PROGRESS OF THE SOLUTION OF THE PROBLEM OF THREE BODIES. 131 
oscillations of the particle about this orbit, when the initial conditions of 
its motion are not exactly such as to cause it to describe the periodic orbit. 
These oscillations represent those inequalities in the moon’s motion which 
depend only on the eccentricity of the lunar orbit and the ratio of the 
mean motions of the sun and moon; and the period of the oscillations 
represents the time between two successive perigees of the moon, so that the 
difference between this period and the period of the orbit gives that part 
of the motion of the lunar perigee which is a function of the mean motions 
of the sun and moon—whence the title of the memoir. 
Let w represent the distance (measured along the normal) of the 
particle from the periodic orbit, at any time ¢ during the performance of 
the small oscillations. Then Hill finds that wis given by an equation of 
the form 
d?w 
dt 
+0w=0 
where 9 depends only on the relative position of the two bodies and the 
particle ; O is therefore a known periodic function of ¢, and can be ex- 
panded in the form 
0=0,+80, cos 2+ 0, cos 444+ ,..., 
where 0,, 0,, 0, ... are pure constants. 
(It ought to be stated here that, since all inequalities in the moon’s 
motion which involve the sun’s parallax are neglected, the distance of the 
two bodies from each other is supposed to be infinite, and the one of them 
at infinity is supposed to possess such an (infinite) mass as would correspond 
to a finite mean motion.) 
The problem therefore is to solve the differential equation 
dw 
det {0 +0, cos 244+ 0, cos 4¢4+ .. .}w=0. 
Equations of this type had been discussed by the founders of dynamical 
astronomy, D’Alembert,'! Lagrange,” and Laplace, and have since been 
discussed by a large number of mathematicians. Tisserand called the 
equation 
dw 
spi {Q,+9, cos 24} w=0, 
which is a particular case of the above, the Gyldén-Lindstedt equation ; 
the name does not seem very appropriately chosen, but as it has now 
‘become established we shall use it here. The same equation occurs in 
the Potential Theory as giving rise to the functions appropriate to the 
Elliptic Cylinder ; it is discussed from this point of view in Heine’s 
‘Kugelfunctionen.’ The more general equation above will be called either 
Hill’s equation or the generalised Gyldén-Lindstedt equation. The theory 
of these equations is a matter of pure mathematics, and the papers in 
1 Opuscules Mathématiques, v. p. 336. 
? ‘Solutions de différents problémes de calcul intégral,’ Miscellanea Taurinensia, 
ili. (@uwvres, i. p. 586.) 
3 Guvres, viii. and ix: 
K 2 
