114 A NEW METHOD OF DETERMINING 
If the eccentric anomaly is taken as the independent variable we have for the 
complete integral 
W.=kh+ hk cos 4 + k, sin 7 — hy ine de 27) (F) dt. 
Introducing the true anomaly instead of the eccentric, we have, 
cos w + e : sin w cos 
>» sny = —___, 
since cos 7 = oS 
a is 
1+ ecosw 
= ky ke : h’ d2 
Wo=khtek + {9 608 © -+- pepsi — hh J (142 =) (FF) dt. 
Neglecting the terms having cos w and p sin w we have in JV the constants h 
and €) hy. 
ho 
The integral of dj; is 
h, om dQ 
ait hth fi) a 
hg 
From the expression for d ; we find 
h Wh? 7 dQ 
dji=—i,l(g)& 
a : 5 h , 
Integrating this, making use of the value of ;', and adding the constants, we have 
h h 
Qo = == 
My It 
0 h? dQ 
=14h)+eh—h f (1+ 2.) (ae dt. 
And since the quantities under the sign of integration do not have any constant terms 
Wwe can write 
yh te 
foe == 1 +k + ek, + periodic terms 
h 5 0 
a = 1 eis + periodic terms 
