1921.] Gifford.—Variations in Size of a Comet’s Head. 251 
proportional to the distance from the centre, the diameter of the head, 
instead of varying directly as the cube of the distance from the Sun, would 
vary approximately as the square of that distance. As the central con¬ 
densation increases the effectiveness of the mutual attractions in causing 
diminution of volume diminishes. When the density at every point is 
inversely proportional to the square of the distance of the point from the 
centre, the mutual attraction of the particles is unable to produce any 
further condensation, and the diameter is proportional to the distance from 
the Sun. If the central condensation is greater than this the energy within 
the swarm prevents the con¬ 
vergence of the orbits from 
having its full effect, and the 
diminution of volume is less 
* 
than it would be if each 
particle described an unper¬ 
turbed orbit round the Sun. 
If, on the other hand, the 
density were to increase out¬ 
wards, as in a planetary 
nebula, the variation in 
volume would be more sur¬ 
prising than it usually is. 
For example, if the density 
at any point within the swarm 
were directly proportional to 
its distance from the centre, 
the diameter of the swarm 
would vary approximately as 
the fourth power of the dis¬ 
tance ; and so on. 
Thus we see that the 
effectiveness of gravitation in 
reducing the diameter of a 
comet’s head is greater the 
higher the density towards 
the outside of the swarm and 
diminishes steadily as the 
central condensation is in¬ 
creased. Comets’ heads in 
which the nucleus is strongly 
marked should not show such 
marked changes of volume as 
those in which the matter is 
more evenly distributed, and 
observations of the changes in diameter of the heads of different comets 
should give us information as to the distribution of density within 
them. 
Effect of Differences in Period of Revolution for Different Particles .—To 
find the probable effect of the different periods of revolution of different 
particles in the swarm let us take once more Encke’s Comet as an illustra¬ 
tion. Its greatest distance is 4-08 astronomical units ; and its least distance 
0-34 astronomical units ; so its major axis is 4-42 astronomical units, or 
Fig. 3. 
The point x = F42, y = 281, outside the 
limits of the diagram, comes above the curve, 
y — 226, corresponding to x = 142. 
