369 
of Edinburgh, Session 1879-80. 
that the square of its ratio to that of the comet may he neglected), and 
in a direction making a given angle with the tangent to the orbit. The 
result is that such particles will lie approximately on a semi-parabola, 
the vertex being at the head of the comet. When the ejection is 
towards regions outside the orbit, the parabola lies behind the head 
of the comet ; but if the ejection be inwards the parabola precedes 
the head. This parabola diminishes in parameter as the curvature 
of the orbit increases.* There can be no doubt that here we have a 
very striking resemblance, if no more, to the form usually assumed 
by the tails of comets ; and for comets with many tails (like that of 
1744) we require only a greater number of definite directions of 
maximum ejection. Why this ejection is generally (though by no 
means always) outward, (for several comets have had two tails, of 
which one was turned towards the sun) we cannot attempt to explain 
till we know at what part of the group of masses the impacts are 
most likely to take place. 
The present theory differs altogether from that of Olbers, Bessel, 
and others, in assuming the fragments which form the tail to have 
but little velocity relatively to the nucleus, while the received theory 
assigns them very rapid motion along the tail : — Olbers says as 
much as a million miles per day. The one theory endeavours to 
represent the motion as a result of the received law of gravity ; the 
other introduces the hypothesis of a solar repulsive force often 
* These conclusions are found to follow easily from the very simple investi- 
gation for a circular orbit. For the approximate differences of radius-vector, 
and angle-vector, at the time t , of the comet and of a particle projected at time 
q, with relative velocity^?, from its head, in a direction making an angle ip with 
the tangent, are — 
Here a is the radius of the orbit, and a> the angular velocity in it. 
If co (t - ti) be a small angle x-> whose third and higher powers inay be 
neglected, these expressions take the form — 
from which we easily deduce the results stated above. It appears that in the 
majority of large comets xp is nearly a right angle,. 
and 
