70 A NEW METHOD OF DETERMINING 
In terms of the eccentric anomaly we have, at once, 
Tr 2 9 © 
(“) = 1—2ecose + € cose 
= 1-4 $6 — 2ecose + 3’ cos 2e. 
Substituting the values of cos «, and cos 2¢, we have 
r\2 5 a) Q) (3) 
(2) = 1+ 3¢—4J, cos g— 4-J., cos 2g — 4-J;, cos 3g — ete. 
a 
To find an expression for the factor 8 Lis z sin (f’ + Il’), for brevity, we let 
sin I sin J 
= . cos 9’ cos IT’, — .sin Il’, 
a a 
: 7’ sin f’ Y 5 
and from the known expressions for a sae A cos jf’, we get 
gia JT Fo ; (0) (2) s (1) (3) : 
= (S00 (jae IN) = [ee + Sy, | ¢, sing’ + 4 [ey =5 Ja ¢, sin 29’ + ete. 
a a 
(2) 
(0) (a) (3) 
— $2G + ee —vJ, |e,cosg’ + le ae | c, cos 2g’ + ete. 
we 
In the same way, if 
: ae 
CG = Sin T cos @ cos il, e, = 2+ sin UH, 
we find 
Shall ~P _s : : _(0) (2) : a) 6) : 
— poly =p) = J, +J, |essing + $| J +2 | sin 2g + ete. 
a ~ 
6 
athe (0) (2) @  _6) (6) 
— $ee,-+ [a —J, |c¢,cosg+3 | ahs + Jo. | c,cos 2g + ete. 
By means of the expressions for the factors 
r\2 f in I yr’. % j in I - S 5 
(Z)i, — ; 5 sin (f’ + Il’), Pelee : . sin(f + I), 
