1921.] Gifford.—Variations in Size of a Comet’s Head. 
249 
the swarm. Such points will trace out orbits nearly parallel to one another 
but converging towards perihelion. 
As the swarm passes round the Sun the separate meteorites composing it 
crowd together, much as a large field of runners becomes congested whilst 
turning a sharp corner (fig. 1). 
If all the orbits are of the same excentricity, with the Sun in one focus 
and the major axes in the same straight line, the radial separation of any 
two orbits in any direction is proportional to their distance in that direc¬ 
tion from the Sun. Thus if 
— = 1 -j- e cos 6 and — = 1 + e cos 0 are 
r r 
the polar equations of two such orbits, and r i, r 2 the distances from the 
Sun at which any radial line cuts these orbits, 
. n — r 2 
r 2 
h 
= a constant. 
h 
r 2 
That is, if we neglect the mutual gravitation of its members, the diameter 
of the swarm is proportional to its distance from the Sun, and in particular 
the diameter of the swarm at perihelion is to that at aphelion as the 
perihelion distance is to the aphelion distance ; that is, AB : CD : : SB : SC. 
But it is clear that in the case of an ordinary comet we cannot neglect 
the mutual attractions of the meteorites ; for, if we could do so, since the 
orbits have slightly different major axes, the periods of revolution of the 
particles would differ slightly, and the materials of the head would gradu¬ 
ally be spread out around the orbit. This tendency is counteracted by 
the fact that at every point those meteorites winch are farthest, from the 
Sun are moving inwards through the swarm, and vice versa. The different 
particles are continually interchanging orbits, and this fact may enable 
the comet to escape for a long period the fate, which must ultimately 
overtake it, of being stretched out to such an extent that its identity is 
lost. 
Furthermore, the diminution of diameter so far accounted for is much 
less than that actually observed. For instance, in the case of Encke’s 
Comet the perihelion distance is to the aphelion distance as 0*34 : 4*08— 
that is, at 1 : 12. We might expect the perihelion diameter to be one- 
