OF ARTS AND SCIENCES. 55 



/, ( 9 ) = {~ g r - jr^rY 1 (i^r - g^irY* .... 



(i^gr - g~^~g^- 1 (i^f - g^=~grY r+1 • • • • 



/r (fr) /r far) 



Then the most general expression for the matrix <£ satisfying the 

 equation O </> = <f> 12' is 



4> = So'?" 1 ^ - 9'i)' tl " p - 1 «i Mi fl'i («' - ^) r } 



+ 2o* -1 {(0 - ^)^- r - 1 « 2 M 2 Q", (fi' - ;7 2 )'} 



+ ....+ ^-'{(n - ^) M '- r - 1 n« M o'« («' - SO), 



where M 1} M 2 , .... Mj, are arbitrary matrices. 



If O' = ft, the expression to which this expression for <£ reduces 

 gives the most general matrix commutative with Q. 



If O' = 0, and the matrices M in the above expression for <f) are 

 so taken (as is possible) that the determinant of <f> is not null, then 

 the expression for <f> gives the most general solution of the reduc- 

 tion of the given matrix fi to its canonical form. 



