‘oasis 
ON THEORETICAL DYNAMICS. 37 
sometimes be obtained by means of integrals which appear at first foreign 
to his method. Before going further, it may be convenient to remark that 
the author restricts himself to the case in which H is independent of the 
time, and where, consequently, the condition of vis viva is satisfied ; it was, 
however, remarked by Liouville that the analysis, slightly modified, applies 
to the most general case where H is any function of ¢ and the variables, and 
it is possible that when the entire memoir is published (it is given in ‘ Liou- 
ville’s Journal’ as an extract), the theory will be exhibited under this more 
- general form. 
67. To give an idea of the analytical results, the equations are considered 
under the form 
dp; dH dq; dH. , 
mara 7 a ial to i=n) 
(where, as already remarked, H is independent of ¢). The integrals admit, 
therefore, of representation in the canonical form a, 3, a,,a,,.-+. a2n—2 
where a(=H) is the equation of vis viva 6(—G—A) is the integral conju- 
gate to this, and the only integral involving the time, and the remaining 
integrals a, and a,, a; and a,.. aon—3 and agn—2 are conjugate pairs, we 
have (a,, a) (=(4, H))=0, (4,, 6) (=(a,; G) )=0, (a,,.¢,)=1, (a, ay) 
=0,.. (a), don—2)=0. 
The integrals a,,a,.-. a,—2 verify the linear partial differential equa- 
tion 
i=n(dHdZ dH dé 
j eis has — }=0 H, Z)=0 (1), 
: > dq; dp; dp; dq; or ( > 4) ( ) 
which is also satisfied by =H, and of which the general solution is ¢= 
$(H, a,, a,..+ c2n-2), while, on the contrary, the first member of the equa- 
tion (1), becomes unity for ¢=G, in other words (H,G)=1. The equa- 
_ tion (1) replaces the original differential equations ; it is to the equation (1) 
_ that the theorems of Poisson and Bertrand may be supposed to be applied, 
and it is this equation (1) which is studied in the memoir, where it is 
_ shown how the order may be diminished when one or more integrals are 
known. 
_ In the first place, the integral a=H which is known, may be made use of 
to eliminate one of the variables, suppose p,; the result is found to be 
l=n—1/dp, df dp, o) ae 
| Sat (aes dp. an ey) a9 
which has the same integrals as the equation (1), except the integral of vis 
viva £=H;; it is this equation (2) which would have to be integrated if 
only the integral a were known. 
__ Suppose now there is known a new integral a,; this gives rise to the 
partial differential equation 
l=n(da,dg da, =)= a 
tf (Fe dp; dp; dq; =0 or (a, ¢)=0 (4), 
which is satisfied by Z=H, G, a,, a, a4-++G2n—2, but not by a,, which gives 
(a,,a,)=0. The equation (4) is satisfied =H, and it may be therefore 
transformed in the same manner as the equation (1) was, viz. p, may be 
expressed in terms of the other variables and of a. The author remarks 
that it will happen, what causes the success of the method, that this opera- 
tion, the object of which is to get rid of the solution =H, conducts to two 
different equations, according as £=G or ¢= any other integral of the. 
