^{page 407 note *} The 7th and 8th lines of p. 486 have unfortunately been transposed by the printer. Also, in the first determinant of the footnote on the same page the first b ₁ should be b ₀.

^{page 407 note †} Proc. Roy. Soc. Edin., xxvi. p. 362, pp. 364–366.

^{page 408 note *} By making the observation that the v's are neatly expressible as determinants the whole matter may be put much more simply. Thus, taking the case where u = (β ₀+β ₁ x)/(α ₀+α; ₁ x+α; ₂ x ²), we see at a glance that

and therefore from the data that

or, what is the same thing,

From these two equations on solving for α; ₀: α; ₁: α; ₂ and substituting in α; ₀ + α; ₁ x + α; ₂ x ² we obtain

^{page 410 note *} Continuing the case of the previous footnote we should prefer to begin with

and then proceeding exactly as before we should arrive at

The function sought would then be

^{page 411 note *} The proposition Borchardt is concerned with is of course that The equation f(x)=0 has as many pairs of imaginary roots as there are changes of sign in any one of the three series mentioned.

^{page 412 note *} Proc. Roy. Soc. Edin., xxv. pp. 939–942.

^{page 416 note *} The term “invariant” is first used in this paper.

^{page 417 note *} “Meicatalecticizant,” Sylvester truly says, would have been the more correct word, but even he took alarm sometimes.

^{page 417 note †} The name would have been equally appropriate for the determinants of the preceding paper which have ν in their diagonal.

^{page 420 note *} A preferable form, because making the “catalecticant“ still more prominent, is

Similarly an odd-degreed function may be represented so as to bring the “canonizant” into prominence: for example may be written or

^{page 424 note *} Brioschi (and afterwards Baltzer, § 12, 9) would have done better to change into the similar form of the seventh order, for then the result would have been

in which the determinant on the right has a simpler law of formation than Brioschi's and yet is readily reducible to the latter, and which, as we see on putting n=4, has the further merit of showing that δ _n equals the dialytic eliminant of f(x)=0, f(x)=0.

^{page 427 note *} Brioschi unfortunately neglects the sign-factor. See Proc. Roy. Soc. Edinburgh, xxiii, p. 132, where the footnote might have made mention of the fact that the identity had already appeared in one of Cauchy's own memoirs of the year 1813. (See Journ. de l'ec. polyt., x. cah. 17, p. 485.)

^{page 428 note *} Cayley, A. Mémoire sur la forme canonique des fonctions binaires. Crelle's Journ., liv. pp. 48–58, 292: or Collected Math. Papers, iv. pp. 43–52. If “lamdaic” be not used as a noun, “lamdaic canonizant“ would be better than “lamdaic determinant.”

^{page 430 note *} It is better to note at this stage that the determinant is the product of

and therefore is equal to