THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 13 
To develop the expression for (): we put 
cos J/.sin 1’ = sin A, sin I’ = & sin Kj, ) 
cos Il’ = k cos K, cos Icos Il’ = k, cos K,,J 
and hence 
i= cos f.cos f’.k cos (Il — K) + cos f. sin f’.k, sin (Il — K;) 
— sin f.cos f’.k sin (11 — A) + sin f. sin f’.k, cos (11 — 44). 
Introducing the eccentric anomaly «, we have 
a . A a . 
cos f = — (cos e—é), sin f = —. cos. sing, 
é being the eccentricity, and ¢ the angle of eccentricity ; and find 
._. H= cos «.cos &.k cos (11 — K) — cos ¢. ek cos (11 — K) 
— cos e.¢k cos (11 — K) + eek cos (11 — KX) 
+ cos «.sin ¢.cos ¢’.k, sin (II — A,) —sin ¢’.é. cos ¢’. k, sin (Il — A) 
— sin e.cos ¢.cos @.k sin (1— A) + sin e.é.cos p.k sin (1— XK) 
+ sine. sin e’.cos ?. cos ? .k, cos (Il — K)). 
= 
re d ‘ A A\2 
Substituting the value of ~, me fT in the expression for (=) we have 
(=). = 1+ a’*— 2e.cos « + & cos *« — 2aeek cos (Il — I) 
+ 2a¢k cos (11 — K) cos ¢ — 2ae’ cos @. k sin (II — FV) sin ¢ 
— [2a7e — 2aek cos ((I—K) + 2ak cos (Il — K) cos « 
— 2a cos o.k sin (I] — XX) sin €] .cos 
— [ — 2ae cos 9’. k, sin (11 — K,) + 2a cos > cos 9’. k, cos (11 — Aj) sin ¢ 
+ 2a cos ¢’.k, sin (11 — 4.) cos e] . sin & 
+ a? é€?. cos 7’. 
Putting 71, Go, 72, for the coefficients of cos ¢’, sin +’, cos */, respectively, and 7, for 
the term not affected by cos «' or sin ¢’, we have the abbreviated form 
9 
er = 7) — 71. Cos &’ — By. sin & + yz. cos *e’. (7) 
