Class C: **Directional** relation changing periodically; velocity ratio constant or varying.

In the investigation, therefore, of the comparative motion, of the driver and follower, in an elementary combination, it is unnecessary to consider relations of angular direction, which are already fixed by the connection of each piece with the frame; so that the inquiry is confined to the determination of the velocity ratio, and of tbe **directional** relation, so far only as it expresses the connection between forward and backward movements of the driver and follower.

Williss classification is founded, in the first place, on comparative motion, as expressed by velocity ratio and **directional** relation, and in the second place, on the mode of connection of the driver and follower.

The comparative motion of two points at a given instant is capable of being completely expressed by one of Sir William Hamiltons Quaternions,the tensor expressing the velocity ratio, and the versor the **directional** relation.

When a continuous motion produces a reciprocating motion, or vice versa, or when a reciprocating motion produces a motion not reciprocating at the same instant, the **directional** relation is said to be variable.

It consists of two elements, the velocity ratio, which is the ratio of any two magnitudes bearing to each other the proportions of the respective velocities of the two points at a given instant, and the **directional** relation, which is the relation borne to each other by the respective directions of the motions of the two points at the same given instant.

When a continuous motion of the driver produces a continuous motion of the follower, forward or backward, and a reciprocating motion a motion reciprocating at the same instant, the **directional** relation is said to be constant.

Class B: **Directional** relation constant; velocity ratio varying.

The comparative motion of the first driver and last follower is obtained by combining the proportions expressing by their terms the velocity ratios and by their signs the **directional** relations of the several elementary combinations of which the train consists.

He divides the elementary combinations in mechanism into three classes, of which the characters are as follows: Class A: **Directional** relation constant; velocity ratio constant.