carried by the wheel itself, the wheel being driven by an ordinary cylindrical endless screw.' Fig. 136 shows this form of endless screw, and fig. 137 is an arrangement to show the manner of cutting the spiral thread upon the solid, in which A is a wheel driven by an endless screw B, of the common form; Ca toothed wheel fixed to the axis of the endless screw and geering with another equal toothed wheel D, upon whose axis is mounted the smooth surfaced solid E, which it is desired to cut into Hindley's endless screw. For this purpose a cutting tooth F is clamped to the face of the wheel A. When the handle attached to the axis BC is turned round, the wheel A and solid E will revolve with the same relative velocity as A and B, and the tooth F will trace upon the surface of the solid a thread which will correspond to the conditions. For from the very mode of its formation the section of every thread through the axis will point to the center of the wheel. The axis of E lies considerably higher than that of B, to enable the solid E to clear the wheel A. The edges of the section of the solid through its center, exactly fit the segment of the toothed wheel, but if a section be made by a plane parallel to this, the teeth will no longer be equally divided, as they are in the common screw; and therefore this kind of screw can only be in contact with each tooth along a line corresponding to its middle section. So that the advantage of this form over the common one is not so great as appears at first sight. 217. If the inclination of the thread of a screw to the axis be very great, one or more intermediate threads may be added, as in fig. 138. In which case the screw is said to be double, or triple, according to the number of separate spiral threads that are so placed on its surface. As every one of these threads will pass its own wheel-tooth across the line of centers, in each revolution of the screw, it follows, that as many teeth of the wheel will pass that line during one revolution of the screw as there are threads to the screw. If we suppose the number of these threads to be considerable, Fig. 138. Fig. 140. Fig. 139. D for example, equal to those of the wheel-teeth, then the screw and wheel may be made exactly alike, as in fig. 139; which may serve as an example of the disguised forms which some common arrangements may assume. The old Piemont silk-mill is an example of disguised endless screws. 218. In fig. 140 is represented a method of communicating equal rotation by sliding contact between two axes whose directions if produced are parallel. Aa Bb are the axes, parallel in direction. The axis Aa is furnished with a semicircular piece CAC, forming two equal branches, and terminated by sockets bored in a direction to intersect the axis at right angles. The axis bB is provided with a similar pair of branches dbD, and the whole is so adjusted that their four sockets lie in one plane perpendicular to the axes. A cross with straight cylindrical polished arms is fitted into the sockets in the manner shown in the figure; and its arms are of a diameter that allows them to slide freely each in its own socket. If one of the axes be made to revolve, it will communicate to the other by means of this cross a rotation precisely the same as its own. For let fig. 141 be a section through the cross transverse to the axis, and let AB be the axes, and the circles be those described by their sockets respectively. * Described in Encyc. Méthodique, 'Manufactures and Arts,' tom. ii. p. 31; and in Borgnis, Machines pour confectionner les étoffes, p. 160. Fig 141. D Then if D be a socket of A, the arm of the cross which passes through it must meet the center A; and in like manner if C be a socket of B, the arm CB must pass through B. Also, if D move to d, the new (or dotted) position of the cross will be formed by drawing dA through A, and Bc perpendicular to it through B the other axis; therefore C will be carried to c; and it is easy to see that the angle DAd= CBc. Therefore the angular motion of the axes is the same. Also, every arm of the cross will slide through its socket and back again during each revolution, through a space equal to twice the perpendicular distance of the axes (AB).* A B * This arrangement is essentially the same as that of a coupling invented by the late Mr. Oldham, and introduced by him into the machinery of the Banks of England and Ireland. His form of it is more solid, but not so well adapted for geometrical illustration as that which 1 have given. His axes are each terminated by a disk in which a transverse groove is planed, and the cross consisting of two square bars in different planes has each bar completely buried in the groove of its neighbouring disk. 219. THE simplest mode of obtaining a varying angular velocity ratio, when the rotations are to be continued indefinitely in the same direction, is by the pin and slit, fig. 142, where Aa, Bb are axes parallel in direction, but placed with their ends opposite to each other. Aa is provided with an arm carrying a pin d, which enters and slides freely in a long straight slit formed in a similar arm, which is fixed to the extremity of Bb. If one of these axes revolve, it will communicate a rotation to the other with a varying velocity ratio; for the pin in revolving is continually changing its distance from the axis of Bb. Let C be the center of motion of the pinarm, K the center of motion of the slit-arm, Fig. 142. f D B P the pin, R the constant radius of the pin from C, r the radial distance from K, and let P move to p through a small angle; draw pm perpendicular to CP, then angular velocity of pin: angular velocity of slit If CP revolve uniformly, the angular velocity of KP will vary cos CPK as 1 or if CK be small, as; therefore when the centers of motion are near, this contrivance produces the same law of motion as that of Art. 97. will give the position of KP corresponding to any given position of CP. By altering the direction of the slit, or by making it curvilinear, other laws of motion may be obtained. 220. In the endless screw and wheel (Art. 212), the thread of the screw is inclined to the axis of the cylinder at a constant angle, and the angular velocity ratio of screw and wheel is constant. If, however, the inclination of the thread be made to vary at different points of the circumference, as shown in fig. 143, the angular velocity ratio will vary accordingly. For example, if the threads through half the circumference lie in planes perpendicular to the axis of the screw, the wheel will revolve with an intermittent motion, remaining at rest during the alternate half rotations of the screw. If A, a be the respective angular velocities of the screw and wheel, R, r their pitch-radii, A T it appears, from Art. 208, that tan 4. = a R But as the inclination changes, the teeth of the wheel must be made in the form of solids of revolution, having their axes radiating from the center of the wheel. Fig. 143. Fig. 144. B 221. A simple intermittent motion is effected by a pinion of one tooth A, fig. 144. This tooth will in each revolution pass a single tooth of the wheel B across the line of centers; but during the greatest portion of its rotation will leave the wheel undisturbed. To prevent the wheel B from continuing this motion by inertia through a greater space than this one tooth, a detent C |