by what has just been proved it follows that 

 /i \9v 



e 2 = ^M-{a 2 + b 2 (o-r j2 ) + c 2 (e- g2 y+d,(e-9 2 y+e 2 (o-g i ) i +.. . .}, 



/ 2 \9v 



etc. Consequently 77 is a rational integral function of 6. Let 

 rj=f(0), where f{6) denotes the most general rational integral 

 function of 6 ; then, since 



w ./(0). or 1 = /<Q), 



the most general matrix commutative with fi is 



cf> = urjw- 1 = w. f{6) . ca- 1 = /(fi). 



Worcester, Mass., January 2, 1892. 



