38 REPORT—1857. 
equation (4); so that in the second form of the transformed equation the 
unknown integral =G is also eliminated. This second form is found 
to be 
l=n—-1 (da, dé da, dé 
int (Fe Fam pee) =O (on) =0 (8) 
which is precisely similar to the equation (1) (only the number of variables 
is diminished by two unities), and is possessed of the same properties. Its 
integrals are a, a4, a4... G2an—2, Which are all of them integrals of the pro- 
blem, and give Cae a;)=0. And the theorems of Poisson and Bertrand 
apply equally to this equation; the only difference is, that the number of 
terms in the expressions (a, 3) is less by two unities. A new integral (ag) 
leads in like manner to an equation (8) similar to (5), but with the number 
of variables further diminished by two unities, and so on, until the half 
series of integrals a,a,,a3..-d2,-3 are known; the conjugate integrals 
B, ag) Ay+++ Gan—2 are then obtained by quadratures only, in the method ex- 
plained in the memoir, and which is in fact identical with that given by the 
theorem of Liouville and Donkin. The memoir contains other results, 
which have been already alluded to in a general manner ; some of these are 
made use of by the author in his ‘Mémoire sur les problémes des trois 
corps, Journal Polyt., t. xxi. pp. 35-58 (1856). 
68. Liouville’s note of July, 1855, on the occasion of Bour’s memoir, 
mentions that the author of the memoir had recognized that, according to 
the remark made to him, his formulz subsist with even increased elegance 
when H is considered as a function of ¢ and the other variables. But (it is 
remarked) the general case can be always reduced to the particular one 
considered in the memoir, provided that the number of equations is aug- 
mented by two unities by the introduction of the new variables r and uw, the 
former of them, r, equal to ¢+ constant, so that 
dt 
an 
the latter of them, w, defined by the equation 
du dH 
a 
Suppose in fact that 
V=H+u, 
then, since 7 and w do not enter into H, which is a function only of é and 
the variables g,...p,.-+, we have 
qv de 
and, moreover, the differential coefficients with respect to ¢,q¢,...p,+.. Of — 
the functions H and V are equal. The system may be written 
d_dV  du__aV 
dr dw dr at, 
dp_aVv iy__aV 
dr dq’ = dr ap 
which is a system containing two more variables, but in which V is inde- | 
pendent of the variable r, which stands in the place of & The transforma- — 
