As an example of this we may take the front claw of the common crab, represented in fig. 156. This consists in fact of five separate pieces, A, B, C, D, E, not including the moveable jaw F of the actual claw; each piece is jointed to the next by a hinge-joint. But upon our principles the entire limb may be considered to consist of two principal members C and E; of which the first is jointed to the body of the animal by a universal joint of three axes of flexure, and the second to the first by a joint of two axes of flexure, or Hooke's joint.

For the piece C is united to the claw E by means of an

intermediate piece D, and the axes of the joints which connect them are shewn by the line 5,5 between E and D, and 4,4 between D and C. These axes meet in a point k, and therefore by what has preceded, it appears that E moves with respect to C about the point k, and that it is at liberty to turn round any axis of flexure passing through that point and in the plane 5, k, 4. So that this is in fact a natural Hooke's joint. The compound joint which connects the piece C with the body of the animal is more complex; and to exhibit its arrangement, two projections are given, one upon a plane perpendicular to the other, and intersecting it in the line mn.

We may suppose the claw to be laid down on the table in the upper figure, in which case this becomes the Plan and and the lower the Elevation, although the figures are drawn without any relation to the position of the claw with respect to the body of the animal, but only so as best to exhibit the joints, as will appear presently.

A ring 4 or a is jointed to the body of the animal by a joint whose axis is 1, 1, in the Plan, and 1, 1, in the Elevation. This is jointed to a second ring B, or b, by an axis 2, 2, or II, II; and B is jointed to C by a third axis vertical in the Plan, whose projection is therefore a point 3. It is shewn at III, III, in the Elevation. C is therefore connected to the body of the animal by a compound joint of three axes, whose directions nearly meet, but of which no two are parallel, neither are they in three parallel planes, and therefore, by Art. 310, C is at liberty to move about an axis situated at any angle with respect to the body. The compound joint, in fact, corresponds to the ball and socket joint employed for the shoulder of vertebrate animals. Its motions in different directions are of course limited by the extent of angular motion of which each separate hinge is capable.

The diagram is reduced from a very careful drawing. I found that the axis 2,2 was as nearly as possible in a plane perpendicular to 3, and that when the ring A was placed in its mean position, the axis 1, 1 was also in a plane perpendicular to 3. This determined the choice of the position of the planes of projection.

That of the Plan is parallel to the joints 1,1, 2,2, and therefore perpendicular to the joint 3, which thus becomes a point. The plane of the Elevation is parallel to the point 3.

As to the joints 4,4, 5,5, the joint 4,4 is in the drawing a little overstrained to allow 5,5 to come into parallelism with the plane of the paper; and 4,4, is also not in reality

exactly perpendicular to 3. However, it must be understood that my object here is not to shew the relation of the limb to the body of the animal, but merely the principle of arrangement of the joints.

The claw E is shewn in its extreme outward position with respect to C; in its mean position it would be at right angles to the paper; and in the extreme inward position E and C would come into contact, to allow of which the shape of the intermediate piece and position of the hinges are beautifully adapted.

CLASS B. DIVISION D.

COMMUNICATION OF MOTION BY REDUPLICATION.

312. In the examples of Reduplication contained in the corresponding division of Class A, the strings and the motion of the follower are all parallel, and the velocity ratio constant. If the strings and the paths make angles with each other, a varying velocity ratio will ensue; as in the following example. Let the string be fixed at 4, 157 fig. 157, and passing over a pin B, let it be attached to a point C; let Bb be the path of the pin, Cc that of the extremity of the string, and when C is moved to c, very near to its first position, let B be

carried to b; draw perpendiculars bm, bn, Cp, upon the two directions of the string in its new position.

Then since the length of the string is the same in both positions, we have AB + BC = Ab + bc, that is,

Am+mB+ Bn + nC = Ab + bp + pc,

But ultimately,

Am, and bp = nC; .. mB+ Bn = pc,

1

or Bb (cos bBA + cos b BC) Cc.cos c CB;

=

where the angles are those made by the direction of the string with the respective paths of the pin B and of the extremity C. But by the motion of the system these angles alter, and thus the velocity ratio varies.

If the strings and the path of B become parallel, the

CHAPTER IX.

ELEMENTARY COMBINATIONS.

CLASS C. DIRECTIONAL RELATION CHANGING.

313. In the combinations which have occupied our attention in the preceding Chapters, the directional relation of the pieces has remained constant; but, as I have already explained (Art. 21), there exists a numerous class of combinations, in which the directional relation changes periodically, or in other words, that while one piece pursues its own path with a constant direction of motion, the other piece periodically changes its direction, travelling back and forwards through a constant space. From this it follows that the latter piece must necessarily be limited in the extent of its path by the very nature of the combination, but it will also appear that in the greater number of combinations this reciprocating piece is the follower.

314. The velocity ratio of the pieces may be constant, or may vary; but as the driver may be generally supposed to revolve uniformly, the follower, if the velocity ratio be constant, will in that case travel with a uniform velocity to the end of its path, and instantly reversing the direction, will return with a uniform velocity, and so on. This sudden change, for dynamical reasons, is better avoided; and although, as we shall see, it may be effected, yet now that mechanical principles are better understood, those combinations are always preferred in which the reciprocating body is brought gradually to rest, and again gradually set in