124 REPORT—1899, 
little differing from, Bour’s system, by Brioschi! in 1868, and Siacchi? in 
1871 and 1874 ; Vernier * in 1894 published what is substantially only a 
reproduction of Siacchi’s paper of 1874. Amplifications and corrections 
were also made by Mathieu * in 1873-8. 
Previously to 1868 the restricted problem of three bodies had been 
discussed by Scheibner® in 1866. His equations refer to a somewhat 
more general case, but for the restricted problem of three bodies they are 
as follows: Let be the mean motion of the two bodies, and a, y the 
coordinates of the particle, referred to the centre of gravity of the bodies, 
the (moving) x-axis being the line joining the bodies. Also let 
d: 
d 
ata eid 
then the equations of motion are 
dx_ 6H di _ push dy_cH dy__ 6H 
dt si dt tx’ dt ‘én? dt ~ by? 
where H isa certain function of 2, y, & n, and H=constant is the Jacobian 
integral of the system. 
In 1868 Scheibner © reduced the general problem of three bodies to a 
canonical system of the eighth order without using Jacobi’s transforma- 
tion to the two fictitious masses. Let ¢, qs, 7; be the mutual distances 
of the three bodies, and let Aye Pa=y > Ps =\ where T is the 
1 2 
kinetic energy ; let q, be the angle Chik the node (of the plane of the 
bodies, on the invariable plane) makes with one of the principal axes of 
inertia of the bodies at their centre of gravity ; and let py=k cos 7, where 
% is the constant of angular momentum on the invariable plane, and 7 is 
the angle between the plane of the bodies and the invariable plane. Then 
the differential equations become 
dq, °H dp, tH »_4 9 
a ip. aaah Bled U=h 2 2) 
where H is a certain function of the quantities p and g, and H =constant 
is an integral of the system. 
When the motion is in one plane, the system reduces to the sixth 
order, as p, becomes a constant, and q,, now measured from a fixed line 
in the plane, is determined by a simple quadrature. This reduction is 
more symmetrical than one given by Perchot and Ebert ‘ in 1899. 
1 «Sur une transformation des équations différentielles du probléme des trois 
corps,’ C. R. lxvi. pp. 710-14. 
2 «Jntorno ad alcune trasformazioni delle equazioni differenziali del problema 
dei tre corpi,’ Atti di Torino, vi. pp. 440-54; ‘Sur le probleme des trois corps,’ C. &, 
xxviii. pp. 110-13. 
3 «Sur la transformation des équations canoniques du probleme des trois corps,’ 
C. R. cxix. pp. 451-4. 
4 ‘Mémoire sur le probleme des trois corps,’ C. #&. Ixxvii. pp. 1071-4, xxviii. pp. 
408-10; ‘Mémoire sur le probleme des trois corps,’ Liowville (3), ii. pp. 345-70; 
‘Sur Vapplication du probléme des trois corps 4 la détermination des perturbations 
de Jupiter et de Saturne,’ Journal de ? Heole Polytechnique, xxviii. pp. 245-69. 
5 «Satz aus der Storungstheorie,’ Crelle, lxv. pp. 291-2. 
5 * Ueber das Problem der drei Korper,’ Credle, |xviii. pp. 390-2. 
7 ‘Sur la réduction des équations du probléme des trois corps dans le plan,’ 
Bulletin Astronomique, xvi. p. 110-16, 
