130 REPORT —1899, 
In 1872 Newcomb,! assuming that the coordinates of the planets can 
be expressed by trigonometric series, as in Delaunay’s theory, proved 
various properties of the coeflicients, &c., by using the function called by 
Clausius the virial, which is the mean value of the kinetic energy of the 
system. This was extended by Siacchi * in 1873, 
In 1874 Newcomb ? proceeded to justify his assumption regarding the 
expression for the coordinates as functions of the time. He applies the 
transformation of Jacobi and Radau to the equations of (+1) bodies, and 
so obtains a system of the 6nth order. It is assumed that a set of infinite 
series of the forms 
Oey ae . . ° 
p= >K, my (Ay H%QAo+A3Ag + » «+ +43nA3n) 
can be found, where p; is one of the coordinates and \,=/,+0,¢ (the quan- 
tities / being 3n arbitrary constants, and the quantities b and K being 
functions of 3n other arbitrary constants), such that the differential equa- 
tions are approximately satisfied by these series. Newcomb, then, using 
the method of variation of arbitrary constants, replaces these series by 
others of the same form which satisfy the differential equations to a higher 
degree of approximation. Proceeding in this way, it appears that the 
problem of three bodies can be formally solved by series of this kind. 
The year 1877 saw the appearance of a paper * which may be regarded 
as the beginning of the new era in Dynamical Astronomy. The author, 
Mr. G. W. Hill, was at the time an assistant on the staff of the American 
Ephemeris. 
The first of the novelties in this paper is the abandonment of Kepler’s 
ellipse. It had hitherto been usual to take, as the first approximation to 
the orbit of the moon, an elliptic path round the earth ; the orbit, in fact, 
which the moon would actually describe if the sun did not exist to disturb 
it ; the actual path of the moon was then found by calculating the per- 
turbations caused by the sun on this elliptic motion. Hill, however, does 
away with the elliptic orbit altogether, and takes, as the intermediate 
orbit or first approximation to the moon’s path, an orbit which includes 
all the inequalities which depend only on the ratio of the mean motions 
of the sun and moon, but takes account of no other inequalities. This 
difference between Hil] and the older theorists may be otherwise stated 
as follows: the old astronomers first solved the problem of two bodies, 
and then attempted to solve the problem of the three bodies by suitably 
varying the solution so obtained ; whereas Hill begins by solving the re- 
stricted problem of three bodies, and then attempts to solve the problem 
of three bodies by suitably varying this solution. 
Suppose, then, that an orbit for the particle is known, which is periodic, 
z.e. which is such that the two bodies and the particle retake the same 
relative positions after the lapse of a certain interval of time. Then the 
coordinates of the particle can be expressed as sums of sines and cosines of 
multiples of a linear function of the time. We can now consider the small 
' “Note sur un théoréme de mécanique céleste,’ @. #. Ixxv. pp. 1750-3. 
? «Sur un théoréme de mécanique céleste,’ C. R. Ixxvii. pp. 1288-91. 
® «On the General Integrals of Planetary Motion,’ Smithsonian Contribution to 
Knowledge, 1874, pp. 1-31. 
**On the part of the Motion of the Lunar Perigee which isa Function of the 
Seas Motions of the Sun and Moon,’ Cambridge; Mass., Press of John Wilson & 
on. 
