250 
The N.Z. Journal op Science and Technology. 
[Nov. 
twelfth of the aphelion diameter. The observed diminution of diameter is 
enormously greater than this. 
It is necessary, therefore, to seek a further cause, and this is found in 
the mutual attractions of the separate meteorites which compose the head. 
The effect of these attractions depends very largely on the distribution of 
density within the swarm, but it usually consists of a further diminution of 
volume between aphelion and perihelion and a corresponding increase in 
the other half of the orbit. 
To get a rough idea of the magnitude of the effect of mutual attractions 
in altering the diameter of a swarm let us consider some of the simplest 
cases. 
Consider first a swarm of uniform density. Fix attention on a particle 
passing from the point A on the outside boundary of the swarm towards 
the centre 0 (fig. 2). At any point P it is attracted by the mass contained 
within the sphere passing through P 
and concentric with the cluster. This 
mass is proportional to x 3 , if OP = x. 
The force acting on the particle is pro¬ 
portional to X' 
X - « — that 
x z 
is, to x. 
Its motion is therefore simple harmonic 
motion. 
Other particles may be describing 
orbits approximately elliptical with dif¬ 
ferent excentricities, the limiting case 
being that of a circular orbit with 
radius equal to that of the swarm. 
If v be the maximum velocity in the 
case of the simple harmonic motion 
—that is, the velocity when at the 
centre 0—the acceleration a directed 
towards the centre when 
the 
v 2 
particle is at the extreme distance OA = r is given by the formula a = — ; 
T 
V 2 
.*. r — —. 
a 
That is, for any definite velocity at the centre of the swarm the amplitude 
of vibration is inversely proportional to the acceleration, and therefore 
inversely proportional to the attraction. 
Now, suppose that, owing to the convergence of the orbits, the distances 
between the particles are all reduced to half their former values. The 
mass crowded into any sphere within the cluster is eight times what it 
was before. A particle passing through the centre with the original velocity 
will only be able to get to one-eighth of the original distance from the 
centre. This suggests that in such a comet’s head the diameter will be 
directly proportional to the cube of the distance. The measurements given 
by Chambers on page 242 of The Story of the Comets of the diameter of 
Encke’s Comet in 1838 suggest that the diameter was approximately 
proportional to the cube of the distance (see fig. 3, in which the recorded 
values are indicated by crosses). 
These results must be greatly modified if the density within the swarm 
is distributed differently. Thus if the density at any point is inversely 
