30 REPORT—1857. 
The author remarks, that, in seeking the cases of exception to Poisson’s 
theorem, there is obtained a new method of integration, which may lead to 
useful results ; and, after referring to Jacobi’s memoir on the elimination of 
the nodes in the problem of three bodies, he remarks that, by his own new 
method, the problem is reduced to the integration of six equations, all of 
them of the first order ; so that he effectuates one more integration than had 
been done by Jacobi; and he refers to a future memoir (not, I believe, yet 
published) for the further development of his solution. 
58. To give an idea of the analytical investigations, the equations of 
motion are considered under the Hamiltonian form 
dg_dH dp__dit 
dt~ dp’dt~ dq’ 
where H is any function whatever of g,...p,... without ¢, and thena given 
integral being 
a=9$(9,+- p,-+-), 
the question is shown to resolve itself into the determination of an integral 
B=V(q,.--p,-++), such that identically (a, 3)=0 or else (a, B)=1, where 
(a, 3) represents, as before, Poisson’s symbol, viz. 
(a, 8) 
(© Paap) t 
O(a,B) SadB dadB 
09, p) dqgdp dp dq 
The partial differential equations (a, 3)=0 or (a, 8)=1, satisfied by cer- 
tain integrals , are in certain cases, as Bertrand remarks, a precious method 
of integration leading to the classification of the integrals of a problem, so 
as to facilitate their ulterior determination: it is in fact by means of them 
that the several results before referred to are obtained in the memoir. 
59. Bertrand’s note of 1852 in the ‘ Comptes Rendus.’—This contains the 
demonstration of a theorem analogous to Poisson’s theorem (a, /3)=const., 
but the function on the left-hand side is a function involving four of the 
arbitary constants and binary combinations of pairs of corresponding vari- 
ables, instead of two arbitrary constants and the series of pairs of corre- 
sponding variables. 
60. Bertrand’s notes, vi. and vii., to the third edition of the ‘ Mécanique 
Analytique,’ 1853, contain a concise and elegant exposition of various 
theorems which have been considered in the present report. ‘The latter of 
the two notes relates to the above-mentioned theorem of Poisson, and places 
the theorem in a very clear light, in fact, establishing its connexion with the 
theory of canonical integrals. Bertrand in fact shows, that, given any in- 
tegral a of the differential equations (in the last-mentioned form, the whole 
number of equations being 2h), then the solution may be completed by 
joining to the integral a a system of integrals B,, B,...,,_,, which, com- 
bined with the integral a, give to Poisson’s equation an identical form, viz. 
which are such that 
(a, By, )=1 (a, By)=0,.-. (a, Bo) =0- 
This, he remarks, shows, that, given any integral a, the solution of the pro- 
blem may be completed by integrals f,, 3,... Box—~11 Which, combined with 
a, give all of them an identical form to the theorem of Poisson. But it is 
not to be supposed that all the integrals of the problem are in the same case. 
In fact, the most general integral is n=a@(a, (,, B;.+ Bo,—1); and it is at once 
if for shortness 
