PROGRESS OF THE SOLUTION OF THE PROBLEM OF THREE BODIES. 1338 
proved the existence of an infinite number of periodic solutions in the 
general problem of three bodies; and in 1887 Bohlin ! applied an idea of 
Hill’s (viz. using the Jacobian integral to separate off regions of space 
into which the moon cannot enter) to a more general class of dynamical 
problems. In the same year (1887) Hill ? discussed the different systems 
of variables which might be employed in solving a system somewhat more 
general than the restricted problem of three bodies, namely, that of a 
particle of zero mass, attracted by two bodies which move in Keplerian 
ellipses round their common centre of gravity. 
Poincaré’s* memoirs of 1881-6 on curves defined by differential 
equations lead to one result of importance in Dynamical Astronomy. In 
order that the system of » bodies may be stable, two conditions must be 
fulfilled : firstly, the mutual distances must always remain within certain 
limits ; and, secondly, if the system has a definite configuration at any 
instant, it must be possible to find a subsequent instant at which the 
configuration differs from this as little as we please. It follows from the 
investigations of this series of memoirs that, if the first of these conditions 
is satisfied, the second is also. 
In 1883 Lindstedt * resumed the consideration of the problem which 
had been treated by Newcomb nine years before, namely, the expression of 
the coordinates in the problem of three bodies as trigonometric series, 
whose arguments are linear functions of the time. A fuller account ° of 
the work was published in 1884. The author starts from the equations of 
Lagrange’s ‘ Essai sur le probléme des trois corps’ ; the system is reduced 
to four different equations, each of the second order ; and these are solved 
by successive approximation. After the rth approximation has been 
effected, the (r+1)th approximation is obtained by solving four linear 
non-homogeneous differential equations with constant coefficients. This 
ean be done by known methods ; but if the solution is carried out in the 
usual way, termg will arise in which the time ¢ occurs as a factor (these 
are the ‘secular terms’ of the old planetary theory). Lindstedt therefore 
modifies the equations in accordance with a method indicated by himself 
in a previous paper,® and obtains a solution in which ¢ occurs only in the 
arguments of trigonometric functions. It thus appears that the mutual 
distances of the three bodies can be expressed as trigonometric series of 
four arguments, each of which is a linear function of the time. The 
defects of Lindstedt’s memoir in regard to convergence, &c., will be 
noticed in connection with other papers. 
A fresh proof of Lindstedt’s results was given by Tisserand’ in 1884-5, 
' ‘Ueber die Bedeutung des Princips der lebendigen Kraft fiir die Frage von der 
Stabilitit dynamischer Systeme,’ Acta Math. x. pp. 109-30. 
? *Coplanar Motion of two Planets, one having a Zero Mass,’ Annals of Math. 
iii. pp. 65-73. 
* «Sur les courbes définies par les équations différentielles’; Ziowville (3) vii. 
pp. 375-422 ; (3) viii. pp. 251-96 ; (4) i. pp. 167-244; (A) ii. pp. 151-217. 
* «Sur la forme des expressions des distances mutuelles dans le probléme des 
trois corps,’ C. R. xlvii. pp. 1276-8, 1353-5; ‘Ueber die Bestimmung der gegen- 
seitigen Entfernungen in dem Probleme der drei Kérper,’ Astr. Nachr. cvii. 
pp. 197-214. 
5 « Sur la détermination des distances mutuelles dans le probléme des trois corps,’ 
Annales de V Ecole Norm. (3) i. pp. 85-102. 
° «Beitrag zur Integration der Differentialgleichungen der Stérungstheorie, 
Mémoires de ? Acad. de Saint-Pétersbourg, xxxi. No. 4. 
7 «Note sur un théoréme de M. A. Lindstedt concernant le probléme des trois 
corps, C. &. xcviii. pp. 1207-13; ‘ Mémoire sur le probléme des trois corps, Annales 
de lv Observatoire de Paris, Mémoires, xviii. 
