CHAPTER II.

CONSTRUCTIONAL FORMS AND THEORY.

508. In the preceding chapter I have endeavoured to give the history of the Universal joint.' In the present I propose to describe its various forms, and to develope the formulæ by which the laws of the motion communicated from one axis to the other are defined.

Fig. 359.

509. Fig. 359 represents one of the simplest forms of the universal joint for the communication of rotation. But every form of it may be described, in the nomenclature of Hooke, as consisting of five several parts, namely, two axes Aa, Bb, to the respective ends of which are fastened two arms CAc, DBd, which embrace and take hold of the four points or pivots at C, c, D, d, of the medium CDcd. Each of the semicircular arms has two center holes,

a

B

d

into which the sharp ends of the medium are put, which center holes Hooke calls the hands of the arms. The two points C, c taken hold of by the hands of the driving axis Aa, Hooke terms the points. The other two points Dd taken hold of by the second pair of hands he terms the pivots.

He proceeds to insist that great care must be had that the pivots and points lie exactly in the same plane, and that each two opposite ones be equally distant from the center, that the middle lines of them cut each other at right angles, and that the axes of the two rods may always cut each other in the center of the medium cross or plate, whatever change may be made in their inclination.

The shape of this medium may be either a cross (as in fig. 359), whose four ends hath each of them a cylinder, which is the weakest way; or secondly, it may be made of a thick plate of

brass, upon the edge of which are fixed four pivots, which serve for the hands of the arms to take hold of (as in fig. 357). This is much better than the former, but hath not that strength and Fig. 360.

Fig. 361.

steadiness that a large Ball hath, which is the way I most approve of, as being strong, steady, and handsome.' The four figs. 360, 361, 362, and 363, are forms employed by myself.

Fig. 362.

Fig. 363.

B

In the last I have substituted thin boards for the arms and medium, hingeing them in the manner shown in the figure, where O is the intersection of the linear axes, Bb, Aa the axes, BOQ, AOQ triangular boards fixed to the axes and connected by a quadrantal board and hinges. The triangular boards correspond to the arms, and the quadrantal board to the medium of Hooke's nomenclature. But I prefer to term them radial planes and link plane, of which more above.*

510. Fig. 364 is reduced from Hooke's tab. ii. fig. 10 of his 'Description of Helioscopes,' p. 14, to show his complete form of the joint, when applied to astronomical mechanism. This consists in constructing the arms so as to enable their lines of flexure as

* Vide p. 250.

CC to be inclined and fixed at any required angle to the axis of the shaft, the line CC always remaining in the plane which contains

Fig. 364.

A

Ch

the axis and arms. This adjustment is required to set the axis

6

1 1 of the medium cross to the inclination of the sun's declination, so that the arms CC at the end of the first axis, may by their revolution make the line 11 of the cross describe such a cone about the first axis, as the motion of the Sun doth about the axis of the Earth, making the center of the Earth the Apex of that cone, which will be done if the said semicircular arms be moved, and set to the declination of the Sun for that day.' This adjustment is employed, for example, in describing an elliptical dial by the orthographical projection to obtain the lines that divide the Ellipsis of either Tropick,'* also when the joint is employed in carrying round the hand of a clock in the shadow of a style perpendicular to its face, when the inclination of the arms is made to vary daily, by the clockwork alone, in correspondence with the sun's declination.†

But modern science has entirely banished Dialling,' in which the philosophers of Hooke's period revelled, and the only employment of the universal joynt' in a modern observatory, is for the at

tachment of long pendent handles to the adjusting screws of large instruments which would otherwise be inaccessible without ladders; thus employing it as a joint of flexure. It is also used in connecting a series of shafts in machinery, so as to transmit their rotations uniformly from one part of the frame or machine to another, as will be shown below.

511. Hooke's first application of his form (fig. 358 above) of the universal joint was to the construction of a machine to graduate Fig. 365.

sun-dials, as we have seen, and was founded upon the theorem, that if two axes AO, OB, that meet at a point O are connected by that joint, and mounted in a frame, that is so adjusted in position that one of the axes, AO, shall be parallel to the direction of the style of the proposed dial, and that the other, OB, shall be perpendicular to the plane D of that dial, then if the first be moved by a clock once round in twenty-four hours, the other shall move its index on the plane of the dial to which it is adapted, in the same velocity with the shadow of the sun in that plane. Consequently, to graduate a dial-plate the first axis must have an index travelling over the surface of the twentyfour hour plate H, and by setting the index in turn to each of the hours, and at each hour marking the place of the lower

index h on the blank dial-plate D, it will be accurately and easily completed.

In the diagram I have employed my own solid-angular form of the universal joint (Fig. 365 above, and page 249). It is composed of a driving radial plane AOP fixed to the horal axis AO, a follower radial plane OQB fixed to the dial axis BO, and a link plane POQ connected to the respective radial planes by lines of flexure OP, OQ. The angles AOP, AOQ, POQ are right angled at O. OQ is perpendicular to the axis OB, and travels in a plane parallel to the dial D.

Let the axis AO be the edge of the style, therefore the plane AOQ produced contains the sun, and OQ is the shadow of that edge, and indicates the hour line on the dial. The index h which is fixed to the axis OB, and travels over the blank dial D, is parallel to OQ. I have extracted this method of proving the identity of the laws which express the velocity ratio of the two rods connected by the universal joint with the velocity of the shadow of the style over the dial plate, from Hooke's paper On an Instrument for Describing all Kind of Plane Dials;' but have translated it into modern English, and illustrated it with a new diagram.

512. Having shown that the velocity ratios of the universal joint are the same as those of the sun-dial, we may employ for the

former a simple construction for the delineation of the relative successive angular positions of the horary lines of a sun-dial which was first employed by Clavius in 1581,* for dials whose style was * Gnomonices, 1581, pp. 32, 149. Ferguson reproduced as his own the construction G G