relating to " Canonic Roots" 



457 



The Resultant of this matrix means the quantity R which, 

 equated to zero, will indicate the possibility of the simultaneous 

 nullity of all its first minors, so that R will be the factor common 

 to the resultants of every couple of these minors. If we name 

 the columns of the matrix taken in any arbitrary order C,, C 2 . . . C„, 



C/i_i C n , it 



O n _i C n , o 2 , 



a 



and call R' the resultant of Cj C 



R' 



may readily be made out that ..y is equal to a power of the deter- 

 minant obtained by suppressing the uppermost (or x) line of the 

 rectangular matrix C 3 . . . C w _! C n . 



To find R, we may proceed in the general case in the manner 

 indicated in the example following, where n — 1 is made 4. 

 Taking the two extreme first minors and dividing them respect- 

 ively by y and x, we have two equations of the following form 



r determining R, viz. 













y 3 y^x yx 2 x 3 





y 3 



o 



y-x 



yx 





a b c d 

 a! b ! d d' 



=o, 



b 



c 



d 



d 



d r 





a" b" c" d" 





b" 



c" 



d" 



By rejecting, as we have done, the factors x and y from the 

 above equations, certain factors, it is true, are lost to their result- 

 ant (R ; ) ; but it will easily be seen that these factors are each of 

 them powers of one and the same determinant, viz. the deter- 

 minant 



bed 



V 

 b" 



d' 

 d» 



and that their product is contained in the irrelevant factor 



itself a power of that determinant, as above explained. 

 R, we may write down the oblong matrix 



R' 

 IV 



To find 



u 

 v 

 w 



yx 

 c 



d 



d! 



d" 



and make its three first minors respectively equal to u, v, w, i. e. 



yx x l 

 c' d' 

 d" 



,n 



b" 



yx x' 



,H 



d" =V. 



yx x' 

 c d 

 d d' 



w; 



then we shall obtain the equations following, of which the inter- 

 mediate ones result solely from the equations last assumed, but 

 the first and last from those combined with the original two 



