THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
845 
We thus get the following rules for the use of the expansion of cos (k6-\-/3) for the 
determination of It. 
When k=l, omit in © 2 term in cos (1+2) 
in ©3 terms in cos (k —3), cos (1+1), cos (£+3) 
all of © 4 
When £— 2 , omit in © : term in cos (1+1) 
in 0 o terms in cos (k), cos ( 1 + 2 ) 
all of © 3 , 0 ^. 
Then following these rules, we find 
When k— 1 , j 8 = — 
cos (6 —ct) = ( 1 —e 2 ) cos (J2 — 7&) — e + e cos 2 (fl — — ) . . . . (274) 
When k= 2 , /3=— 2 tt 
3e 2 
cos 2(6 —ct)= cos 2(/2 — 73 -) — 2 e cos (fl — ■ ■ ■ ■ (275) 
These are the only series required for the expansion of It or ''T'(a), since by what is 
shown above, cos 3(6 —et) need not be expanded. 
Now multiply (270) by 1+fe 2 ; (271) by-§e(l+fe 2 ); (272) by §e(l+^e 2 ); and (273) 
by fe 2 ; add the four products together, and add fe 2 cos (« + 2 ct), and we find from 
(267) after reduction 
cp(a) = (l — hfe 2 -b i n- L e 4 ') cos (2/2 + «) —fe(l —- 8 -e 2 ) cos (/2 + «+ttt) 
+f-e(l — ” 5 “o"e 2 ) cos (3/2 + a — 77r)-j-h 2 -e 2 cos (412 + a — 2 ct) . . . (276) 
Next multiply (274) by 3e(l+^e 2 ); (275) by fe 2 ; add the two products, and add 
1+fe 2 , and we find from (267) after reduction, 
R=1—fe 2 +fe 4 +3e(l —- 8 -e 2 ) cos (/2— ^-ffe 2 cos 2(/2 — or) . . (277) 
5 Q 
MDCCCLXXX. 
