ON THE SPECIAL PROBLEMS OF DYNAMICS. 215 
114. Jacobi’s memoir “Sur l’élimination des Neeuds &e.” (1843),—The 
problem of the motion of three mutually attracting bodies is in the first 
instance reduced to that of the motion of two fictitious bodies (which may be 
considered as mutually attracting bodies, attracted by a fixed centre of force)*. 
In fact, in the original problem the centre of gravity of the three bodies moves 
uniformly in a right line, and it may without any real loss of generality be 
taken to be at rest; that is, if the w-coordinates of the three bodies are é,, 
E,, &,, then m,é, -+m,é,-++m,£,=0, or é,, é,,&, may be taken to be linear functions 
of two quantities w, and x, And similarly for the y-coordinates and the 
z-coordinates respectively. And (#,, ¥,, 2,), (@5 Yo» Z,) may be regarded as 
the coordinates of two bodies revolving about a fixed centre of force. Hence 
representing the differential coefficients in regard to the time by ~,', &c., and 
treating these as new variables, the equations of motion will assume the form 
dx, dy, _dz, _ dx, dy, _ dz, 
7 ay 7 
ao yy cot X, Y2 %, 
i 1 2 2 2 
where X,, Y,, Z,, X,, Y,, Z, are forces capable of representation by means of 
aforce-function U. Thisis a system of twelve equations; but since X,, Y,, Z,, 
X,, Y,, Z, are independent of the time, we may omit the equation (dt), and 
treat the system as.one of eleven equations between the variables w,, y,, z,, 
Woy Yor Za Ly’ Yy's Z'y Vay Yn Z 3 If this system were integrated, the deter- 
mination of the time would then depend on a quadrature only. But for the 
system of eleven equations we have four integrals, viz., the integral of Vis Viva 
and the three integrals of areas, and the system is thus reducible to one of 
(11—4=) seven equations. This preliminary transformation in Jacobi’s 
memoir explains the remark that the problem, as it stood before him, depended 
on a system of seven equations. 
115. Jacobi remarks that in the transformed problem the three integrals 
of areas show (1) that the intersection of the planes of the orbits of the two 
bodies lie in-a fixed plane, the invariable plane of the system; (2) that the 
inclinations of the planes of the two orbits to this fixed plane, and the para- 
meters of the two orbits considered as variable ellipses, are four elements any 
two of which rigorously determine the two others. 
And then choosing for variables the inclinations of the two orbits to the 
‘invariable plane, the two radius vectors, the angles which they form with the 
intersection of the planes of the two orbits, and lastly the angle between this 
intersection (being as already mentioned a line in the invariable plane) with 
a fixed line in the invariable plane, he finds that the last-mentioned angle 
entirely disappears from the system of differential equations, and is determined 
after their integration by a quadrature. In this new form of the differential 
equations there is no trace of the nodes. The differential equations which 
determine the relative motion of the three bodies are reduced to five equations 
of the first order and one of the second order. The equations in question are 
the equations I. to VI. given at the end of the memoir. It is to be remarked 
that the differential dt is not eliminated from these equations; the last of 
2 
them is SL (ur+ H,7,°) =2U —22 ; and if to reduce them to a system of equa- 
* This is the effect of Jacobi’s reduction ; but the explicit statement of the theorem, 
and actual replacement of the problem of the three bodies by that of the two bodies 
attracted to a fixed centre, is due to Bertrand (post, No. 117). 
