216 REPORT—1862. 
tions of the first order we write 4 (ur? + ,7,7)=86, and therefore _ =2U—2h, 
the system may be presented in the form 
du du, ely OE ei Ge 
WY De snel holaiy Bain Ee Bee 
which if we do, and then omit the equation (=dt), we have a system of six 
equations between the seven quantities wu, u,, i,7,,7,7,,0; when this is 
integrated, the equation (=dt) gives the time by a quadrature ; and finally, 
Jacobi’s equation VII. (do=tan u =) gives by a quadrature the angle before 
referred to as disappearing from the system of equations I. to VI. 
116. But when Jacobi says, “ Par suite on a fait cinq intégrations. Les 
intégrales connues n’étant qu’au nombre de quatre, on pourra done dire que 
l’on a fait une intégration de plus dans le systéme du monde. Je dis dans 
le systéme du monde puisque la méme méthode s’appliqué 4 un nombre 
quelconque de corps,” the language used is not, I think, quite accurate. It, 
in fact, appears from the memoir that it is only on the assumption of the 
integration of the system of six equations that, besides the integral of Vis Viva 
and the integrals of areas, the remaining two integrals are known ; in fact, 
after, but not before the system of the order six has been integrated, the time ¢ 
and the angle © are each of them given by a quadrature. - 
117. Bertrand’s “ Mémoire sur l’intégration des équations différentielles de 
la Mécanique ”’ (1852).—I have spoken of this memoir in No. 56 of my former 
Report. The course of investigation is the inquiry as to the integrals, which, 
combined according to Poisson’s theorem with the integral of Vis Viva or any 
other given integral, give rise to an illusory result. But as regards the appli- 
cation made to the problem of three bodies, it will be more convenient to state 
from a different point of view the conclusions arrived at: and I may mention 
that when the author says ‘‘Je parviens . . 4 reduire la question 4 l’intégration 
de six équations tout du premier ordre, c’est-a-dire que j’effectue une intégra- 
tion de plus que ne l’avait fait Jacobi,” he seems to have overlooked that 
Jacobi’s system of five equations of the first order and one of the second order 
really is, as above noticed, a system of the six equations with another equation 
which then gives the time by a quadrature, and that, at least as appears to 
me, he has not advanced the solution beyond the point to which it had been 
carried by Jacobi*. - 
118. Presenting Bertrand’s results in the slightly different notation in 
which they are reproduced in Bour’s memoir ( post, No. 122), then if (a, y, z), 
(x, Y,, 2,) are the coordinates of the two bodies (the problem actually con- 
sidered being, as by Jacobi, that of the motion of two bodies about a fixed 
centre of force), and representing the functions 2*+y°+2*, #,°+y,°+z,7, 
m*(ac'* oe +2"), m,” Cts a) Beep 3m (wa +yy' +22’), m, (ea +9.) +2,2,' ? 
mx, +y,y' +2,2'),m, (vx, +yy,'+22,'), (we, + yy, +22,) mn, (a'e,'+y'y,'+2'2,) 
by u, %,, U,V, W; W,, 7, 7,, 9, $ respectively, then the last-mentioned quanti- 
ties are connected by a single geometrical relation, so that any one of them, 
say s, may be considered as a given function of the remaining nine. And the 
author in effect shows that the equations of motion give a system 
* These remarks were communicated by me to M. Bertrand—see my letter “Sur 
Vintégration des équations différentielles de la Mécanique,” Comptes Rendus (1863)—and, 
in the answer he kindly sent me, he agrees that they are correct. 
