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.

100 MR BROWN, THE SOLUTION OF A PAIR OF SIMULTANEOUS LINEAR
III.
Although this is probably the simplest form obtainable for u4, it is unsuitable for
calculation. The values of u1,... are all of the form
sum of cosines+ L (sum of sines).
To adapt u4 to calculation it is best to express it in the form
3 (P + Qt)
where P, Q are real. I shall show that
* s1U~ 1s2 0 U- U)8
U,3s + S Us U3 _
-u3+ (U2 8S U2 - (2m+..3....(9).
Since 23 = 0 =3 and f- = 12, we have
1 C12 - u2f3 - u1f23.i2 UIDu2 - u2Dul + u2Ds - ulDs2 + s1Duz - s^Du1
U3S3 83 2 U3s3 i32 2u3S3
S1Ut - 21S2 Du3s s - U2U u1S2 DU1 - U2- Du1 S+ D ( - U1s2)
U3S -m S3. 32 383
U2-U12 fDît3 -DS3 i 3 U 3 L3 82
-SU,-Us2,(Du3 s) 8?31 -s,
U'3S3 U3 S3 S U23
Submitting this to the operation D-1 and transposing we obtain the required expression.
It is easy to see that (9) is of the required form. For when we put - for t, that
is, ~-1 for g, the expressions
U1, U2, S1, s 3, u, s, D-1, D respectively
become S2, S, U 2, U S3 -, -3, - u — 1 -D;
the first term of (9) is therefore unchanged, while the second term simply changes sign.
Hence the first terrn is real and the second a pure imaginary.
IV.
It is necessary to examine the four solutions and especially the one last found a
little more closely. Write
u4 = u3 (P + D-1P,).
The expressions (4) show that P and P1, being both real, will be expressible as sums
of cosines of multiples of the angle 2(n -n')t. As P1 contains a constant term B, D-'P,
contains a term of the form tBt (n-n'), and therefore u4 is of the form
ut3 [Bt (n -n') + a power series in r2.
It is therefore of the same form as u3, except for the part
tBtu3 (n - n').