372 Proceedings of Roy cd Society of Edinburgh. [sess. 
in the first place, the nature of these perturbations on various 
assumptions as to the mass of the planet. If it be found that the 
perturbations would not suffice to prevent a return of the comet, 
in a moderately changed orbit, except on the assumption of a mass 
so great that its influence on other planets could not have escaped 
notice, then we may he sure that, if comets 1264 and 1556 were 
identical, the comet must have returned as an unrecognised comet 
in an altered orbit. If we can identify comet 1556 with such a 
comet seen in the last half century, a beautiful problem presents 
itself : Given an orbit transformed into another given orbit by a 
planet of unknown mass in a position approximately known, 
determine the mass and exact longitude of the disturbing planet. 
The estimation of the general character of the perturbations is 
facilitated in the present case by the following considerations : — 
1. The aphelion radius vector (or the line of apsides) is very 
nearly in the line at right angles to the line of Nodes, 
being only 4|° from it. 
2. The comet’s aphelion being 88 times, and the planet 100 
times, the mean distance of the earth from the sun, and 
the angle between the radii vectores of the two bodies at 
the aphelion being 31°, it follows that, at and about the 
time of greatest disturbance, the perturbations by the 
planet are almost entirely perpendicular to the plane of 
the comet’s orbit, so increasing the inclination and retro- 
grading the line of Nodes. 
3. At any other position of the planet where there is any 
component in the plane of the comet’s orbit, the action 
is such as to increase the longitude of Perihelion 
I have made a preliminary computation of the general character 
and amount of these perturbations, and find that if the new planet 
have the same mass as Jupiter, the orbit of this comet would not 
he so seriously affected as was that of Lexell’s comet by Jupiter ; 
hut the plane of the new orbit would he inclined to that of the 
old one at about 5°, so that the longitude of the Node would be 
retrograded about 12°, and the inclination of the orbit to the 
Ecliptic would be increased by about 3°, and the longitude of 
Perihelion would he advanced slightly. 
But the number of comets affected by this new planet is so 
