PROFESSOR CHALfJS ON THE PROBLEM OF THREE BODIES. 
533 
11. In order that our method of solution may be successful, the differential, 
Qdt 
sin®j9 
(1 — 4e eosp), 
must admit of exact integration to terms inclusive of the second power of e. Now it 
will be seen by referring to the expression in art. 9 for which Q was substituted, that 
this integration depends on the following integrals, which are exact. The integra- 
tion can, therefore, be effected. 
4e 
\ COS p)dt= — -| — 
] sin^jg ^ ^ ' n sinjo ' n 
cot;?-l-4e^ 
dt= — 
J slidp 
n 
cos ip + sq) 
dt r 
] ^ n + s(n — n!) 
sin®/) J ■ 
cos {p + sq) 
* n-\-s{n — d) 
o 
S o 
11 
1 II 1 
cos(2/)4-s3’) 
■It 1 
nco&p cos/)' -(-n'sinj9 sm/)' 
sin^p 
cos/)' 
sin/) 
^cos (/) -|-s/) cos (/)— dt 
pi-^s[n—7il)~' n—s[n 
1 cos sq 
' rd- — sin/) 
+ &C.] 
/ cos(2/) + sg) cos(2?)-sg) \ dt . o Xro 1 
^ '\2n-\-s[7i—'n!)'2n—s{n—'d)j&\\)dp ^ ’ ’ 'J 
= -2-4j5^5|^(|P^+cos^?cot7>) [^= + 1 , ±2, &c.] 
J^-( 
cos (2/)-l-S/) COS S/ 
dt 
= 2. 
2 cos [p + sq) 
^{n — 'n!){2n + s{n — 'd)'^ sin/) 
2n-\-s[n — ri!) s{n—n') J sin^/)’ 
r / , / , • ■ , , dt „ cos(/y-t-5/) 
J 2 . (w COS /) COS (y 4-^^) + (w' - w ) ) SI n ps\n{p-\- sq ) ) = — 2 sjn^ • 
Since the factors which multiply tlie two last differentials are different according 
as s is positive or negative, it is important to remark that these integrals have been 
obtained witliout giving to s the -f- and — signs. The factors which multiply the 
three preceding differentials do not contain s. 
From the results of these integrations the following equation is readily obtained : 
^ Q,dt ^ ^ \ f r ‘i r 9 
J -4e cos p) = 
, {ae t?Ao , ^ X , c?^Ao \ 1 
2 .; 
Le 
cos sq 
id — s^{n — n')® sin/) 
+ sq sinp+cos sq cos;,)4^ 
4 A 2 
