842 
MR. G. H. DARWIN ON THE SECULAR CHANGES IN 
(e°) cos (2<?) = ( e o){2}+(e)[{l} + {3}]+(e»)[{0} + {2} + {4}] 
+ (e 3 )[{-l} + {l} + {3} + {5}]+(e^)[{-2} + {0) + {2} + {4} + {6}] 
(e) cos (30) = (e){3}+(e 2 )[{2} + {4}]-f-(e 3 )[{l] + {3} + {5}J 
H“ ( e4 )[ {0} + {2 } —|- {4} —j— {6} ] 
(e) cos (0) = (e){l} + (e 2 )[{O) + {2}]+(e 3 )[{ — l} + {l} + {3}] 
+ (e^)[{-2} + {0} + {2} + {4}] 
(e 2 ) cos (4^) = (e 2 ){4} +(e 3 )[{3} + {5}] + (e 4 )[{2} + {4} + {6}] 
(e 2 ) cos (0) = (e 2 ){0} 
(e 3 ) cos (5^) = (e 3 ){5) +(e 4 )[{4) + {6}] 
(e 3 ) cos (^) = (e 3 ){l]+(ef{0} + {2}] 
In these expressions we have no right, as yet, to assume that { — 2} and { — 1} are 
different from {2} and {1} ; and in fact we shall find that in the expansion for <t>(a) 
they are different, but in that for It they are the same. 
Then adding up these, and rejecting terms of the third and fourth orders by the first 
rule of approximation, we have 
*(«) = [( e °) + (e 2 ) + (e*)] {2} + [(e) + (e 3 )][ {1} + {3} ]+[(e 2 ) + (e^)][{0} + {4}] 
+ (e 3 ){-l}+(e 4 ){-2} 
It will be observed that {5} and {6} are wanting, and might have been dropped 
from the expansions. Also {0} and {4} are terms of the second order, therefore 
wherever they are multiplied by (e 4 ) they might have been dropped. Hence 
(e 3 ) cos (50) need not have been expanded at all. A little further consideration is 
required to show that (e 3 ) cos (0) need not have been expanded. 
(e 3 ) cos (0) is an abbreviation for -g-e 3 cos (0—a — Szj), and therefore in this 
case {1} = cos (12 — a — 3 tjt) and [2j= cos (2/2 — a —4zrsj ; but in every other case 
{1}= cos (/2 + a + sy) and {2} r= cos (212 +a). Hence the terms {1} and {2} in 
(e 3 ) cos (0) are of the third and fourth orders and may be dropped, and {0} may also 
be dropped. Thus the wRole of (e 3 ) cos (0) may be dropped. 
With respect to { — 2} and {— 1}, observe that {2} in the expansion of cos (^0-4-/^) 
stands for cos [2/2 + (^ 1 —2)trr+yS 1 ]; and {— 2] in the expansion of cos stands 
for cos [2/2 — (A + 2 )ut—/ 3J ; and k lt are either 1, 2, 3, or 4 ; and /3 1? are multiples 
of a constant. Hence {2} and {— 2} are necessarily different, but if (3 X and /3 2 
were multiples of nr they might be the same, and indeed in the expansion of R 
necessarily are the same. 
