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.

104 MR BROWN, THE SOLUTION OF A PAIR OF SIMULTANEOUS LINEAR
But the effect of putting Ç-1 for ' in à is only to interchange an even number of
rows and columns and therefore to leave A unaltered. Making this change in the last
equation we find
A = - (-u, Ds2 + u2Ds, - zu4Ds + s4Du3) C2.
Whence, by addition,
2A = - {/i2 - 2m (s2U1 - ZS1) -f34 + 2m (4U3 - u4Ss3)} C12
= (C - (2- 34),12 =- 222,
in virtue of (7) and (12). Hence A=-(7~2.
A I s2A + ~2À
Finally, AQ. = 2A, etc.
and Q 1=- D-' (s2A + 2uA), etc.
C12
And the particular integral corresponding to the right-hand members, A, A, is
u = - {Z)1D-l (s2A + ut2A) - u2D-l (slA + îtnA)
12
- u.D- (s4A + 4A) + uD-1 (3A + A )}......(13),
s = C {lD-~ (s2A + Z2A)- s2D-' (sA + t1A)
^12
-- s3D- (s4A + u4A) + s4D-1 (sA + 23A)}.
It is easy to see that s is derivable from u (as it should be) by putting g-~ for g. In
fact, the coefficient of u1 in the first term is conjugate to that of u2 in the second term,
that of i3 in the third term is a pure imaginary and that of u4 in the last term is real.
VI.
In the applications of this result to the Lunar Theory A is always an expression of
the form
Sqii'+ i +S i-., i =0, 1, + 2,...
where r, qi, qi' are known constants; A is derived from A by putting ~-1 for ~. Thus A,
A are conjugate complexes whose real and imaginary parts are respectively sums of cosines
and sines. The corresponding particular integral should in general be of the same form.
Hence a difficulty arises owing to the fact that t4, 84 contain t in a non-periodic form. I
shall now show that in general all the non-periodic parts disappear from the particular
integral.
Put
u = u4' + LBu.t (n - n'),
s4 = s4' + tBs3t (n - n').