46. But in rolling cones, the perimeters of every corresponding pair of circles move in the same direction at the point of contact, although the circles are not in the same plane; for the radii of contact are perpendicular to the line of intersection of the two planes which contain the circles, and consequently the tangents at the point of contact coinIcide with that line and with each other. On the contrary, this is not the case in rolling hyperboloids.

For the circles whose radii are CE, AE, lie in planes whose intersection is the line Ee; and the tangents to these circles at the common point E, plainly cannot coincide either with that line or with each other; so that the motion of the surfaces is not in the same direction, and the rolling action is imperfect, and the more so the greater the angle ECp, that is, the greater the distance gk between the axes; for as this distance diminishes, the hyperboloids approach to a pair of cones whose common apex is h.

47. In practice, as in the case of cones, a thin frustum only is required of each hyperboloid, and these frusta include so small a portion of the curve surface, that a frustum of the tangent cone at the mean point of contact may be substituted without sensible error; and this may be found as follows:

given in any particular case, by the last proposition; observ

ing that in Fig. 14. AE, EC are the mean radii of the frusta; g, k, the centers of the generating hyperbolas, and gh, hk their semiaxes-major.

Let PTt be the tangent at P; then, by the known properties of the hyperbola,

48. This third case of axes, neither parallel nor meeting, admits of solution by means of the cones of the second case; thus*:

Let Aa, Bb be the two axes, take a third line intersecting the axes at any convenient points C and D respectively; and let a short axis be mounted so as to revolve in the direction of this third line between the other two axes.

Now a pair of rolling cones, e, f, with a common apex at c, will com

Б

B

16

A

A

municate motion from the axis Bb to the intermediate axis; and another pair g, h, with a common apex at D, will communicate motion from the intermediate axis to Aa; and thus the rotation of Bb is communicated to Aa by pure rolling contact.

Let A, A, a, be the respective angular velocities of the axes Bb, CD, Aa; and R, R,, r the radii of the bases of their cones, those of the cones, f, g, being the

same;

• From Poncelet, Mecanique Industrielle, p. 300.

exactly as if the cones e, h, were in immediate contact.

To apply these Solutions to Practice.

49. Theoretically we have now the complete solution of the problem in all the three cases; having shewn how to find a pair of cylinders in the first case, and of conical frusta in the other cases, by which a given angular velocity ratio will be effected. If these solids could be formed with mathematical precision, then, the axes having been once adjusted in distance so that the surfaces should touch in one position, they would touch in every other position; but in practice this is impossible, and various artifices are employed to maintain the adhesion upon which the communication of motion depends.

The surface of one or both rollers may be covered with thick leather, which by giving elasticity to the surface enables it to maintain adhesional contact, notwithstanding any small errors of form.

One of the axes may be either made to run in slits at its extremities instead of round holes, or else it may be mounted in a swing frame. Both methods allowing of a little. variation of distance between the two axes, the contact of the rollers will in this way also be maintained, notwithstanding small errors of form.

If the weight of the uppermost roller is not sufficient to produce the required adhesion, or if the rollers lie with their axes in the same horizontal plane, then weights or springs may be employed to press the axes together. The practical details of these methods belong rather to the department of Constructive Mechanism than to the plan of the present work.

50. But the most certain method of maintaining the action of the surfaces is to provide them with teeth. The plain cylindrical or conical surfaces of contact are exchanged for a series of projecting ridges with hollow spaces between. These ridges or teeth are distributed at equal distances from each other on the two surfaces, and generally in the direction of planes passing through the axis, so that when the driving wheel is turned, its teeth enter in succession the spaces between those of the follower. They are SO adjusted that before one tooth has quitted its corresponding space the next in succession will have entered the next space, and so on continually; consequently, the surfaces cannot escape from each other, and there can be no slipping, notwithstanding slight errors of form.

The action of this contrivance falls partly under the head of rolling contact, and partly under that of sliding contact; for the teeth considered separately act against each other by sliding contact, and the forms of their acting surfaces must be determined, as we shall see, upon that principle.

On the other hand, the total action of a pair of toothed wheels upon each other is analogous to that of rolling contact. Equal lengths of the two circumferences contain equal numbers of teeth, and therefore equal lengths will pass the line of centers in the same time, if measured by the unit of the space occupied by one tooth and a hollow between. In fact, the adhesion which enables the surface of 'one plain roller to communicate motion to another arises from the roughness of the surfaces, the irregular projections of one indenting themselves between those of the other, or pressing against similar projections; and the contrivance of teeth is merely a more complete developement of this mode of action, by giving to these projections a regular

form and arrangement. I shall proceed therefore to explain in this Section all that relates to the general action, arrangement, and construction of toothed wheels; leaving the exact form of the individual teeth to the next Section, and observing, that this arrangement corresponds to the ordinary practical view of the subject; for all that belongs to the complete action or construction of a pair of toothed wheels is always referred to a pair of corresponding plain rollers, or rolling circles, which are termed the pitch circles, or geometrical circles.

51. Geering is a general term applied to trains of toothed wheels. Two toothed wheels are said to be in geer when they have their teeth engaged together, and to be out of geer when they are separated so as to be put out of action; and generally speaking, a driver and follower, whatever be the nature of their connexion, are said to be in geer when the connexion is completely adjusted for action, and out of geer when the connexion is interrupted.

52. Toothed wheels with few teeth are termed pinions. This phrase is merely to be considered as the diminutive of toothed wheel; and there is no impropriety or ambiguity in calling a pinion a toothed wheel, if more convenient.

53. The teeth of wheels may be either made in one piece with the body or rim of the wheel, or they may be each made of a separate piece and framed into the rim of the wheel.

The first method is employed in cast-iron wheels of all sizes, from the largest to the smallest; also for brass or other metal wheels in smaller machinery, which are formed out of plain disks by cutting out a series of equi