596 HANSEN ON THE PERTURBATIONS OF BODIES IN 
The coefficients of the powers and products of the coordinates 
z and y of this expression are whole and rational functions of 
A, B, and =: By virtue of the foregoing expressions of A and 
1+ ¢cosf’ f 1 
a’ (1 — e?) ee 
whole and rational functions of sin f' and cos f’, and may conse- 
quently be reduced to the form 
ay + a, cos f' + a,cos2f'+.... + a, cospf 
+ 6, sinf' + B,sn2f'+....+ 6, snp f', 
where the coefficients are whole and rational functions of the 
eccentricity e! of the planet and of the mutual inclination of the 
orbits of the comet and planet. Consequently there do not re- 
sult any series that go on to infinity according to the powers of 
I and e’. If we denote the coefficients in general, when reduced 
to the above form, by C;;, we have then 
Q= Say’ Cyr 
neglecting the terms in which k + /is <2. We see readily that 
in the last term of the development of C;,; that has been pointed 
out, » =2(k+ 71) +1. Call now the eccentric anomaly of the 
comet w, then is 
B, and of the value , these coefficients are all 
e=cosu—e; y= V¥1—e.sin4u; 
and consequently ay’ is a whole and rational function of sin u 
and cos uw, in which the coefficients are also whole and rational 
functions of e and “1 — e. Consequently zy! is reduced to 
the following infinite series :— 
ay! =y, + 7, COSU+ yo COS 2U+4.... + ¥n 47008 (kK + Du 
or = e)Sinu + « sin 2u+.... + 4&4 ,sin (kK+/))4u, 
according as 7 is an even or an odd number. In this expres- — 
sion the coefficients are also whole and rational functions of e— 
and “1—e?. By the multiplication of this expression for a* y! 
into the above expression for C,,;, there results finally, 
OQ => K; jy cos @u + 7f!) + 2Z; 9 sin (tu + df"), 
in which, infinite series proceeding according to the powers and 
products of the eccentricity and inclination, are altogether 
avoided, and consequently the natural convergence of the dis- 
turbance-function is not at all vitiated, or at all events in the 
smallest possible degree. 
