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.

106 MR BROWN, THE SOLUTION OF A PAIR OF EQUATIONS, ETC.
where [lr] denotes the constant term in the expansion of J as a sum of cosines. Now
s2A + u2A and s1A +,1A are conjugate. Hence
[s2A + z2A]o = [s1A + 1A]o = [(si + S2) A]o
Thus the non-periodic part is
(1, - 2) [(Si + s2) A] (n - n')t. C2............................. (15).
In the applications to the Lunar Theory, the part of the complementary function
used is obtained by putting Q3=0=Q4, and the constants in uz, u2 are so adjusted
that we can put Q=1= =Q. I shall show that (15) is equivalent to a small addition
&c to c in the index of ' in
11 +?t2 = ie+ir2i1C +.is/2i+1-c
squares and higher powers of &c being neglected.
Put c + c for c in the last expression.* It becomes
u1 é8c + 2t2 -8C
Remembering that = exp. (n- n') t and expanding in powers of 8c we obtain
21i + h2 + (1i - u2) ctL (n - n') t.
Comparing with (15) it is evident that we can put
3c = [(s1 +,s) A]o + C2.
This is nothing else than the general form of the expression which I obtained in
a paper, "Investigations in the Lunar Theory *." For
12 =f/2 = [f,] = j (2j + 1 + m + c) ej2+ Sj (2j - 1 - m + c) e_2,
on substitution of the values (4) in fi. Also s, + s8 is the same as the expression there
denoted by Se. The comparison of A with the remainder of the equation of the paper
just referred to will follow from what precedes that equation. The general case is given
in my memoir on "The Theory of the Motion of the Moon, etc.t." No useful purpose
will be served by giving further details of the comparison of the two forms for c.
The final conclusion is that the non-periodic terms either disappear of their own accord
*or belong to a part of the complementary function which is not to be included in the
general development. The last part of this investigation-concerning ec-is of course only
applicable to cases similar to those which occur in the Lunar Theory where we proceed by
continued approximation and where we require to have only periodic terns. In the general
problem the non-periodic terms will remain.
* American Jour. Math. Vol. xvII. p. 336, equation (16). + lMe. R. A. S. Vol. LIII. p. 75.