750 
ME. G. H. DARWIN ON THE SECULAR CHANGES IN 
Then making use of (76) and (77), and remembering that cos iz=P 2 —Q 2 > smi=2PQ, 
we may write equations (75), thus 
cli d 
( 2 P^n-=-[2^W I +2^W I]I + KP 3 +3^)W 1 +i(3P 3 +^)W 3 +W 0 ] 
+ (P>-Q>) 
dW n , dW 3 - 
di 
cl o' 
u b J 
+/P SW ) . (78) 
chi d d "W dW 
f t = 
df 
dg 
• (79) 
It is clear that by using these transformations we may put ijj=ip' = 0, x~X before 
differentiation, so that \Jj and y again disappear, and we may use the old development 
of W. 
The case where Diana and the moon are distinct bodies will be taken first, and it 
will now be convenient to make Diana identical with the sun. 
In this case after the differentiations are made we are not to put JW=N' and e=e. 
The only terms, out of which secular changes in i and n can arise, are those depend¬ 
ing on the sidereal semi-diurnal and diurnal tides, for all others are periodic with the 
longitudes of the two disturbing bodies. Hence the disturbing function is reduced to 
W n and W 2 . Also dW^/dN' and dWJdN' can only contribute periodic terms, 
because N — N' is not zero, and by hypothesis the nodes revolve uniformly on the 
ecliptic. 
Then if we consider that here p' is not equal p, nor q to q, we see that, as far as is 
of present interest, 
W n = 2F cos 2f P^Q^p 2 —q 2 ) 2 — 2p 2 <? 3 ] [(y/ 2 —/ 2 ) 2 — 2p' 2 q' 2 ^\ 
Wo =2G cos g PZQ^pZ-QyiipZ-qty-Zpyjip'Z-qy^py*-] 
Also the equations of variation of i and n are simply 
Then if we put 
(2PC)»|=(- P3 -^ 
~ dW a 
d£ 
dW 3 
dg _ 
dn dW dW 3 
dt di dg 
/= 2P l Q\{p 2 - rf - 2pY][(p' 2 - q'~Y - 2 p'Y 2 ] 
= -§- sin 4 z(l —f silk/) (1 — f sin 2 /) 
l r =P^^(P 2 -^) 2 [(^-g 2 ) 2 -2pY][(^-W) 3 -2p¥ 3 ] 
= j sin- i cos 2 1(1 —f sin 2 / (1 —f sin 2 /) J 
(80) 
