4M Mr. J. J. Sylvester on a remarkable Modification 



(the mode of formation of which is self-apparent), these succes- 

 sive coaxal determinants will be 



1^1 A, 



Ai 1 

 1 A, 

 

 

 





 ^10 



1^3 1 



IA4I 

 1 A^ 



t. e. 



If Ajj AjAj""" 1, AjAgAjj"- Aj-~* Ag, A^A^AgA^'^'AjAg'^'AjA^ 

 — A3A4 -f- 1, AjAjAgA^Ag AjAjAg Aj A^Ag — A3A4A5 — Aj A^Ag 



H-Ag + Ag + Al. 



It is proper to introduce the unit because it is, in fact, the value 

 of a determinant of zero places, as I have observed elsewhere. 

 Now I have demonstrated directly in this very Magazine (August 

 1852), under cover of the umbral notation, that the signaletic 

 value of a regularly ascending series of principal coaxal deter- 

 minants formed from any symmetrical matrix is unaffected by 

 any such transposition whatever of the lines and columns of the 

 matrix as does not destroy the s)rmmetry about the principal axis. 

 Hence, then, beginning from the lower extremity of the axis A5, 

 and reading off the ascending series of coaxal minors from that 

 point, we obtain the reverse series, 



1> A5, A5A4— 1, A5A4A3 — A5 — Ag, A5A4A3A2 — A5A4 

 A5A2 AgAg +1, A5A4Ag AgAj A5A4Ai AgAgAg AgAgAi 



-A5A4A3+A1+A3+A5. • 



Hence we see that the denominators to the convergents of 

 ] 



s; 1 



4__ 



A/ 



beginning with 1, form a series signaletically equivalent to that 

 similarly formed from the fraction 

 1 



A5.I 



A4^1 



As^l 



lO'l jbiiX 



A._l. 

 A,' 



