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.

DIFFERENTIAL EQUATIONS, WHICH OCCUR IN THE LUNAR THEORY. 97
with their first integral,
F-Du.Ds+ m2(u+s)2+ =0C,
admit a particular solution,
U =_ u = O ai 2i+l, 8= s0 = a_i '2i-1,
containing two arbitrary constants; these constants are the quantities denoted by n, t0
above. The coefficients ai are functions of n and the known constants present in the
differential equations.
Put U=Uo+U1, S = o+S1,
and, after expansion in powers of 1i, sl, neglect squares and products of these quantities.
Omnitting the suffix, and giving proper meanings to M, N, the resulting equations
become those denoted by (2) above.
The first integral F= C becomes
aF aF
= u21 + s=O0.
auo0 as
If, however, we had deduced this first integral directly from (2), it would have been
= C', where C' is an arbitrary constant. When the equations (2) are considered independently the constant C' must be retained.
Three independent solutions of (2) are known. In finding the principal part of the
motion of the lunar perigee Dr Hill* gave one of them, namely, U =Du,, s = Ds, and
obtained the forms of the other two; the coefficients of the latter have been obtained
by myself. It is therefore only necessary to find a fourth solution, linearly independent
of the other three, in order to obtain the generai solution.
II.
The Fourth Integral of the Equations.
(D + m) u + Mu + Ns=.............................. (3),
(D - m) s + Ms + u = 0.................................(3').
The known integrals may be denoted by
-1 = E2 e++C S1= o? E-.2i -1+C
u 2 =:ii 'i2ii+-c S =:i' _ 2i-l-C+
- ii2i+l —, 3-_ i= — 2i —1-C. )....................(4).,3 = Ei (2i + 1) a2i+l, S3 =,i (2i - 1) a_2i-l
* Acta Math. Vol. viII. pp. 1-36. + Mem. R. A. S. Vol. LIII. p. 94.
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