... C.W. Borchardt¹

S. Cohn already devoted himself to this paper of the ill. Jacobi, but taken away by an untimely death he did not left a manuscript ready for printing. [Note non reproduced in the complete works. T.N.]

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... p. 297-320)²

``De investigando ordine systematis aequationum differentialum vulgarium cujuscunque'', reproduced in C.G.J. Jacobi's gesammelte Werke, fünfter Band, herausgegeben von K. Weierstrass, Berlin, Bruck und Verlag von Georg Reimer, 1890, p. 193-216, translated from latin by F. Ollivier (CNRS, LIX UMR CNRS 7161, École polytechnique, 91128 Palaiseau CEDEX, mél. francois.ollivier@lix.polytechnique.fr) with the help of Alexandre Sedoglavic (LIFL, UMR CNRS 8022, Université des Sciences et Technologies de Lille, 59655 Villeneuve d'Ascq CEDEX, mél. sedoglav@lifl.fr). T.N.

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... canonical³

The system called here canonical or in canonical form is the same as one that in ``Theoria novi multiplicatoris'' is said to be in normal form (J. de Crelles tome $29$ p. $369$, cf.. C.G.J. Jacobi gesammelte Werke, vierter Band, p. 501) but it completely differs from the one that Jacobi qualifies of canonical in ``Nova methodus aeq. diff. partiales primi ordinis integrandi'' (J. de Crelles tome $60$ p. $122$, cf. Jacobi, g. Werke, fünfter B., p. 128). B.

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... system.⁴

We recognize here what Ritt called the ``differential analog of Bézout's theorem''. T.N.

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...#tex2html_wrap_inline1605#⁵

This is what we would call Kaehler's differentials: $\delta u = \sum_{i} \partial u/\partial x_{i} \delta x_{i}$, producing the tangential linearized system. T.N.

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... constants⁶

This affirmation stands only in the generic case; however it is far from beeing obvious. T.N.

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... expression⁷

This notations stands for the shape of the expression, not for its particular value. So, the coefficients $A_{i}$ are a priori different in $(\xi)_{k}$ and $(\xi')_{k'}$ appearing in $v_{j}$ and $v_{j'}$ with $j\neq j'$, even if $i=i'$ and $k=k'$. T.N.

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... denotes⁸

Same remark for $[\lambda]_{m}$ as in note 5 for $(\xi)_{m}$. T.N.

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... decreases⁹

Jacobi is here more precise than in proposition II, for he asserts that the order is always less than $H$ if the determinant vanishes. Ritt seemed to doubt that this bound is still valid outside of the generic case, where the determinant is not zero (cf. J.F. Ritt, Differential algebra, AMS, New York, 1950, p. 136), but the example of Ritt (loc. cit. p. 140) takes for hypothesis the order of two components and not the order of the polynomials defining them. Jacobi's bound remains conjectural. T.N.

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... form¹⁰

See ``De aequationum differentialum systemate non normali ad formam normalem revocando'', published by A. Clebsch, C.G.J. Jacobi's gesammelte Werke, fünfter Band, herausgegeben von K. Weierstrass, Berlin, Bruck und Verlag von Georg Reimer, 1890, p. 485-513. T.N.

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... table¹¹

We would probably today say matrix, but the word was only introduced in 1850 by Sylvester, that is about at the time this posthumous paper was written, Jacobi beeing dead in 1851 (cf. Dieudonné, Abrégé d'histoire des mathématiques, Hermann, Paris, 1978, p. 96). T.N.

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... sets¹²

Despite its anachronism, ``set'' has been used to translate complexus. T.N.

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... series¹³

The one containing a maximum in $C$. T.N.

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... series¹⁴

(Containing $N$.) T.N.

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... maxima¹⁵

See above the definition of the third class page .

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...#tex2html_wrap_inline2587#¹⁶

The original text has $f'$ instead of $f$, that does not make sense: a probable typographical mistake. N.d.T.

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