40 REPORT—1857. 
equation, and the general theory of the connexion of the integration of a 
system of ordinary differential equations, and of a partial differential equa- 
tion of the first order, a theory, however, of which Jacobi can only be con- 
sidered as the second founder; seventhly, the notion (arising from the 
researches of Lagrange and Poisson) and ulterior development of the theory 
of a system of canonical integrals. 
I remark in conclusion, that the differential equations of dynamies (in- 
cluding in the expression, as I have done throughout the report, the gene- 
ralized Lagrangian and? Hamiltonian forms) are only one of the classes of 
differential equations which have occupied the attention of geometers. The 
greater part of what has been done with respect to the general theory of a 
system of differential equations is due to Jacobi, and he has also considered 
in particular, besides the differential equations of dynamics, the Pfaffian 
system of differential equations (including therein the system of differential 
equations which arise from any partial differential equation of the first order), 
and the so-called isoperimetric system of differential equations, that is, the 
system arising from any problem in the calculus of variations. In a sys- 
tematic treatise it would be proper to commence with the general theory of 
a system of differential equations, and as a branch of this general theory, to 
consider the generalized Hamiltonian system, and in relation thereto to 
develope the various theorems which have a dynamical application. — It 
would be shown that the generalized Lagrangian form could be transformed 
into the Hamiltonian form, but the first-mentioned form would, I think, 
properly be treated as a particular case of the isoperimetric system of dif- 
ferential equations. 
List of Memoirs and Works above referred to. 
Lagrange. Mécanique Analytique. Ist edition. 1788. 
Laplace. Mécanique Céleste, t.i. 1799. 
Poisson. Sur les inégalités séculaires des moyens mouvemens des planétes. 
Read to the Institute 20th June, 1808.—Journ. Polyt. t. viii. pp. 1-56. 
1808. 
Laplace. Mémoire..... Read to the Bureau of Longitudes 17th Aug. 
1808.—Forms an Appendix to the 3rd volume of the Mécanique 
Céleste. 1808. 
Lagrange. Mémoire sur la théorie des variations des élémeuts des planétes 
et en particulier des variations des grands axes de leurs orbites. Read 
to the Bureau of Lengitudes the 17th Aug. and to the Institute the 
22nd Aug., 1808.—Mém. de I'Instit., 1808, pp. 1-72. 1808. 
Lagrange. Mémoire sur la théorie générale de la variation des coustantes 
arbitraires dans tous les problémes de la Mécanique. Read to the 
Institute 13th March, 1809.—Mém. de I’Instit., 1808, pp. 257-302 
(includes an undated addition), and there is a Supplement also without 
date, pp. 363, 364. 1809. 
Poisson. Mémoire sur la variation des constantes arbitraires dans les ques- 
tions de Mécanique. Read to the Institute 16th Oct., 1809.—Journal 
Polyt., t. viii. pp. 266-344. 1809. 
Lagrange. Seconde mémoire sur la variation des constantes arbitraires 
dans les problémes de Mécanique, dans lequel on simplifie l’application 
des formules générales a ces problémes. Read to the Institute 19th 
Feb., 1810.—Mém. de I’Instit., 1809, pp. 343-352. 1810. 
