Theory of Canonical Forms and of Hyperdeterminants, 401 

 multiplied by the determinant .^ ,^^ ^^ . _,^,i ^^.^ ^,^ 



&c. &c. -^ 



This last above written determinant may be shown from the 



t(t+i ) 

 method of its formation to be equal to {Ip — mn) 2 ^ ?. e. to \ipj.ty, 

 because ^—m7i=l. Again, since »- V\ 



a^''=zM-\-d'-^m.cv'-'\y + kc. . . . -hm' .y' "^ VfJ. / 



+m''~'^ .py'' 



&c. = &c. \ 



y'^z^n' .a;' + m'~Kpx"-^ . y + . . . -i-p'-y', 



the resultant of (^)'^' • • • (j-)*- ^"^ obtained by treating 



x'j x'~^.y, . . .y' as the eliminables, will be equal to the result- 

 ant of the same functions when w", x'^'^.y,... y" are taken as 

 the eliminables* multiplied by a power of the determinant. , , 



l~^,n; ...; m!'~^.p 



^bifiQ%in ()r\^hikii ^\bi\^ ,4- .t-if/tl' •*■ 



a J » . , f p y 



which determinant, like the last, is unity. Thus, then, we have 

 succeeded in showing that the resultant obtained by eliminating 

 x^, x*-"^ .y, . . .y^ between 



\d^) ' \d^/ dv '"' \dv/ 'ff^m^Qaa&J 



is equal to the resultant obtained by eliminating (a/y, x^''" 

 * y . . . ?/'^ between 



'^ (£rr. (I)- (i)n...(|)'p', •"- 



Or, which is evidently the same thing, the resultant obtained by 



eliminating x'', x''~^ .y . , ,y^ between 



* For the statement of the general principle of the change of the vari- 

 ables of elimination, see my paper in the March Number, 1851, of the 

 Cambridge and Dublin Mathematical Journal. 



