^{page 33 note *} Lagrange. Réflexions sur la résolution algébrique des équations. Nouv. Mémoires…Acad…. Berlin, 1770, 1771: or Œuvres complètes, iii. pp. 205–421 (227–229).

^{page 36 note *} With Cayley the assertion

included 6 equations, whereas with Trudi it only includes 3, namely, the first 3 of Cayley's 6: and with Cayley the assertion

was meaningless, whereas with Trudi it includes 2 equations. Since in the former case Trudi's 3 equations are known to necessitate the other 3, there is clearly no good reason for refusing to profit by the new usage. What is common to any two arrays which Trudi may equate is the excess of the number of columns over the number of rows: and evidently if his excess be δ, the number of included equations is δ + 1.

^{page 39 note *} See under Recurrents.

^{page 39 note †} A most natural and helpful notation for such a remainder would be

Thus, in the case here used for purposes of illustration, the remainders would be written

^{page 42 note *} It is worth, noting that it was in this connection that the word “syzygetic” was first used, the full title of the memoir of 1853 (which clearly had considerable influence on Trudi) being “On a theory of the syzygetic relations of two rational integral functions, comprising an application to the theory of Sturm's functions, and that of the greatest algebraical common measure.”

^{page 44 note *} See Art. 5 of “On a theory of the syzygetio relations…”

^{page 46 note *} Crelle's Journ., xxviii. p. 269.

^{page 51 note *} Crelle's Journ., XV. (1835) p. 108, where however m = n.

^{page 52 note *} Gordan (1870) in quoting the two from Baltzer says that mn of the. primary minors of the former eliminant are secondary minors of the latter. (Math. Annalen, iii. p. 356.)

^{page 54 note *} This is in accordance with the statement in § 13 of the complete memoir, and is some what different from that first published.