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. 105
Then,u4, s4' are periodic. The sumi of the third and fourth terms of (13) becomes
- t3D-1 (4'A + u1A) + u4D-1 (s3A + JA)
- [utD-1 {(s3A + uA) t} + u3tD-1 (sA + U3A)] tB (n - n').
The first line of this expression is in general periodic. The second line becomes, on integrating its first term by parts,
u3BD-2 (sA + f3A).
The non-periodic part thus disappears.
When we perform the double integration involved in this last expression, we obtain
u3 [fo + C'i (n - n') t + periodic part}
where CO, C7 are arbitraries. The terms containing Co, C, are simply parts of the complementary function and may be considered as contained in Q3u3 + Q4u4. The particular
integral may therefore be written
1 = C(sA + 9A) - [uDD (lSA - lUA) + u1D-' (sA + UA)
- I3D-1 {s4A + u4'A - BD-1 (sA + t3A)}].....(14),
which is its final forni.
VII.
In general this particular integral consists only of periodic terms. There are, however, two cases in which non-periodic terms may arise. If =an odd integer, that is,
if A is of the form q^, '2i+1, the integrals multiplied by 2t4' and t4n might give rise to
terms of the form at where a is a constant.
In this case, sA + u3A is of the form,/3i (2i -~2i) and therefore its integral will
be periodic. The last term of (14) is of the form
- uD- costs. + power series in 2)
- -U3 (tk + k' + power series in 2),
k, k' being constants, the former definite and the latter arbitrary. The terms - u3(tk +k')
may be written
Ic kI- 4'
- k'u - {utBt (n - n) + U4} Bt (n - n') + Bt (n - n')
The first two terms of this may be considered as included in the part Qu + Q4ut4 of the
complementary function; the last part is definite and periodic. Hence no non-periodic
part remains.
The second case of non-periodicity occurs when
A = iqi C2i+1+C + {iqi' 2i+1-C.
Here the first two terms of (14) may give rise to the non-periodic part
{uot (n - n') [sA +?uA], - utt (n - n) [s A + ulA]o} - C2,
VOL. XVTIII. 14