PROGRESS OF THE SOLUTION OF THE PROBLEM OF THREE BODIES. 127 
invariable plane is the plane of wy, and where a, dy, a3, 0, bo, b3 are con- 
stants subject only to the conditions 
Then the problem can be reduced to the Hamiltonian system of the 
eighth order, 
dq, __ 6H dp; Aes __¢H 
py alll okie) the eon 
di ép? dt &q; Start dln Patri 8s) 
where ( . 
SH MtM; 9 PPAQe +98 —T) ei ee +N 
H= [Prgng! AP = 249645 =f ee {Po(%4 D190) +kb,} 
D< WMoM1- 
{ 25(ay—Dsq)—?3(a2—bago)  — Eas 
In this, 74 is the constant of angular momentum. Bruns then reduces 
this to a system of the sixth order by eliminating the time and using the 
integral H=—A; writing H=H,po+H., where H, and H, do not 
H,+h 
involve p , and putting K=— wwe have 
1 
Bo tec OK. (5541,9) 8) 
dgo &p, dg ey; 
which is the required system. 
It may be noted that a particularly simple case of Bruns’s transforma- 
tion is afforded by putting a, = —1, ag=1, a,=0, 6; =—1, b,=0, 63=1 ; in 
this case g is simply the ratio of the two vectors which join the projection 
of m, to the projections of mz and mz, respectively on the invariable plane. 
Kiaier’ in 1891, starting from Jacobi’s transformation, likewise reduced 
the problem to a canonical system of the sixth order. 
The differential equations of the restricted problem of three bodies 
were discussed by Tisserand * in 1887 and by Poincaré* in 1890. Both 
authors reduce the problem to a canonical system of the fourth order ; 
Tisserand takes variables defined by means of the elements of the 
instantaneous ellipse described by the particle round one of the bodies, © 
while Poincaré uses the instantaneous ellipse described by the particle 
round the centre of gravity of the system. 
§ Il. Certain Particular Solutions of Simple Character. 
Lagrange‘ in 1772 had shown that the equations of motion of the 
problem of three bodies can be satisfied by two particular solutions of a 
very simple character ; in one case the three particles are always at the 
vertices of a moving equilateral triangle and in the other they are always 
on a moving straight line. We shall generally call these respectively the 
motions of Lagrange’s three equidistant particles and three collinear 
particles, 
* “Sur la réduction du probléme des trois corps au systéme canonique du sixiéme 
ordre,’ Astr. Nach. cxxvi. pp. 69-76. 
* «Sur la commensurabilité des moyens mouvements dans le systéme solaire,’ 
Bull. Astr. iv. pp. 183-92. 
* © Sur le probléme des trois corps,’ Acta Math. xiii. pp. 1-270. 
* «Essai sur le probleme des trois corps,’ Priz de V Académie de Paris, iz. 
