to permit of complete turning. This is founded on the fact that one side of a triangle cannot be greater than the sum of the other two. From these two laws together it is shown that it is impossible to have two O's alternating with two U's.

Next it is pointed out how the U motion may be provided for by constructing a circular slotway in one piece, and shaping the other piece to fit the slot way, so that by imagining the radius of curvature of the slotway to be indefinitely increased a relative movement of reciprocating sliding motion, represented by the symbolical letter I, will be substituted for the swinging motion U. A slide being conceived to be a swing through a zero angle about an infinitely distant centre, the previously mentioned laws will apply to associations containing I motions, and it will follow that a combination of three slides and one swing is precluded by the first law.

If four slides are associated, in which all four of the links of the original mechanism are to be conceived to be infinitely long, an indeterminate motion will result comparable to the motion which would be possible if five bars were joined by pairs in a closed circuit. One of the slides may be suppressed, and a definite motion will result from three slides.

If the foregoing analysis be compared with that due to Reuleaux, to which it bears a close resemblance, it will be seen that Reuleaux conceives that the elementary essential components of machines are the pairs of consecutive links which are in mutual contact, whereas it is here proposed that the relative motions of consecutive links should be regarded as the essential elements or components of a machine movement. Whilst the pairs of surfaces of contact of consecutive pieces should be formed to suit the kind of relative motion which those pieces are required to undergo, yet the forms of those surfaces do not themselves entirely govern the character of the motion.

Reuleaux assumes that what he calls a turning motion and the I motion are entirely governed by the forms of the surfaces of mutual contact, but shows that to ensure a more complex motion a restraint is required to be imposed by means external to the two links. Those additional means of constraint have to be included with that due to the forms of the surfaces of mutual contact in the conception of a complete pair, and often the whole mechanism is required to complete one pair contained in it.

Reuleaux does not attempt to discriminate between a turning and a swinging pair; the same pair of surfaces of mutual contact is suitable for both; the difference consists of a difference only in the rest of the mechanism, yet the difference in the two motions is most apparent, and is very important, both kinematically and also from the practical engineer's point of view.

No advantage is derived from analysing a machine into parts such as pairs if it requires the whole machine to complete one of the parts.

The enunciation and the explanation of the influence of the first law previously mentioned, of the constancy of the sum of the four angles of a quadrilateral in governing the association of the OU and I motions in one mechanism, is one of the important original features of this paper.

The influence of the second law, viz., that the two sides of a triangle are together not less than the third, in limiting the association of the O, U, I motions, is now also for the first time pointed out, though Reuleaux and others have, without formally enunciating the law, made use of the fact to determine the proportions necessary for certain suggested movements.

By the application of these governing laws one is able to write an exhaustive list of all the possible combinations in one simple mechanism of the three simple O, U, I motions, and to explain why other combinations are precluded.

Fourteen distinct combinations are possible, and only fourteen. They are exhibited by the following formula, in which a large O associated with a small o signifies that in one case adjacent links turn relatively to one another so as to continuously increase the angle between them, and in the other to continuously diminish the angle. The double signifies that two complete revolutions accompany one complete to and fro swing or slide.

4 Group.

6 Group.

2 Group.

U Group.

छ

Following Reuleaux, the author applies the principle of the "inversion of the kinematic chain," considering it to be a continuous of links in a closed circuit containing a sequence of sequence elementary motions. In explaining what is meant by inversion, it is pointed out that relatively to an observer or user of a machine one piece is fixed. This is called the frame of the machine. Each one of the four links may in turn be made the fixed or frame link, and

although the relative motion of the four links will in all cases remain unaltered, the absolute movement, or movement relatively to the user of the machine, will in general be different for each fixing, and constitute a new machine movement. Changing the fixed or frame link is called the "inversion of the chain."

The author makes use of the term "primary pieces," originally suggested by Rankine for those links which are in sequence with and directly connected to the frame link, and shows that if, after inversion, the new primary pieces have the same kind of motions as the previous primary pieces had, the consequent machine movement is not a new one, but a repetition of a previous one.

From the mechanisms

and

only can four different

machine movements be obtained by inversion. From the others 3, 2, or only 1 can be derived.

They are distinguished from one another in the formula by using a thick line for the frame link. Thus

signifies a machine movement like that employed in the crank and connecting rod engine.

is exemplified in the oscillating engine much used in paddlewheel steamers.

is found in Stannah's pendulum pump, and

is the movement adopted by Rigg in the design of his high speed engine. The intimacy of the relation of this engine to the preceding ones is here for the first time indicated. In all, thirty-two and only thirty-two distinct machine movements can be derived from the fourteen previously enumerated mechanisms by inversion.

It is shown that Reuleaux's principle of inversion can be applied with more advantage and consistency if a machine movement is analysed into its component motions than if a machine is analysed into its component pairs, and the notation lends itself to a very clear exhibition of the effect of inversion.

The author next discusses the relation of cams and spur-wheel mechanisms to the foregoing kinematic chains, showing that they are the result of the suppression of one of the previous four links and the amalgamation of the two adjoining simple motions into one more complex. A comparison is also made with belt gearing and expressive formulæ suggested.

The author then passes to the consideration of machines the parts of which do not move parallel to one plane.

Reuleaux was the first to show that if the links of the previously mentioned kinematic chains be bent to the form of great circles of a sphere the axes of the connecting pins will be radial, and the previously mentioned machine movements will be possible under the modified circumstances.

In spherical motion the counterpart of what is a slide in plane motion could be obtained by a swinging motion about a pole of which the bent link is the equator. The motion is to be conceived as due to the use of two bent links, the length of one of which is a quadrant of a great circle of the sphere.

In these so-called spherical mechanisms, Law I has to be modified as follows:

The sum of the four angles of the spherical quadrilateral varies, having a value of 37 for a maximum and 27 for a minimum.

This and Law II, which is the same as before, will preclude the same combinations in spherical mechanisms which were precluded in plane mechanisms.

Law I explains at once why in Hooke's joint, which is the spherical counterpart of Oldham's coupling, the angular velocity-ratio of the connected shafts is not constant, whereas in Oldham's coupling it is.

The author points out that the kinematic chain containing three slides cannot be adapted to give a movement on a sphere. The virtual construction would consist of a spherical triangle between the links of which no relative motion is possible, and there is not room on the sphere for a movement at each joint of a bent quadrilateral, the length of each side of which is equal to a quadrant. But a three-slide mechanism can be adapted to give motion on the surface of a cylinder, and it is the only one of the fourteen kinematic chains which can be so adapted, and examples of it are found in the various helical motions so largely used. (The letter V is used to represent helical motions.) This method of showing the relation between screw motions and plane motions is a novel feature of the paper.

The remaining mechanisms consist of those in which the axes of the turning and swinging motions neither meet nor are parallel. They include the motion which occurs at a ball-and-socket joint represented by . The method of classification according to the proposed scheme is summarised as follows:

All simple machine movements may be ranged in four divisions, viz. :

1. Consisting of plane mechanisms, in which the pieces move in or parallel to the surface of a plane.

2. Spherical mechanisms, in which the pieces move in or parallel to the surface of a sphere.

3. Cylindrical mechanisms, in which the pieces move in or parallel to the surface of a cylinder.

4. The remainder, to which the name conoidal mechanisms is given, in which the axes of the swinging and turning motions neither meet nor are parallel.

The mechanisms in each of these divisions are classed in two subdivisions.

Sub-division S, with surface contact of consecutive links.

Sub-division P, with point contact of consecutive links.

The mechanisms of sub-division S of divisions 1 and 2, 1, and 2, will consist of those in which O U I motions only are used. Those of 3, will include the helical or V motion, and Those of 4, will include the motion

and-socket joint..

requiring the use of a ball

To the pairs of links which have the relative motions O, U, I, V, Reuleaux has given the name lower pairs. Reuleaux claimed two characteristics for lower pairs, viz. :—

1. Definiteness of motion derived from the surfaces of mutual contact themselves.

2. The possibility of distributing the contact over an area which may be extended as much as desired.

If it is desired to differentiate between the O and U motions, Reuleaux's turning pair cannot possess the first characteristic.

The second characteristic is of considerable value in relation to the liability to abrasion and wear, but the advantage of greater immunity against wear has to be purchased at the cost of a more complicated construction and a more restricted character of movement.

As examples

The mechanism consisting of a pair of spur wheels turning in bearings which are at a fixed distance apart will belong to 1p.

A pair of bevel wheels will belong to 2p.

The so-called cylindrical cam motion will belong to 3p, and the worm-and-worm wheel mechanism to 4p.

The mechanisms in each of the eight sub-divisions are still further sub-divided into combinations. The combinations of 1,, 2, and 3s, are exhaustively enumerated, and it is suggested that an extension of the methods of applying the geometrical laws would lead to the preparation of an exhaustive list of the possible combinations in the other sub-divisions. The combinations are still further sub-divided into inversions according to Reuleaux's principle of the inversion of a machine.

Further than this there will be varieties of any inversion differing in the details of the construction and uses of the machine movement.

Lastly, the author proceeds to show how the foregoing considerations assist in the analysis of compound mechanisms. It is assumed