ference of the circle A E C, in rolling on the line M N. The curvature of the teeth of the pinion is an involute as before.

SYSTEM COMPOSED OF A WHEEL AND TANGENT, OR ENDLESS SCREW.

Fig. 2.-In the construction of this variety of geering, we must first fix upon the number of teeth in the wheel, and the distance of its centre from the axis of the screw. Then conceive a plane passing through the axis EF of the screw, parallel to the face of the wheel, and let C be the centre of its primitive circle. If now a perpendicular C G be drawn from C upon E F, and C A be taken as the radius of the pitch circle B A D of the wheel, the difference A G will represent the radius of a cylinder, which may be termed the primitive cylinder of the screw; and a line M N drawn through A, parallel to E F, will be a generatrix of that cylinder, which will serve the purpose of determining the form of the teeth.

The section having been made through the axis, the question obviously resolves itself into the case of a rack driving a pinion; consequently the curve of the teeth, or rather thread, of the screw should be simply a cycloid generated by a point in the circle A E C, described upon A C as a diameter, and rolling upon the straight line M N. It is to be remarked, further, that the outlines of the teeth are helical surfaces described about the cylinder forming the screw, with the pitch A b equal to the distance, measured upon the primitive scale, between the corresponding points of two contiguous teeth. These curves have been drawn on our figure, but being for the most part concealed, they are expressed by dotted lines. The teeth of the wheel are not, as in ordinary kinds of geering, set perpendicularly to the plane of its face, but at an angle, and with surfaces corresponding to the inclination and helical form of the thread of the screw. In some instances, the points of the teeth and bottoms of the spaces are formed of a concave outline adapted to the convexity of the screw, in order to present as much bearing surface as possible to its action. In this kind of geering, for obvious reasons, it is invariably the screw that imparts the motion.

Fig. 3 represents an edge elevation of the wheel, projected as in previous examples.

SYSTEM COMPOSED OF AN INTERNAL SPUR-WHEEL DRIVING A PINION.

Plate XVI., fig. 1.-The form of the teeth of the driving wheel is in this instance determined by the epicycloid described by a point in the circle A E C, rolling on the concave circumference of the primitive circle MA N.

The points of the teeth are to be cut off by a circle drawn from the centre of the internal wheel, and passing through the point E, which is indicated, as before, by the contact of the curve with the flank of the driven tooth.

The wheel being supposed to be invariably the driver, the curved por tion of the teeth of the pinion may be very small. This curvature is a part of an epicycloid generated by a point in the circle M A N rolling upon B A D.

SYSTEM COMPOSED OF AN INTERNAL WHEEL DRIVEN BY A PINION.

Fig. 2. This problem involves a circumstance which has not hitherto come under consideration, and which demands, consequently, a different mode of treatment from that employed in the preceding cases. The epicycloidal curve A a, generated by a point in the circle having the diameter A O, the radius of the circle MA N, and which rolls upon the circle BAD, cannot be developed upon the flank A b, the line described by the same point in the same circle in rolling upon the concave circumference MAN; and for this obvious reason, that that curve is situated without the circle B A D, while the flank, on the contrary, is within it. It becomes necessary, therefore, in order that the pinion may drive the wheel uniformly according to the required conditions, to form the teeth so that they shall act always upon one single point in those of the wheel. This may be most advantageously effected by taking for the curvature of the teeth of the pinion the epicycloid A d described by the point A in the circle M A N, rolling over the circle B A D. It will be observed that, as in the preceding examples, the tooth E of the pinion begins its action upon the tooth F of the wheel at the point of contact of their respective primitive circles, and that it is unnecessary that it should be continued beyond the point c, because the succeeding tooth H will then have been brought into action upon G; consequently the teeth of the wheel might be bounded by a circle passing through the point c. It is, however, one of the practical advantages which this species of geering has over wheels working externally, that the surfaces of contact of the wheel and pinion admit of being more easily increased; and by making the teeth somewhat longer than simple necessity demands, the strain may be diffused over two or more teeth at the same time. The flanks of the teeth of the wheel are formed by radii drawn to the centre O, and their points are rounded off to enable them to enter freely into the spaces of the pinion.