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. 99
Substituting the last result in (6'), we find
DL 2) (es)
L+2 -s
which, on integrating, gives
L Lo Lo
(:~S)- () 2 ~ U2 C3 + U 31 + t3 C12)2
where Lo is a new arbitrary constant.
Thence =(Q)+L -, etc.,
Thenee QI = (QI) + L~D-1 (u C23 + u2C,31 + u3C&2)2
in which (Qi) is a new arbitrary constant and D-1 denotes an integration, i.e. the operation
inverse to D.
If, finally, we now let Q1, Q2, Q3, Q4 represent four arbitrary constants, the general
solution of (2) is
u = Q121, + Q2u2 + Q3U3 + Q4U4,
S = QlSl + + Q Q33 + Q4S4,
where 4 = jUjD-'( 3 + u + uC2)2' j = 1, 2, 3.
(u1 C03 + 22 C31 + US:)2
This result is true whatever particular solutions are represented by
U1, 8 ~12, 82 2 3, 83
as long as they are linearly independent. As, however, the expression for u4 can be
very much simplified by using the values given earlier, I shall immediately proceed to
the special case under consideration.
It is easy to show that C3 0 = C23. For, looking at the forms assumed, we see
that u1, s, contain the factor tc, us, s the factor ~-' and u3, 3s have no such factor.
Hence f23 has the factor r', f3l the factor g'-". As c is supposed incommensurable with
unity, the equations (7') are only possible if C03 =O and C03 = 0.
Hence we have
u2 = uD u2Du3 - u Du + 2-u3Du - 1Du + u U, D, - u-, Du,,4CI22 = UlD-1 U27)83-U3D~2 + UsD-1
U3 Un U32
The first two terms of the right-hand side are integrable and become
U2 U1
1i - - n2 -
that is, zero. Whence considering C122 as absorbed in the arbitrary Q4, we have
114 = n3D- (uiDz-2Dui).(8).
U US -D~ -1 (U U2~-' ' t'l'0 )................................. (8).
We may similarly show that
s4 = s3 D- (s Ds2 - sDs,)
13-2