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.

96 MR BROWN, THE SOLUTION OF A PAIR OF SIMULTANEOUS LINEAR
I.
In order that the series which occur may be all algebraical instead of trigonometrical,
we use the conjugate complexes ut, s, where
u = x+yt, s= x-yt.
We also put
W = exp. (n - n') (t - to),
d 1 d
dC L (tn-n')dt'
ni
m = n — to = O t.
n -
The generality of the results is not affected by the last supposition.
The simultaneous equations then take the form
(D + m)2 +Mt + Ns = A `
-..........................o..........(1),
(D - m)2 + Ms + Nu = A
where M, N are of the form pii,,
A is of the form pij-2i+1+T + p'/ 2i+1-T, J
M=M.
The bar placed over a letter or expression denotes here and elsewhere that t has been
changed to -, that is, ~-l put for.
To obtain the particular integrals of equations (1), it will first be necessary to obtain
four independent particular integrals of
(D + m)2 u + Mu + Ns = 0 )
(D- m)2s + Ms + =....................
Denote these integrals by
' = j, s=Sj, j=l, 2, 3, 4,
so that if Qj denote an arbitrary constant, the general solution of (2) is
=.j =jQj sj, j= 1, 2, 3, 4.
By supposing the Qj to have variable instead of constant values we can then proceed
to find a particular integral of (1) and thence their general solution.
In order to make certain of the later arguments clear it is necessary to indicate the
manner in which the equations (1) arise.
The equations
3 Ku
D2u + 2m Du + - m2 ( + s) - - = 0,
2 (us):
D2s - 2m Ds + m2 (u + s) ( = 0,
2(us)