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.

102 MR BROWN, THE SOLUTION OF A PAIR OF SIMULTANEOUS LINEAR
for the fourth solution, these two forms must be the same except as to a constant
factor. Hence
kauo D- (u1D2 -u2DuI)
K 213 U~ DU- - u-DLU - )
ln U 32
This relation is a somewhat remarkable one. In investigations where the arbitrary
x a~y
constants are varied-and there are many such-we have a means of obtaining ', aY
an' an
(which are the most troublesome to find) when the numerical value of the ratio n'/n has
been used in finding x, y. A direct proof of this relation is desirable. This and the
theorems which I have given elsewhere* are probably particular cases of some much more
general theorem. Thus, of the four integrals of the linear equations two only are required
for the development of the lunar theory, the other two arising from additions to the
arbitrary constants in the particular solution of the original equations.
V.
Having obtained the solution of
(D + m)2u + Mu + Ns = 0,
(D- m)2s + Ms + Nu = 0,
in the form u=2Qjuj, s= Qjsj, j=1, 2, 3, 4,
the next problem is to find the solution of
(D + nm)2t + Mu + Ns = A,
(D- m)2s + Ms + u = A,
where A, A are functions of the time.
Following the usual method of varying the arbitraries we have,Du^j. DQ = A, YDsj.DQj = A (1.
ujDQj = 0,:sjDQj = 0
These must be solved in order to find the variable values of the arbitraries. The only
difficulty is to find these values in forms sufficiently simple to be of use.
The expressions at the end of II. show that we can derive s,4/s, from u,/u, by putting
-~1 for Ç and changing the sign. For ul, s2 interchange as do u2, sl, while D changes
sign: u3 becomes -S3. Since
t4 = u3 (P +'QO),
we have 4= s3 (- P + Q).
Hence U4S3 - s4 3 = 2,3s3P
= S2 s - S2................................(11)
by the result obtained in III.
* P?'oc. London Math. Soc. Vol. xxvIII. pp. 143-155.