surfaces will come successively in pairs into contact on the common contingent line Cc. But as the axes of rotation are not parallel, the relative motion of each pair of lines during the short time of their mutual action is compounded of a direct approach and recess, combined with a sliding motion parallel to their common direction as will appear below.

Fig. 24.

43. The nature of these hyperboloids and their mutual action are best explained by models, in which the solid is represented by two equal disks E, e fixed to the axis Aa, and provided with a series of equidistant notches l, m, n, p &c. In the circumference an equal number of holes, 1, 2, 3...... are bored, one opposite to each notch, as shown in fig. 24, which represents the outside of the disk.

E 1

m

2

n

A string passed through E and secured with a knot inside is carried over the notch 7, and thence to the corresponding notch in the opposite disk, which places the string in the position of the generator. This string is laced backwards and forwards, always carrying over the notch of the disk as m to the outside, then through the hole (1) next on the inside from 1 to 2, and on the outside over n to the opposite disk and so on. When completed this forms a skeleton frame. If two such skeletons are mounted in contact in a proper frame, in the relative positions of the figure, and revolved by hand, the respective strings will come into successive contact, sliding lengthwise in opposite directions. In practice these solids are required as above stated for the construction of toothed wheels whose axes are neither parallel nor meeting and only a comparatively thin frustum or slice of the solid is required. The successive lines on the surface are replaced by teeth which must be in the same direction as the lines to enable

them to come into successive contact. Also these wheels generally require to be in pairs, of which the teeth are different; but the dimensions and relative proportions of two hyperboloids required to communicate the rotation of one axis to another in any ratio can only be effected by formulæ and constructions, which may be obtained as follows.

44. In fig. 25 the two hyperboloids are shown in contact. EE is the axis of the larger, and FN that of the smaller, the farther part of which is concealed by being necessarily passed behind the larger one. Its general outline is, however, shown by the dotted lines.

The circle of least diameter in the center of the length of the hyperboloid, assuming its extremities to have equal radii, is termed the gorge circle.

MPN is the common perpendicular of the axes EE', FN, and also contains the radii MP, NP of the gorge circles which touch at a point P, in this common perpendicular.

CC, is the contact line of the two hyperboloids, and composed of two generators of the respective surfaces which coincide along their whole length.

Now the condition required for the contact of two curved surfaces at any two points belonging respectively to these surfaces, is that the direction of the two normals shall coincide into

one right line when the two surface points come together. Manifestly this condition is fulfilled in the contact point P of the gorge circles which are not in the same plane.

To show that the same condition is complied with at every other point of the common generator, it must be observed that, through every point of the surface of a hyperboloid, as at C, it is possible to draw two lines CC, CC,, both of which will coincide with the surface throughout its length, and consequently each separately would generate the surface by revolving about the axis EME'.

The projections of these generators on the base circle C,DC, are obtained by projecting the upper extremity C on the base at D, drawing DC, to meet the lower extremity. This line will touch the projection p,E'p, of the gorge circle at p.

A line similarly drawn from D to the extremity C, of the companion generator CC, will touch the projected circle at p2. Join these lower extremities by a line CC. We have now an isosceles triangle DC, C, with apex D and base C, C.

This triangle, of which the two legs are.in contact with the solid, determines the position of the tangent plane at their concourse at C.

A plane, DE, CE, passing through CD and EE, will bisect the angle CDC2, and also pass through the intersection C of the two opposite generators. But, as CC, is common to the two curve surfaces, and C a point of contingence, the normal CA

must be perpendicular to the plane C, CC, at the apex, and in the same direction with the normal CB of the other hyperboloid.

45. Fig. 26 shows the small central circles, or gorge circles (as they are termed), in action.

EM, FN are the respective axes, MN their common normal, P the point of contingence of the circles. PPP, is the line of contact of the two solids, along which their respective generators are also represented in coincidence.

Let the larger gorge circle move through a small angle PMm, so as to carry the radius MP into the position Mm. This will

remove the point P of its generator into the position m, and the whole generator of the larger hyperboloid into the position mm very near to the first. By the contact of the surfaces the generator of the smaller hyperboloid will be carried along with the first generator, and the motion being small, the two will remain in longitudinal contact. But the point of the second generator is carried about the axis NF in the direction Pn, and thus the whole generator is removed to the position n1n, where P11, P ̧ ̧ are parallel to Pn. Thus the motion of the generators through a small distance is performed with a coincidence of direction, accompanied by a longitudinal sliding, measured by the ratio of mn where Pp is the shortest distance between the successive Pp' positions.

Manifestly the motion of the larger gorge circle through the small angle mMP compels the smaller gorge circle to describe the small angle PNn. Hence as the angular velocities of two bodies are measured by the magnitude of the angles they describe simultaneously, let be the angular velocities of the greater and

w

perpendicularly through the common normal at the point P, and therefore parallel to the axes of the hyperboloids, which are projected upon this plane at Pa, Pb. PC is the position of contact of the generators, ACB the common normal of the hyperboloids. at their upper extremities.

Through the extremity C of the united generators, pendicular to them, a plane Aa bB is passed.

and per

In this combination it is evident that the intersection lines of the latter plane with the previously explained elements of the figure, describe upon it two similar right-angled triangles A Ca, B Cb, in which Aa=MP=R and Bb=NP=R1.

Let the angle CPa=a. CPb=a1.

In the plane a CP draw a line mn parallel to PC, and from P lines Pm, Pp, Pn respectively perpendicular to Pa, PC, Fb. This triangle, mPn, supposed small, is manifestly the same as the triangle mPn in fig. 26, for, in both mn is parallel to the coupled generators PC and mP, nP, pP, are in planes respectively perpendicular to the two axes and the generator.

Through Clet a plane CeE pass, intersecting normally the axis MA in E. Therefore CE (H) is the radius of the greater hyperboloid. A second plane, CfF, through C perpendicular to the axis NB, contains the radius CF of the lesser hyperboloid CE= √ Ee2 + Ce2= √ R2 + PC2sin'a where PC is the half-length from the gorge circle of the generators (= G) and similarly

H2= Ee2 + Ce2= R2 + G2.

when H is the greater radius of the hyperboloid.

and G the half length of the generator.

If from any point of the line PC normals be drawn to meet the axes MA NB, they will be in one right line and in the constant ratio of MP to PN. If drawn very near together, they constitute a ruled surface, bounded by AM, BN, and generated by a right line, which travels in contact with those lines and with PC, but always at right angles to the latter.

It is manifest that the same surface would be generated by a right line whose extremities rest against AB and MN, and travel