^{page 358 note *} The two sets of equations are

and

The former is substantially the interpolation-problem which goes back to Newton, and which may therefore for distinction's sake be associated with his name: the latter being first found solved by Lagrange (Recherches sur les suites récurrentes … Mém. de l'acad. de Berlin, 1775, pp. 183–272; 1792, pp. 247–299: or Œuvres complètes, iv. pp. 149–251; v. pp. 625–641) may be called Lagrange's set, provided we remember that he also gave a solution of the other. The first to deal with both of them in more or less general form by means of determinants was Cauchy (1812): but in saying so a mental reservation must be made in view of Cramer's mode (1750) of continuing Newton's work.

^{page 358 note †} Newton, , Principia, lib. iii.Google Scholar lemma v.: also Arithmetica Universalis, probl. lxi. Lagkange, , Journ. de l'éc. polyt., ii. cah. 8, 9, pp. 276Google Scholar, 277: or Œuvres complètes, vii. pp. 285, 286. Cauchy, , Journ. de l'éc. pclyt., x. cah. 17, pp. 73Google Scholar, 74: or Œuvres complètes, 2^e sér. i.

^{page 361 note *} See also Crelle's Journ., xxv. pp. 178–183 (1842), and Grunert's Archiv d. Math. u. Phys., xxiii. pp. 235, 236 (1854).

^{page 362 note *} A still better form for the right-hand number is

^{page 363 note *} Not the posthumous book with this title edited by Prouhct and published in 1857.

^{page 364 note *} Sylvester, . On rational derivation from equations of existence …‥ Philos. Mag., xv. (1839), pp. 428–435Google Scholar: Collected Math. Papers, i. pp. 40–46.

Sturm, . Démonstration d'un théorème d'algèbre de M. Sylvester. Journ. (de Liouville) de Math., vii. (1842), pp. 356–368.Google Scholar

^{page 365 note *} It may be noted in this connection that

if φ (x) = (x − a ₁) (x − a ₂) … (x − a _n)

^{page 368 note *} The product φ(a₁)·φ(a₂)…φ(a_n–1) is arrangeable as a square array of binomial factors, being in fact, save as to sign, the product of all the denominators in the double alternant, and is thus seen to be symmetrical with respect both to the a's and to the t's. If therefore we multiply each row of the alternant by the product of the denominators of the row, or each column by the product of the denominators of the column, we multiply the alternant by (−1)^{(n−1)(n−2)}φ(α₁)·φ(α₂) … φ(a _n−1). The two determinants thus resulting have elements which are the product of n − 2 binomial factors, and are equal to

If, on the other hand, we multiply each element of the alternant by we obtain the product of the two Π's as reached by the ordinary multiplication-theorem of determinants.

^{page 372 note *} In this connection papers by Cayley (1846) and Borchardt (1845) are referred to, but no mention is made of Sylvester's (1839).

^{page 381 note *} Previous suggestions of such a determinant appear in Binet's paper of 1837 and Joachimsthal's of 1854.

^{page 382 note *} See footnote to page 365.