9. The thing indicated may be of such a nature that instead of motion it may be required to exhibit rather the periods of its being in action or out of action, opened or closed, bolted or unbolted, and so on; as in the case of clicks, bolts, or valves; in which cases lines may be used in the above manner, but words must be added in explanation of this new employment of the signs. The line should be on the right side when the piece is out of action, unbolted, or open, and on the left side when in the reverse state. Dotted lines will be employed if the piece rests in both states; and if it be necessary to exhibit the time occupied by the motion of transition from one state to the other, this can be done by a short continuous line at the beginning of each; thus if a valve fly open suddenly and close gently, it will be represented as in the margin. (The detent K is an example of this rule.)

If any other modifications of movement should present themselves, it will not be difficult for any one who has rendered himself familiar with the symbols and method just explained, to contrive others adapted to the new combinations which may present themselves.

362. As an example of the way in which very minute circumstances of motion are shown in this manner, it may be remarked, that the motion of the saw-frame, excentric, and click-lever, is necessarily continuous; but that the motion given to the ratchetwheel by the click does not begin at the instant the change of motion in the click takes place. The click must first move through a small space until it abuts against the tooth of the ratchet-wheel which is ready to receive it. On the other hand, it is evident that the ratchet-wheel and the click will both cease their motion in that direction together. When the click moves backwards the ratchet-wheel with the pinion and wood-carriage will remain at rest until the saw begins its cut, when they will be driven slightly backwards until the ratchet-tooth abuts against the end of the detent. All these accidents of motion in the ratchet-wheel and its connected pieces are exhibited by the notation, as will appear by comparing the motion lines of G with those of F. It is true, that in the actual machine these small motions are reduced exceedingly by giving a great number of teeth to the ratchet-wheel; but I have exaggerated them to show the susceptibility of the notation, which when applied to complex machinery is of the very greatest service; more especially in assisting in the invention or improvement of machines.

363. The system of motion lines is not intended to exhibit accurately the law of motion of the pieces, as in the graphic

representation of Art. 13, although it is founded upon the same principle; but merely its general phases.

When the simultaneous motions are required to be precisely exhibited, their motion curves may be, however, exactly laid down and compared, by placing them side by side; their parallel axes of abscissæ then become the indicating lines of Babbage's system. In this case, however, I am inclined to think the second. method (Art. 14) is preferable, in which the ordinates are proportional not to the velocities but to the spaces; of the use of which I have already given an example in Art. 298.

364. I have found some advantages in the amalgamation of the system of Babbage with that of which an explanation has been given in Art. 332.

For in defining trains of mechanism in the present work, I have shown that they consist of principal pieces moving each according to a given path, and connected one with the other in succession by means of drivers and followers, which are attached to these moving pieces. Now the drivers and followers carried by any one of these pieces must all move according to the same law, since they move as one piece; and a single indicating line with its velocity numbers and motion curves is quite sufficient for every such piece: whereas, as we have seen, in the notation just exhibited, every part of the machine has such an indicating line and figure attached to it, and consequently all the parts that are united together merely repeat the same indication as B, C and E; or G and H, in page 288. In the next page I have shown the Saw-mill under the form of Notation which I have been in the habit of employing, and which it will be seen at once differs only from that of page 288 by being united with the old clockmakers' form already explained; by which means the genealogy, so to speak, of the motion is perhaps more clearly perceived, and the number of indicating lines reduced.

365. To represent a machine in this form, rule as many parallel lines as there are principal moving pieces in the train, writing the name or nature of each in the first column. Upon each line write all the followers and the driver which are carried by the piece to which it belongs; taking care to place every follower vertically under its own driver, if possible. Every follower may be connected with its driver by an arrow formed according to the rules in Art. 360, or by a simple line. The arrow is only necessary if the nature of the machine renders it necessary to place some of the followers above their drivers. The connecting lines might also receive additions, by which the nature

of the connection, as by sliding, wrapping, link-work, &c. might be shown; but the names of the parts are generally sufficient for this purpose; and there is a great mischief in unnecessarily multiplying symbols. Numbers attached to toothed wheels are their numbers of teeth, to pullies their diameters in inches, to cranks and excentrics their throw in inches, unless otherwise stated. In the column of Velocity the numbers attached to revolving pieces show their angular velocity in turns per minute, and to sliding pieces their linear velocity in inches per minute, unless otherwise stated in words. In the column of Comparison of Motion, the rules in Art. 361 are followed, but that when two or more pieces move together in a system, one indicating line is made to serve for them all by connecting those to which it applies by a bracket. Thus the variation of motion in the ratchet-wheel spindle and the wood-carriage being the same, one line is used for them both. Columns may be added for the pitch of the wheels, or any other particulars that may be required.

It rarely, however, happens that the whole notation is necessary. For some machines the table of the origin of motion is required, for others that of the comparison of the motion; and of the system of the latter, and of its utility when properly applied, it is impossible to speak too highly.

CHAPTER XV.

REDUPLICATION.

366. THIS term I ventured, in my first edition, to apply to a mode of modifying motion which depends upon a totally different principle from the sliding, rolling, and wrapping connections to which our previous pages have been devoted. It is principally employed in the construction of tackle of all sorts, used on shore for raising weights, and in the rigging of ships.

367. If an inextensible string AfgB be passed over any number of fixed pins or pullies, as ƒ and g, and if the extremities

Fig. 261.

១

A,B of the string be compelled to move each in the direction of its own portion, Af, gB of the string, then the motion. of one of these extremities will evidently be communicated unaltered to the other, and every intermediate portion of the string will move with the same velocity. This is unaffected by the form of the pins over which the string passes, and they may therefore be fixed cylinders or else pullies, that is to say, wheels mounted on revolving axes, which are generally substituted for fixed pins, for the mere purpose of reducing the friction of the string in passing over them.

A

B

If however some of the pins (or axes of the pullies) be attached to a piece capable of motion, and the string be passed back and forwards over the fixed and movable pins alternately, this re

duplication will cause the several intermediate portions of the string to move with different velocities, and the movable piece will receive a velocity compounded of these in a manner which we will proceed to investigate. Thus let the string, fig. 262, be