Sometimes a single line is employed, which being fastened at d is coiled three or four times round the roller, and then carried on to c; the coiling is sufficient to enable the cord to lay hold of the roller in most cases, as for example, in the common drill and bow.

A

191. But the constancy of the ratio is interfered with in both these contrivances, by the varying obliquity of the straight parts of the cords which connect the pieces, as well as by the tendency to heap up the successive coils in layers upon each other, thereby increasing the effective diameter of the rollers. This is remedied by cutting a screw upon the surface of each roller, which guides the cord in equidistant coils as it rolls itself upon the cylinder.

d

105

Thus, fig. 105, let A give motion to B by a cord cd, in the manner already shown in fig. 103, but let screws be cut upon the surface of the rollers; then during the motion of A the extremity of the straight portion of the cord will be gradually carried to the right as it is wound up, and vice versa; and this motion will be constantly proportional to the rotation, and at the rate of one pitch of the screw to each complete turn of the cylinder.

To cause the straight portion ed to move parallel to itself, the screw cut upon B must be of such a pitch that the endlong motion of d may be the same as that of c. Now since the velocity of the surfaces of the two cylinders are equal, and every revolution of either screw carries the cord endlong through the space of one pitch, let m × circumferences of Anx circumferences of B, and let C, c be the respective pitches of their screws; R, r their radii, then we must have mC = nc,

192. In the combination of fig. 104, the screw roller will prevent the irregular heaping up of the cord on the band, but will not correct the varying obliquity of the cord. This may be got rid of thus.

Let B, fig. 106, be the sliding carriage, CD, HK the 106

sides of the frame which supports the roller, E the roller formed into a screw. This roller has a screw F cut on its axis, of the same pitch as that of E, and passing through a nut in the frame CD; the other extremity of the roller is supported by a long plain axis G, passing through a hole in the frame HK; the cord being tied at b to the carriage, and at the other end to the screw-barrel E; it follows, that when the latter is turned round, it will travel at the same time endlong by means of the screw and nut F, exactly at the same rate, but in the opposite direction, as the end of the cord is carried along the barrel by its coiling; consequently the one motion exactly corrects the other, and the cord b will always remain parallel to the path of the slide B*.

A similar and contrary cord being employed to connect the other end of the slide with the band, will enable the roller to move the slide in either direction.

193. A well made chain of the common form, with oval links, will coil itself with great regularity upon a re

* From a machine by Mr. Holtzapfel.

volving barrel, if a spiral groove be formed upon the surface, of a width just sufficient to receive the thickness of the links. As shewn in fig. 107, the links will alternately place themselves edgewise in the groove and flat upon the surface of the barrel.

194.

107

When the revolving piece is required to move

only through a fraction of a revolution, the combination is made more simple.

Thus let A represent a revolving piece or quadrant, whose axis is B, b, and whose edge is made concentric to it, and let CD be the sliding piece represented as an open frame for clearness only, but supposed to be guided so as to move

D

B

108

in either direction along the line CD produced. If cords or chains be attached at c, d, to the quadrant and at e, f, to the sliding frame; and a third cord be attached contrariwise to the quadrant at h and the frame at g, then either the motion of the quadrant or the frame will communicate motion to the other in a constant ratio, and in either direction at pleasure.

195. WHEN two arms revolving in the same plane are connected by a link (Art. 32), their angular velocities are inversely as the segments into which the link divides the line of centers. This relation is constantly changing, as the arms revolve, unless the point of intersection T (fig. 6), be thrown to an infinite distance, by making PQ parallel to AB, in all positions, which can only be effected by making the arms equal, and the link equal in length to the distance between the centers. In this case the angular velocities will become equal, and their ratio consequently constant.

109

a

196. This produces the arrangement of fig. 109. D, B are centers of motion, Bd = Df the arms, df (= BD) the link. If Bd be carried round the circle, BdfD will always be a parallelogram, and consequently the angular distances of Bd and Df from the line of centers the same, and their angular velocity the same.

But in any given position of one of the arms Bd, there are two possible corresponding positions of the arm Df, for with center d, and radius df, describe an arc which will necessarily cut the circular path of ƒ round D in two points f and A; therefore AD is also

A

B:

a position of the arm corresponding to Bd, in which the link d intersects the line of centers in a point C; and if Bd be moved, the point C will shift its place, and consequently the angular velocity of AD will not preserve a constant ratio to that of Bd.

It appears, then, that this system is capable of two arrangements, one in which the angular velocity ratio is constant, and the other in which it is variable, according as the link is placed parallel to the line of centers, or across it.

But if the motion of this system in either state be followed round the circle, it will be found that when the extremity d of the arm Bd comes to the line of centers, either above or below, at a or s, the extremity of the other arm will also coincide with that line, since the link is equal to BD, and therefore to ap or st. In these two phases (Art. 17) of its motion the two positions fd, Ad of the link coincide, and at starting from either of these phases, the link has the choice of the two positions. If, for example, the arms be at Ba and Dp, then as a moves towards d, p may either move towards f, in which case the link will remain parallel to BD, until the semicircle is completed, or else p may move towards A, and then the link will lie across BD, until the semicircle is completed by d coming to s, when a new choice is possible. But in any given position of Bd intermediate between Ba and Bs, it is impossible to shift the link from one position to the other without bending it.

The two phases in which the arms coincide with the line of centers, are termed the dead points of the system.

197. When this contrivance is employed to communicate a constant velocity ratio, some provision must be made to prevent the link from shifting out of the parallel position into the cross position, when the arms reach the dead points.