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.

98 MR BROWN, THE SOLUTION OF A PAIR OF SIMULTANEOUS LINEAR
If Q1, Q2, Q3 be three arbitrary constants, then
u= jQjuj, s= jQysj, j= l, 2, 3...........................(5)
is a solution of the equations. Owing to the introduction of Q,, Q2, Q3, we can consider
ui,... 83 completely known; c is a constant which is supposed incommensurable with
unity.
To discover the fourth integral, the method of the Variation of Arbitrary Constants
is used in the usual way, by assuming that
'uiDQ, + u2DQ2 + u3DQ, = 0.
By substituting (4) in the differential equations we find
Du,. DQ1 + Du,. DQ2 + Du3. DQ3 = O,
j (sj D2Qj + 2Dsj. DQj - 2mnsDQj)= 0............................ (6).
Put u2Du3-u3Du2 = a1, etc.
Then DQ1 DQ2 DQ3 _
Then = L, suppose.
~a, 02 a3
Substituting in (6), the equation for L may be written,
(a=s) DL + 2LD (sas) - L (SsDa + 2mcas) = 0........................(6'),
where Eas = as, + a2s2 + a3s3, etc.
The last term of this equation can be shown to be zero. Substitute ul, si and u2, s,
successively in (3): multiply the resulting equations by s2, si respectively and subtract.
We thus obtain
(D + 2m) (s2Du, - i Du2) + (m2 + M) (s2ul - u2s) = 0.
Also, treating (3') in a similar manner,
(D - 2m) (u2Ds, - uDDs2) + (m2 + M1) (u2s - sul) = 0.
The sum of these two equations is integrable and gives
s2Du, - tl1Ds2 + u2Ds1 - s1Dz2 + 2m (s2U1 - u'2si) = C12,
where C12 is a constant. It should be noticed that this constant is not arbitrary since
the values of u1, si, u2, s. were definitely fixed, so that CI, may be treated as a known
constant.
Denote the last equation by
f 2= C.............................................
We find in an exactly similar manner
f23 = C23î f31= C31...................................... (7 ).
Multiply these three equations by ui, u2, us and add. Noticing the meanings attached
to aC, a2, a3, we obtain
u1 23 + 23 C31 + U3 12 = $_s.
Similarly 0 = u1Df23 + u2Df1 + u2Df2i
=,sDa + 2m$as.