790 
MR. G-. H. DARWIN ON THE SECULAR CHANGES IN 
D% — (/Cl + K. z ) ( k 2 4- a) j Sj-j + (** + dt }j S1 
+ tlie same with 2 and 1 reversed 
D 4 2/=(/Ci+/c 2 )(« ; a +a)|Ui—T x j+(^ k:i L +—) lJi4- &c. 
^ ) ^ = (' C 1 + K: 3)( ,<: 2 + /S)|s i — 1 Sjj +( / K 1 - + rf —^S 1 + &C. 
^ i V — ( K i J r K i){ K i J r / 3 ) IT] +( # c i 2 +^) t 1 + & c - 
The last three of these equations are to be found by a parallel process, or else by 
symmetry. 
If the right hand sides of (198) be neglected, we clearly obtain, on integration, the 
first approximation (183) for 2 , y, £, rj. This first approximation was originally 
obtained by mere inspection. 
We now have to consider the effects of the small terms on the right on the 
constants of integration L x , L. 2 , L(, L/ introduced in the first approximation. 
The small terms on the right are, by means of the first approximation, capable of 
being arranged in one of the alternative forms 
cosl , sin] , ,, „ 
. \-Kd-j-t i W + the same with 2 lor 1 
smj 1 cosj 1 
Now consider the differential equation 
/7i >> /72 
j' fl + (a e +5 2 )— -j-a 2 b 2 x=A cos (at-j-rj)-j-Bt cos (at-\-y) . . . (199) 
First suppose that B is zero, so that the term in A exists alone. 
Assume x—Ct sin ( [at-\-y 7 ) as the solution. 
Then 
cPx 
dt* 
C{—aH sin (at- j-^) + 2 a cos (at-\-y)} 
fl^T 
~— = C{aH sin («t-f- 77 )— 4a 3 cos (at-\-r))} 
By substitution in (199), with B— 0 , we have 
C{— 4a 3 + 2 a (« 2 + b~) } = A 
Therefore the solution is 
*=- 2 ^hw in < ai +’)) 
