when Rr is by hypothesis very small. This inconvenience has been sufficient to banish the contrivance from practice, for although it is represented in all mechanical books under the name of the Chinese windlass, it is never actually employed.

BY SLIDING CONTACT.

402. Aa, fig. 208, is an axis upon which are formed

two screws B and D, whose pitches

208

are C and c respectively. B passes

b

MB

AD

through a nut b fixed to the frame, AM

and D through a nut d, which is

capable of sliding parallel to the axis of the screw*.

Now when a screw is turned round it travels with respect to its nut through a space equal to one pitch for each revolution, consequently one turn of Aa will cause it to move with respect to b through the space C. But the same motion will cause the nut d to move with respect to its screw through a space c. The nut d, therefore, receives two simultaneous motions, for by the advance of the screw Aa through the fixed nut b, the nut d is carried forwards through the space C, but by the revolving action of the screw Aa it will be at the same time carried backwards through the space c; its motion during one rotation of the screw Aa is therefore equal to the difference of the two pitches C-c. If C be greater than c this will be positive, and the nut will advance slowly when the screw Aa advances; but if c be greater than C, the nut will move slowly in the opposite direction to the endlong motion of the screw. If C=c then Cc 0, and the nut d receives no motion, which is indeed obvious. All this supposes that the threads of the two screws are both right-handed or both left-handed. If

=

• This contrivance is claimed by White, (Century of Inventions, p. 84,) and also for M. Prony, by Lanz and Betancourt, (Essay, D. 3).

one be right-handed and the other left-handed, each revolution of the screw Aa will cause the nut d to advance through a space = C + c.

F

B

209

D E

403. In fig. 209*, Ff is a screw which passes through a nut g, this nut is mounted in a frame so as to be capable of revolving but not of travelling endlong in the direction of the axis of the screw. So that if the nut were turned round, and the screw itself prevented from revolving, this screw would receive an endlong motion in the usual manner, at the rate of one pitch for each revolution of the nut. A toothed wheel E is fixed to the nut, and engaged with a pinion C, which is fixed to the axis Aa, parallel to the screw. To the screw is also fixed a toothed wheel D, which engages with a long pinion B upon the same axis Aa which carries the pinion C. When Aa revolves therefore, it communicates rotation both to the screw and to the nut. If B and C, D and E were respectively equal, it is plain that the nut and screw would revolve as one piece, and consequently no relative motion take place between them; but as these wheels are purposely made to differ, the nut and screw revolve with different velocities, and thus a motion arises between the nut and its screw, which causes the latter to travel in the direction of its length, with a velocity ratio that may be thus calculated.

Let the letters BCDE applied to the wheels, represent their respective numbers of teeth, and let P be the pitch of the screw. Also, let the synchronal rotations of the axis Aa, the nut and the screw, be L L,, and L, respectively,

• This combination occurs in White's Century of Inventions.

But the endlong motion of the screw depends upon the relative rotations of the screw and nut, and not upon their absolute rotations. Now it is obvious, that if the screw make L, rotations, and the nut L, rotations in the same direction, that the screw and nut will have made L, - L rotations with respect to each other, and therefore that the screw will have advanced endlong through a space

which may be made very small with respect to L.

This combination is applied to machinery for boring, for the motion of a boring instrument consists of a quick rotation combined with a slow advance in the direction of its axis, which is precisely the motion given to the screw Ff Nothing more is therefore required than to fix the boring tool to one end of this screw.

The long pinion B (Art. 390) is employed for the obvious purpose of maintaining the action of B upon D during the endlong motion of the screw, and this endlong motion is in fact the difference of two motions that are simultaneously given to the screw. For Aa revolving, if B and D were removed the rotation of the nut would cause the screw to travel endlong with one velocity, and if C and E were removed instead of B and D, then the rotation of the screw in its fixed nut would cause it to travel endlong with another velocity; but these two causes operating simultaneously, the screw travels with the difference of these velocities.

404. A slow relative rotative motion of two concentric pieces may be produced, as in fig. 210, in which Dd is a fixed stud, B an endless screw-wheel revolving upon the stud, and C a second endless screw-wheel revolving upon the tube which carries the preceding wheel B. A is an

D

B

210

endless screw so placed as to act at once upon both wheels*. Now if these wheels had the same number of teeth they would move as one piece, but if one of them has one or two teeth more or less than the other, this will not disturb the pitch of the teeth sufficiently to interfere with the action of the endless screw. And as the revolutions of this screw will pass the same number of teeth in each wheel across the plane of centers, it follows that when one wheel has thus made a complete revolution, the other will have made more or less than a complete revolution by exactly the number of deficient or excessive teeth.

Let B have N teeth, and C, N + m teeth, then since the same number of teeth in each wheel will simultaneously pass the plane of centers, Nx N+m teeth of each will pass during N rotations of C, and N + m of B, which are ⚫ therefore their synchronal rotations, and their relative rotations in the same time are N + m

N = m.

This contrivance is used in counting the revolutions of machinery, for by attaching an index to the tube which carries B, and graduating the face of C into a proper dialplate, b revolves so slowly with respect to C, that it may be made to record a great number of rotations of A before it returns again to the beginning of the course. Thus if B have 100 teeth and C 101, the hand will make one rotation round the dial during the passage of 100 × 101 teeth of either wheel across the plane of centers, that is, during 10100 rotations of the screw. Also the same hand b may read off sub-divisions upon a small dial attached to the extremity of the fixed axis d.

405. This contrivance does not strictly belong to the problem we are at present considering, but it has a kind of

• From Wollaston's Odometer, for registering the number of turns made by a carriage-wheel.

natural affinity with it that induced me to give it a place here. Similarly, a thick pinion upon an axis parallel to Dd, may be employed to drive the two wheels in lieu of an endless screw, but the relative motion will not be so slow*. But by employing two pinions of different numbers of teeth to drive the two wheels a very slow (relative motion may be obtained; thus, if in fig. 209, the screw and nut be suppressed, and the wheel E be the dial-plate, and the wheel D carry the index, as in fig. 210, then we have found

406. A train of mechanism the axes of which are carried by an arm or frame which revolves round a center, as in figs. 211, 212, 213, is termed in this work an Epicyclic train.

The two wheels which are at each end of such a train, or at least one of them, will be always concentric to the revolving frame.

Thus in fig. 211, CB is the frame or train-bearing arm, a wheel A concentric to this frame geers with a pinion b, upon whose axis is fixed a wheel E that geers with a wheel B. And thus we have an epicyclic train A

(Art. 233)

b-E

B,

This combination occurs in a clepsydra, by Marcolini, described in the notes to the ninth book of Vitruvius, by Dan. Barbaro, 1556. Vide also Art. 256,