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.

DIFFERENTIAL EQUATIONS, WHICH OCCUR IN THE LUNAR THEORY. 103
Again, as the first integral obtained in II. is equally applicable to u4, s4, we have
34 =f34 = s4Du3 + 4Ds3 - u3Ds4 - s3Dt4 + 2m (s4u, - u4S3),
which, by inserting the expressions for U4, 84 just given, becomes
C34 = - 2 (S3Du3 - u3Ds3) P - 2tu3s3DQ + 2m (s2ul - us),
or, using the values of P, Q obtained in III.,
34 = - (S3DU3 - 3DS3) SU - 0C
+ (s1u2 - us2) (2m+ 3 -— Ds3) + 2m (s2i- $- S
whence c34 = - C..........................................(12).
We can show as in II. that C4=O0= 24.
Solving equations (10) we obtain
DQj= a
where
A= Du1, Du2, Du3, Du,
Ds,, Ds2, Ds3, Ds4
1 U1, U, U3, ut4
S1, 82, S3, S4
A1 = A, Du2, Du3, Du4
A, Ds, D Ds Ds4, etc.
0, U23, U3, U4
0, 82, S3, S4
In the determinant A the first minor of Du, is
DS2 (U3s4 - S3u4) + Ds3 (u4s2 - s4u2) + Ds4 (U2Is - S2u3),
= S2f34 + 83f42 + 84f23,
= 82 C34 + S3 C42 + St C23
Also, the first minor of Ds, is similarly
-(u2 C34 + t3 C42 + 24 C23).
The other minors of the elements in the first two rows of A are similar, the suffixes
following a cyclical order. We have thus all the minors of the elements A, A in the
determinants Aj.
Remembering that C34 =-C and that all the other constants Cij are zero, we obtain
A = - (sA + u2A) C12,
A2 = (s A + u1A) 0C2,
A3= (s4A + 4A) 0,,
A4 =- (s3A + usA) C12,
and A = - (s2Du1 - s1Du2 - s4Du3 + s3Du4) C0.