566 
Proceedings of the Royal Society 
1. The locus of the second foci of the paths of all projectiles 
leaving a given point, with a given velocity, in a vertical plane, is 
a circle. 
2. The direction of projection for the greatest range on a given 
line, passing through the point of projection, bisects the angle 
between the vertical and the line. 
3. Any other point on the line which can be reached at all, can 
be reached by tivo different paths, and the directions of projection 
for these are equally inclined to the direction which gives the 
maximum range. 
4. If a projectile meet the line at right angles, the point which 
it strikes is the vertex of the other path by which it may be 
reached. 
5. The envelop of all possible paths in a vertical plane is an 
ellipse, one of whose foci is the centre of the earth, and the other 
the point of projection. (In the simpler case this is a parabola.) 
The proofs of these propositions are extremely simple. Thus, 
let E be the earth’s centre, P the point of projection, A the point 
A 
which the projectile would reach if fired vertically upwards. With 
centre E, and radius EA, describe a circle in the common plane of 
projection. This corresponds to the common directrix of the para- 
bolic paths in the ordinary theory. If F be the second focus of 
