^{page 244 note *} Or in Labey's French Translation, ii. pp. 370–378.

^{page 254 note *} The fact that these equations imply |αβ′γ″δ′″|= ±k ² is not alluded to.]

^{page 256 note *} Observe A is not the cofactor of a, viz., |β′γ″δ‴|, but

.

I have drawn attention elsewhere to the fact that at this point a passage occurs which contains Jacobi's first printed reference to determinants. The words are “Valores sedecim quantitatum A, B, . . supprimimus eorum prolixitatis causa; in libris algebraicis passim traduntur, et algorithmus, cuius ope formantur, hodie abunde notus est.”

^{page 260 note *} For the modern reader the following substitute for the missing demonstration will suffice:—

If the cofactors of the elements in the four-line determinant above given be denoted by [11], [12],.…, then from the equations

we have

Multiplying in these lines by α, α′, α″α′″ respectively we see that

and therefore that each of them is equal to

and thus equal to

But by the rule for differentiating a determinant the denominator here is the differential- quotient of the determinant with respect to G′ and this because of the theorem

is equal to - (G′- G″) (G′- G′″)(G′ + G): consequently

^{page 267 note *} We know from a later theorem (Jacobi, 1833) that when A₁ is not 0 the identity is

^{page 271 note *} A short account of Cauchy's memoir is given in the Bulletin des Sciences -Math., xii. (1829), pp. 301–303, by C. S(turm), who says, “M. Cauchy a bien voulu observer, en terminant son article, que j'étais parvenu, de mon côté, àdes théorémes semblables aux siens, sans avoir connaissance de ses recherches. Le Mémoire.de M. Cauchy, et le mien, dont je donne plus loin un extrait, ont été offerts le même jour à, l'Académie des Sciences.” A few pages further on in the same volume we come to an article entitled “Extrait d'un Mémoirs sur l'intégration d'un système d'équations différentielles linéaires, présente à l'Academie des Sciences le 27 Juillet 1829, par M. Sturm.” The abstract occupies nine pages (pp. 313–322), and though it does not contain explicit statement of the two theorems referred to by Cauchy, the theorems themselves are evidently implied. There can be little doubt, therefore, that the memoir here condensed is that which was presented on the same day as Cauchy's.. Probably the full memoir was never printed: Professor Gibson of Glasgow, who has kindly made a search, has failed to find trace of it.

^{page 275 note *} In leaving these preliminary deductions, it may be worth remarking that the like results which flow from the second given substitution and its associated condition are not taken entirely ibr granted by Jacobi, but are given with equal fulness, tho two series indeed appearing in parallel columns.

^{page 275 note †} v. next page.

^{page 276 note †} Jacobi writes the nine equations in one column: they are better arranged in three, however. Cayley at a later date would have preferred to write more luminously

^{page 277 note *} Nowadays we should rather put

^{page 288 note *} We may formulate fur use here the following theorem in modern dress: If |αβ′γ″| be an orthogonant, then