840 
MR. G. H. DARWIN ON THE SECULAR CHANGES IN 
By differentiation we find that, when e = 0, 
d6 rj-f) 
= 2 sin (42— nr), —sin 2(42 — —=—f sin (42 — sr)fi-“ 2 ~ s i n 3(42— ct) 
de 
de 3 
yy= —11 sin 2(42 —sin 4(42 —sr), (y i =2 — 2 cos (42 —ct) 
d0 \3 
de 
= 6 sin (42 — nr) — 2 sin 3(42 — nr), =6 — 8 COS 2(42 — ter) + 2 COS 4(42 — trr) 
Ye C0S — 1 cos 3(42 —nr), (^jA yj=5 sin 2(42 — ct)— | sin 4(42 — m) 
—V———— cos 4(42 — -t) 
de~) ~ 8 8 C0S 4(/2 rfe fZe 3 ’ 
f+8 COS 2(42 — cos 4(42 — nr) 
To expand cos (k6-\-(3) by means of Maclaurin’s theorem, we require the values 
of the following differentials when e = 0 and 9=S2 :— 
I cos (te+P)=-k sin (W+/3)f 
^ cos (i0+/8)= —F cos (k0+/3)(~J-h sin (F>+/3)f| 
S cos (W+/3)=^ 3 sin (W+/S)(g) : -3F cos sin (F?+0)ff 
£cos (F?+/3)=Fcos (W+^)(f ) 4 +6F sin (W+/3)(f j^-SFcos (k0+P)(~d) 
-4F cos (M+flf sin (M+/3)0 
Now when e = 0, Jc9-\-f3= };£1 -f-/3, and the values of the differentials and functions ol 
differentials of e are given above. Then if we substitute for these functions then 
o 
values, and express the products of sines and cosines as the sums of sines and cosines, 
and introduce the abridged notation in which A’42+/3+s(42 — ct) is written (k-\-s), we 
have 
