ORBITS OF LARGE INCLINATIONS AND ECCENTRICITIES. 595 
vergence is so much diminished that it is necessary to give up 
the use of the infinite series thence arising. This takes place to 
a much greater extent when eccentricities and inclinations simi- 
lar to those of the orbit of a comet are in question. It is there- 
fore necessary, in the solution of the problem before us, both in 
the disturbance-function and in all the other functions whose 
expansions are required, to avoid infinite series proceeding ac- 
cording to the powers of the eccentricity and the inclination of 
the orbit of the comet. 
The total disuse of such tN¥1N1TE series is the basis of the 
method which is here represented. 
For this purpose let 
H=Acosf+ Bsinf, 
where f represents the true anomaly of the comet, and 
A= cos? I cos (f'— 2k) - sin? 5 I cos (f' + 2N) 
B= cost > Pan (if S2e,> sin? Pent ee oN), 
in which /’ represents the true anomaly of the planet, I the 
mutual inclination of the orbits of the comet and planet, and 
N + K represents the distance of the perihelion from the ascend- 
ing node of the orbit of the comet reckoned on the plane of the 
orbit of the comet. Call now the disturbance-function ©, and 
the masses of the sun, of the comet, and of the planet respectively 
M, m, m’, then have we, in the case in which r <7’, 
/ 2 
= ane cade ac ke. }. 
Substitute now the above expression for H in the preceding 
values of U,, U;, &c., and these again in the expression for 0, 
and make 
x= —cos f; y =—sin f, 
then 
B fBinsig ' ; 
x 3 3 >) + vy” 75 AB 
Guys Sra 1 
m! ich (a8 ¥, 7 
o= = a5 3 a (15 3 ‘ 
5 5 a 
a0 S(Gaw—3a) v7 fe(Ge 3 8) 
+ &c. 
