182 REPORT—1899. 
which it has been developed will not be reviewed here; the result 
important for our purpose is that an integral can be found in the form 
w= > a, cos{(e+2r)t+a}, 
%™=-COO 
where c depends on the coefficients in the equation, and a, and a depend 
on these coefficients and on two arbitrary constants of integration. For 
the determination of c Hill devised the following beautiful method :— 
(oe) 
Putting e’=Z, we have O= ™\& 9,2?", where the quantities 6, are 
g ys q 
n=—CO 
ee} 
constants, Hill assumes that w is of the form w= > b,f°*?" and sub- 
n=—-CO 
stitutes this value of w in the differential equation. Since the whole 
coefficient of each power of £ must now be zero, an infinite number of 
equations are obtained, which involve the 6’s linearly ; on eliminating the 
é’s a determinant with an infinite number of rows and columns (an idea 
first introduced by Kotteritzsch in 1870) is obtained, which involves only 
¢ and the known quantities 6,. This determinant, equated to zero, 
furnishes the value of c, and consequently the motion of the lunar 
perigee. 
The convergence of the infinite determinant was not considered by 
Hill ; this gap in the work was filled by Poincaré ' in 1886. 
Hill’s paper was reprinted,’ with some additions, in 1886. 
In 1877, Adams,’ referring to Hill’s paper, remarks that he had himself, 
many years previously, investigated the motion of the moon’s node by a 
method similar to that used by Hill for the perigee, and had found the 
same infinite determinant. 
Tn 1878 Hill4 published in a more complete form the derivation of the 
periodic solution, which in its character of intermediate orbit had been 
the foundation of his previous paper. The solution is found by actually 
substituting, in the differential equations of the restricted problem of 
three bodies, expansions of the desired form with undetermined coefticients ; 
these coefficients are then determined as functions of a parameter m, which 
depends on the ratio which the period of the periodic solution bears to 
the period of revolution of the two principal masses round each other, 2.e. 
on the ratio of the mean motions of the sun and moon. By varying m, 
different periodic solutions are obtained ; the last one of Hill’s solutions 
(the orbit of maximum lunation) has cusps at the points where the elonga- 
tion from the sun is a right angle. 
Hill’s work soon led to further developments. In 1883-4 Poincaré,’ 
using a theorem due to Kronecker in the general theory of functions, 
1 ‘Sur les déterminants d’ordre infini, Bulletin de la Soc. Math. de France, 
xiv. pp. 77-90. 
2 On the part of the motion of the Lunar Perigee, which is a function of the mean 
motions of the Sun and Moon,’ Acta Math. viii. pp. 1-36. 
3 ‘On the motion of the Moon’s Node in the case when the orbits of the Sun and 
Moon are supposed to have no eccentricities, and when their mutual inclination is 
supposed to be indefinitely small, Monthly Notices, Roy. Ast. Soc. vol. xxxviii. 
. 43-9. 
he. ‘Researches in the Lunar Theory,’ Amer. Journ. Math. i. pp. 5-27, 129-48, 
245-61. 
> «Sur certaines sclutions particuliéres du probléme des trois corps,’ C. 2. xevil. 
pp. 251-2; Bull, Aste. i, pp. 65-74. 
