We have already explained pp. (25-31) that there are two types of uniform three-dimensional space, viz., finite or elliptic space, in which the geodesies or straight lines are closed and of constant length, and infinite or hyperbolic space, in which the geodesies are unclosed and of infinite length. Euclidean or parabolic space may be regarded as a special third type, or as the dividing case between the two general types; but it is best to regard it as a variety of hyperbolic space, owing to the reality of the infinite. In elliptic space all geodesies through a point either return without intersecting again, or have at most one other fixed point of intersection. In hyperbolic space it does not appear necessary that a geodesic should consist of only a single branch, though this is of course the simplest case; if the geodesies have several branches, all geodesies through a point will have a fixed point on each branch, and two geodesies which intersect will have as many points of intersection as they have branches. When we consider two-dimensional space we find that, even in the elliptic type, the geodesies may have any number of intersections, and may be unclosed and infinite. The proof to the contrary on p. 31 does not apply when the geometry is confined to a surface.

In problems where more than two bodies are simultaneously in collision, the rules usually stated in the text books are not, in general, sufficient for their solution. It has been suggested to me that I should continue the subject of Collision by a discussion of the method required in such cases.

In what follows I confine myself to the case of smooth spheres impinging on one another, or on fixed bodies, so that rotations do not fall to be considered.