«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 (PDF version.)
About the research of the order of a system of arbitrary ordinary differential
equations
« De aequationum differentialum systemate non normali ad formam normalem revocando », reproduced in C.G.J. Jacobi's gesammelte Werke, fünfter Band, herausgegeben von K. Weierstrass, Berlin, Bruck und Verlag von Georg Reimer, 1890, p. 485-513. (PDF version.)
De la réduction en forme normale d’un système non normal d’équations différentielles. (PDF version.)
The complete works of Jacobi are available on the web site of Bibliothèque Nationale de France.
The manuscripts from which came these two publications are kept in the archives of the science academy in Berlin (Jacobis Nachlaß, Archiv der Berlin-Brandenburgischen Akademie der Wissenschaften), as well as other unpublished documents on the same subject.
The editing work was first done by Sigismund Cohn, to whom Carl Wilhelm Borchardt had asked to examine a collection of manuscripts in view of a possible publication. Borchardt had to finish the work after Cohn’s death in 1861. A letter from Cohn to Borchardt (document II/13a) gives some details about the way he worked and the available documents. We give a French translation of it. Cohn describes 4 distinct documents. The first corresponds to document II/23b and has been used in the publication « De aequationum… », that initially appeared in the Vorlesungen über Dynamik von C. G. J. Jacobi, published by Alfred Clebsch in 1866 and was later reproduced in volume V of Jacobi’s complete works. The second corresponds to the document II/13b; II/13c is a copy by Cohn with notes by Borchardt. We have reproduced the abstract in German by Borchardt and the beginning of Cohn’s transcription that is related to the different normal forms that a given system may have. Our French partial translation of II/13c. The text «De investigando ordine systematis… » comes from that transcription. It was first published by Borchardt in Crelle’s journal in 1865. It was in fact made by Cohn from a collection of various partial manuscript of which he tried to make something as coherent as possible.
The last two documents are II/23a and II/22. The first seems quite close to the text of de «De investigando ordine systematis… ». The second studies the link between the choice of a normal form and the multipliers of a given system. The last multiplier method was at the origin of Jacobi’s work on the order of a system and its reduction to a normal form.
The editing process of these texts has required a great work, which his reflected by many partial versions kept in the archives. The document II/24 mixes the text of these two papers. We have reproduced the beginning and the end of that document. Our French translation of II/24. The document II/25 contains an abstract of «De investigando ordine systematis… », followed by two handwritten pages in Latin by Borchardt hand, that were not reproduced in the published version. Our French translation of II/25. The document I/63 seems to be the definitive version (deprived of its lost first 8 pages) of the published text «De investigando ordine systematis… », with comments for the typographer by Borchardt.