286 
A MEMOIR ON THE PROBLEM OF DISTURBED ELLIPTIC MOTION. 
[21_ 2 
which give 
. sin fdf-«»+a+<?«»/ *+1±£mf da= o, 
J J 1 — e 2 a 
e cos fdf+ 
and we thence obtain 
da = 
sin f de 
de + 
2e sin / 
i Vl — e 2 
\/l — e 2 dii 7 na 2 Vl — e 2 dii 7 
dtH ;—^ — dt, 
na dr 
1 — e 2 
2 ae 
da — 
e sin f d\> 
Vl — e 2 dii 
na 2 e d]> 
dt, 
dii 7 , 
-j— dt + 
dr 
2 (1 + e cos f) 2 dii ^ 
na 
(1-e 2 ) 1 d\> 
7 Vi - e 2 sin f dii 7 , e + 2 cos f+ e cos 2 f dii 7j 
de = -j-dt-i di, 
na a?' na 2 Vl - e 2 dj> 
17 nae dr 
(2 + e cos/) sin / dil 
na 2 Vl—e 2 d]> 
the last of which equations, combined with 
(1 + ecos/) 2 , (2±ecos/)sin/, 
n _ ■ + 1 -e ! ’ 
di, 
gives 
, _ (1 - e 2 ) (- 2e 4- cos/+ e cos 2 /) do , (2 + e cos/) sin/ dil , 
9 nae (1 + e cos/) dr f na 2 e d|? 
The fourth equation of the formulae for the variations, viz., d]> = 0, gives 0 = dvr + df, 
and therefore dix = — df, that is, 
d'nr = — 
Vl — e 2 cos/ dii ^ + (2 + e cos/) sin/ dil ^ 
nae 
di 
na 
2 Vl -1 
dji 
and the complete system becomes therefore 
da = 
2esin/ dQj, 
n Vl - e 2 ar 
2 (1 + e cos/) 2 did 
na (1 — e 2 )- cty 
di, 
de = 
V? = 
(for = 
Vl — e 2 sin/ dii ^ ^ e + 2 cos/+ e cos 2 / dii ^ 
na dr na 2 Vl-e 2 # 
di, 
(2 + e cos/) sin/ dii 
na 2 e d}> 
Vl — e 2 sin/ dii ^ + (2 4- e cos/) sin/ dii ^ 
(1 — e 2 ) (— 2e + cos/+ e cos 2 /) dii , 
?iae (1 4- e cos/) dr 
nae 
d? 
na'e 
dji