^{page 153 note *} There is a curious oversight here. In a footnote, Cauchy says “Le Mémoire dont il est ici question a été imprimé en partie dans le xvii^e. Cahier du Journal de l'Ecole Polytechnique.” Now, as a matter of fact, there is no memoir bearing this title. The well-known memoirs contained in Cahier xvii. are headed “Mémoire sur le nombre des valenrs ….” and “Mémoire sur les fonctions qui ….” The second part of the latter, it is true, bears the approximate designation, “Des functions symétriques alternées ….”; but the notation in question occurs in both parts.

It is also not clear what was intended by the words ‘imprimé en partie’ in Cauchy's footnote.

^{page 158 note *} In later notation the derived equations would of course be written—

^{page 162 note *} I may state in passing that in 1869 when lecturing on the subject I found it very useful to write

in place of

and then indicate the number of times the function had to be differentiated with respect to any one of the variables by writing that number on the opposite side of the vinculum from the said variable; thus

meant the result of differentiating once with respect to x, thrice with respect to y, and twice with respect to z. Using this notation to illustrate Jacobi's example, we see that if it were given that

we should have

but that if it were given that

then we should not be certain as to the meaning of as it would stand for

according as u, or y was to be considered constant.

^{page 167 note *} Using this theorem upon itself we have

provided that on the right f is expressed as a function of x, f ₁,f ₂,…, f_n, and f ₁ as a function of x, x ₁, f ₂, …,f_nand ultimately

provided that in every instance on the right fk is expressed as a function of x, x ₁, x ₂, …, xk, f_k+1,…,f_n.

A theorem like this ultimate case Jacobi enunciates and proves quite independently at the end of his memoir (v. § 18). The one, however, is seen to include the other if we note the simple fact that

^{page 183 note *} A curious interest attaches to this result. On tha right-hand side are two determinants whose elements are differential-quotients; but the first, B, being a minor of the second, the total number of different elements is simply the number in the second determinant, viz., (n + m + 1)². On the left-hand side is a compound determinant of the (m+l)^th order, each of whose elements is a determinant of the (n + l)^th order; nevertheless the number of different elements is again (n + m + 1)² and not (m + 1² (n + 1)², because all the (m + 1)² elements of the compound determinant have the n² elements of B in common, and of the 2n + l which border these n ² elements, only one, viz., the cofactor of B, is different throughout, each of the 2n others being repeated m +1 times, so that the total number of different elements is n ² + (m + l)² + 2n(m + 1²) (m + 1). Further, on both sides the degree in these (n + m + 1)² differential-quotients is the same, being clearly (n + 1) (m + 1) on the left, and mn + (n + m + 1) on the right. It is thus at once suggested to us that the identity is not necessarily an identity connecting differential-quotients only, but is true of any (n + m + 1)² elements whatever; and the suggestion is readily verified when the ubiquitous presence of B as a coaxial minor raises the suspicion that the identity must be an ‘extensional.’ The case where n=2 and m = 3 is given on p. 215 of my text-book (Treatise on the Theory of Determinants) in the form—

where it is viewed as an extensional of the manifest identity

The theorem in its general form may be enunciated as follows:—If from the determinant | a1,n+m+1 | there be formed all minors of the (n + l )^thorder which have | a1n | for the cofactor of their final element, and these be orderly arranged in square array, the determinant of this square array of the (m +1)^thorder is equal to

^{page 186 note *} The fact that these identities can be derived in the way here indicated from another which the preceding footnote has shown to be true, not merely of functional determinants but of determinants in general, is convincing proof that they also (i.e., the derived identities) are not restricted to any special form of determinant. Using the fundamental identity as enunciated in the footnote, and taking the special case of it where n=4 and m=2, and where therefore the given determinant may be written | a₁b₂c₃d₄e₅f₆g₇ |, we have

Now in each of the determinants forming the fmt row on the left here, e occurs as an element, in the second row f ₂ similarly occurs, and in the third row g ₃, while on the right these only occur in | a₁b₂c₃d₄e₅f₆g₇ |. Consequently, equating cofactors of e₁f₂g₃ we have

which when put in the form

is a case of the first derived theorem.

The original theorem, it should be noted, is true for all values of n and m; the derived holds only when m<n,—in fact, if we do not, in seeking to obtain the latter, take m<n in the former, we shall fail in our aim. Thus, taking n = 2=m in the former, the given determinant being | a₁b₂c₃d₄e₅ |, we have quite correctly

but while c₁ occurs in each element of the first row on the left, and d₂ similarly in the second row, e₃ does not so occur in the third, and consequently the cofactor of c₁d₂e₃ on the left takes a different form from that given by Jacobi.

The first derived theorem in its general form may be enunciated as follows:— If there be two determinants D and Δ of the n^th order such that the last n – m columns of D are the same as the first n – m columns of Δ, and if there be formed a square array of new determinants by supplanting each of the first m columns of D by each of the last m columns of Δ, the determinant of this square array of the m^thorder is equal to

To illustrate the second derived theorem we may equate cofactors f₁ g₂where we formerly equated cofactors of e₁f₂g₃ the result clearly being

The next of the series would be got by equating cofactors of g₁

^{page 187 note *} This is not the same as putting, with Jacobi,

for the determinant on the left being of the (n + l)^th order there should be n + 1 terms on the right instead of m +1.

^{page 189 note *} Of course tins theorem also is not limited to determinants having differential-quotients for their elements. The general enunciation may be put as follows:—If m determinants of the n^thorder all have the same n – 1 columns in common, and vanish independently, then every determinant of the n^thorder whose n columns are chosen from the m + n − 1 different columns must vanish likewise. (Vide Proc. Roy. Soc. Edin., xviii. pp. 73–82.)

^{page 189 note †} This proposition, and that from which it is derived, are again propositions which hold regarding determinants in general, the class to which they belong being that which concerns aggregates of products of pairs of determinants,—a class, the first instances of which occur in Bezout (1779). In connection with Jacobi's remark regarding the case where m = n, it is worth while to note Sylvester's enunciation in Philos. Magazine (1839), xvi. p. 42.Google Scholar

^{page 191 note *} No indication is given of where Lagrange published the theorem attributed to him. As for the particular way in which x is given as a function of x ₁, x ₂, …, x _m+1, whether by the equation f=0 or not, it is clear that this cannot affect the truth of the result, because the latter contains no f at all, the requisites for validity being (1) that f ₁, f ₂, …, f _m+1 are functions of x, x ₁, x ₂, …, x _m+l; (2) that x is a function of x ₁, x ₂, …, x _m+1; (3) that on the left the f's have by substitution been freed of x before differentiation; (4) that on the right this has not been done, but they are there differentiated as if x were a constant. The subsidiary result

is certainly true whether f = 0 or not, all that is requisite (see p. 176 above) being that f _i+1 on the right shall be what f _i+1 becomes by substituting for x its value in terms of f, x ₁, x ₂x _m+1, and that in the differentiation of it f shall be viewed as constant.

^{page 192 note *} A combination of the successive substitutions is impossible, by reason of the fact that in the second case the equation

is not meant, as in the first case, to include the definition of a _(i+1), which has thus to be defined by a supplementary equation.

The result of the first substitution is very noteworthy, in view of previous footnotes.

^{page 195 note *} The memoirs referred to seem to be—

Euler, L.—De formulis integralibus duplicatis.… Nov. Comm. Acad. Petrop. (1769), xiv. i. pp. 72–103.

Lagrange, J. L.—Sur l'attraction des sphéroïdes elliptiques. Nouv. Mim. Acad. … Berlin (1773), pp. 121–148; or (Euvres, iii. pp. 619–658.

Equally worthy of note is one of date December 1839Google Scholar, but not published till 1840, viz.—

Catalan, E.—Sur la transformation des variables dans les intégrales multiples. Mém. couronnés par l'Acad. de Bruxelles, xiv. (2) pp. 1–47.