Memoirs presented to the Cambridge philosophical society on the occasion of the jubilee of Sir George Gabriel Stokes, bart., Hon. LL. D., Hon. SC. D., Lucasian professor.

V. On the Solution of a Pair of Sinmultcneous Linear Differential
Equations, which occur in the Lunar Theory. By ERNEST W. BROWN,
Sc.D., F.R.S.
[Received 1899 July 14.]
IN the calculation of the inequalities in the Moon's motion by means of rectangular
coordinates a certain pair of differential equations is continually requiirig solution. The
left-hand melnbers are linear and always the same; the right-hand members are known
functions of the independent variable-the time-and vary with each class of inequalities
considered. It has been the practice to obtain the required particular integral by assuming
the solution (the form of which is known) and then to determine the coefficients by
continued approximation. This method is troublesome to put into a form which a computer can use easily and is besides peculiarly liable to chance errors; a large number
of processes would have to be learnt before the computer could proceed quickly and
securely. The main object of this paper is to put the solution into a form which will
avoid these difficulties, but I believe that some of the results may be found to be of
a more general interest. Further, the question of the convergence of the series used
to represent the coordinates in the Lunar Theory may be somewhat narrowed. In fact
it being granted that the series forming the 'Variation' inequalities and the elliptic
inequalities depending on the first power of the Moon's eccentricity are convergent, it
is not difficult to demonstrate, by means of equation (14) below, that all the terms
multiplied by a given combination of powers of the eccentricities, inclination and ratio
of the parallaxes, that is, all the terms with a given characteristic, form a convergent
series.
The equations to be considered are
d2x dy
dt 2 dt+ +L'y= R,
2n' + L'x + L"y = R',