Royal Astronomical Society. 229 
I. Translation of a Letter from Professor Hansen to G. B. Airy, 
Esq., the Astronomer Royal, on a New Method of Computing the 
Perturbations of Planets, whose Eccentricities and Inclinations are 
not small. Communicated by G. B. Airy, Esq., A.R. 
«« Sir,—I hasten to communicate to you a piece of astronomical 
intelligence of some importance. You are aware that all the methods 
that we possess for calculating the perturbations of the planets sup- 
pose that the eccentricities and inclinations are small, and that for 
those of the celestial bodies, which move in orbits very eccentric 
and very much inclined, we have been hitherto obliged to calculate 
the differentials of the perturbations for a great number of points of 
the orbits, and to integrate them by mechanical quadratures. I have 
just now discovered a method by which we can calculate the abso- 
lute perturbations,—that is to say, the perturbations for any time 
whatever, whatever be the eccentricity of the ellipse and the incli- 
nation of the orbit. For a first example of this method, I have cal- 
culated the perturbations of the comet of Encke produced by Saturn. 
The series to which my method leads are of such rapid convergence, 
that the perturbations of the longitude contain only forty-six terms, 
and the perturbations of the radius vector and of the latitude, some- 
what fewer than this. I have reason to believe that it is impossible 
to reduce them to a less number of terms. Instead of writing here 
all the terms explicitly, allow me to represent generally the value for 
the time of perihelion passage. 
«* Here then is the first result of this kind, in which n dt repre- 
sents the perturbations of the mean longitude; wu those of the hyper- 
bolic logarithm of the radius vector, expressed in seconds, of the 
above-mentioned comet; g' the mean anomaly of Saturn; and¢ the 
time, of which the unit is a Julian year. 
ndt = —0-06—1°7152t-+ 1°56 sin g! —14°23 cos g! 
+ 23°41 sin 2 g'+20°65 cos 2 g! 
— 6°39 sin 3 g'+ 8°52 cos 3 g! 
2°65 sin 4 g'— 2°89 cos 4 g! 
‘43 sin 5 g'— U'96 cos 5 g' 
‘32 sin 6 g'+ 0°55 cos 6 g! 
u = —1°05—O'1611¢ + O° 
33°10 cos 2 g!—29°85 sin 2 g! 
‘01 cos 3 g’—11°47 sin 3 g! 
— 3:41 cos 4 g'+ 3°90 sin 4 g! 
+ 1°74 cos 5 g'+ 1°11 sin 5 g! 
b + 0°32 cos 6 g'— 0°82 sin 6 g! 
“In the Astronomische Nachrichten, vol. ix. No. 211, M. Encke has 
published for three periods the separate perturbations of this comet 
for each planet, and for the time of perihelion passage. We are 
therefore able to compare these perturbations with their preceding 
general value. But in this comparison it is necessary to remark, 
that in the calculation of the perturbations by mechanical quadra- 
tures, there arises in the perturbations of the epoch of the mean 
[++ ++1 
1 
0 
0°61 cos g! + 8°86 sing! 
3 
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