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.

MR BROWN, ON THE SOLUTION OF A PAIR, OF EQUATIONS, ETC. 95
where
L, L" cos
L~, are of the forms iqi sn (2i + 1)(n - n)(t - t),
R, of the forms iqi cos {i (t- to) + 7 (t - t)} (n- n'),
to, tl, 7, n, n,, qi being known constants, and i taking all positive and negative integral
values; r is either an integer, in which case t = to, or is incommensurable with an
integer.
The corresponding particular integral required is, in general,
yX = i Pi COS {i (t - to) + r (t - tl)} (n - n').
y -p'i sin
If we substitute this solution in the differential equations and equate to zero the
coefficients of like periodic terms, we obtain an infinite series of linear equations with an
infinite number of unknowns. The series are assumed to be convergent and in most cases
the coefficients diminish rapidly as i increases. Nevertheless, it is frequently found
necessary to proceed as far as i = + 5, demanding the determination of about 20 unknowns
from the same number of equations.
In the determination of the latitude the equation
d2z
dt2 + L1 z = R",
occurs; L1, R" are of similar forms to L, R', respectively. If zi, z, be two particular
integrals of
d2z
+ L1 z = O,
it is known that the particular integral required is
z. C= Z2 l R dt - z fz R" dt,
where C is a constant given by
C = z dz dz,
è1t -Z2dt
I shall show in what follows how we may obtain a similar expression for the solution
of the simultaneous equations above, having a sufficiently simple form to be of use in
computations. Later the significance of the solutions is explained and certain exceptional
cases occurring in the Lunar Theory are treated. The results obtained have in fact been
used in the calculation of the terms of the third* and fourth orders in relation to the
eccentricities, the inclination and the ratio of the parallaxes.
* I.emon. 1R. A. S., Vol. LIII. pp. 163-202.