The N.Z. Journal of Science and Technology. 
42 
[Fee. 
attractive force ^ directed towards a fixed point S, AS being perpen¬ 
dicular to AT. The initial distance r Q of the body from S = AS. 
If v = 0, the path will be a straight line, the limiting form of the ellipse 
of which A and S are at once the vertices and the foci. 
If v 2 < - , the body will describe an ellipse of which S is the farther 
focus. r 
If v 2 — it will describe a circle with S as centre—that is the ellipse 
of which A is one vertex and both the foci coincide with S. 
2 
If v 2 > ^ < -y, it will describe an ellipse with A as vertex and S the 
nearer focus. 
2 
If v 2 — —, it will describe a para-bola. 
f 
2 
If v 2 > —, it will describe an hyperbola, which will be rectangular if 
«» = '- •- A +- 1 = ^ ( a /2 + 1 ) 2 . 
r v 2_1 r 
If v 2 = oo, the straight line AT will be the limiting form of the 
hyperbola. 
The velocities for the circle, the parabola, and the rectangular hyper¬ 
bola are as 1 : V 2 : ^2 -(- 1. 
Now, - is the kinetic energy of each unit mass acquired in approaching 
from an infinite distance to a point at distance r, or the kinetic energy 
which is required to carry the unit mass from a point at a distance r to 
an infinite distance against the action of the force. This is called the 
parabolic kinetol, since any body which possesses this quantity, of kinetic 
energy per unit mass will move in a parabolic path whatever the direction 
of its initial motion may be. Otherwise the orbit will be an ellipse or 
hyperbola, according as the initial kinetol is less than or greater than the 
parabolic kinetol. 
To find the eccentricity, the lengths of the axes of the curve, and the 
distance between the bodies at the instant of nearest approach :— 
(i.) First let the initial kinetol be less than half the parabolic kinetol. 
The path is then an ellipse, with A as one vertex and S the farther focus. 
The initial distance r 0 = a{ 1+e), where a is the semi-major axis and e the 
eccentricity. 
At every point of the curve 
v 
2 
2,a 
r 
> 
a 
or 
r — W 
r 
which is simply the symbolical statement of the fact that the excess of the 
parabolic over the actual kinetol is constant and is equal to the parabolic 
kinetol at a point which is at a distance from the focus equal to the major 
axis. 
Its actual energy at any point, therefore, is just sufficient to carry the 
body away from the focus to a distance equal to the major axis, if the 
direction of motion is changed so that it moves directly away from the focus. 
This is, of course, the energy that is acquired in falling to the point from 
the circle whose centre is the focus and radius the major axis. 
