844 
MR, G. H. DARWIN ON THE SECULAR CHANGES IN 
When Jc= 1, /3=a + rrr 
COS (0 + a + nr) = (1 — e 2 ) COS (/2 + a + zcr) —J- e (1—fe 2 ) COS (2/2 + a) — e COS (a + 2trr) 
+|e 2 COS (3/2 + a—m) . (272) 
When 7—4, /3=a — 2nr 
COS (40+a— 2ccr) = C0S (4/2 +a — 2m) — 4e COS (3/2 + a — ra^+^e' 3 COS (2/2 +a) . (273) 
These are all the series required for the expression of <t>(a), since cos (a + 2cx) does 
not involve 0, and by what has been shown above cos (50+ a— 3m) and cos (6 —a — 3m) 
need not be expanded. 
We now return again to the series for It or 'P'(a), and consider the nature of the 
approximations to be adopted there. 
With the same notation 
(e°) cos (0) = (e°){0} 
(e) cos (ff)=(e) {1} + (e”)[{0} + {2}]+(e 3 )[{ -1} + {1} + {3}] 
+ (e*)[{-2} + {0j + {2} + {4j] 
(e 2 ) cos (2^)=(e 3 ){2} +(e 3 )[{ 1} 4-{3}]+(e 4 )[{0j + {2} + {4}] 
(e s ) cos (3^)=(e 3 ){3j + (e*)[{2} + {4]J 
Since It is a function of 6—m, therefore after expansion it must be a function of 
/2 — m, and hence {1} must be necessarily identical with {— 1}, and {2} with { — 2}. 
Adding these up, and dropping terms of the third and fourth orders, 
R = [( e °) + (e 3 ) + (e 4 )] (0} + [(e) + (e 3 )] {1} + (e 3 ) { — 1} + [(e 2 ) + (e 4 )] {2} + (e 4 ) { — 2} 
Here {0} is a term of the order zero, {1} of the first order, and {2} of the second. 
Therefore by the first rule of approximation {2} and {—2} may be dropped when 
multiplied by (e 4 ). 
Also {3} and {4} may be dropped. 
Hence as far as concerns the present problem 
(e°) cos (0) = (e°){0} 
(e) cos (0) = (e){l)+(e 2 )[{O} + {2}]+(e 3 )[{-l} + {l}]+(e 4 ){O} 
(e 2 ) cos (20) = (e 2 ) {2} + (e 3 ) {1} + (e 4 ) {0} 
and (e 3 ) cos (30) need not be expanded. 
And the sum of these expressions is equal to H. 
