TABLE V. RIGIDITY OF ROPES.

Table of the values of the constants D and E, according to the experiments of Coulomb (reduced to English measures). The radius R of the pulley is to be taken in feet.

No. 1. New dry cords. Rigidity proportional to the square of the

circumference.

No. 2. New ropes dipped in water. Rigidity propor

tional to the square of the circumference.

Circumference of

the Rope in Inches.

Value of D in lbs. Value of E in lbs.

1.6

2:56

1-7

2-89

No. 3. Dry half-worn ropes. Rigidity proportional to the square root of the cube of the circumference.

No. 4.

Wetted half-worn cords. Rigidity proportional

to the square root of the cube of the circumference.

No. 5. Tarred rope. Rigidity proportional to the number of strands.

Number of Strands. Value of D in lbs. Value of E in lbs.

To determine the constants D and E for ropes whose circumferences are intermediate to those of the tables, find the ratio of the given circumference to that nearest to it in the tables, and seek this ratio or proportion in the first column of the auxiliary table to the right of the page. The corresponding number in the second column of this auxiliary table is a factor by which the values of D and E for the nearest circumference in the principal tables being multiplied, their values for the given circumference will be determined.

PART III.

THE THEORY OF MACHINES.

143. THE parts of a machine are divisible into those which receive the operation of the moving power immediately, those which operate immediately upon the work to be performed, and those which communicate between the two, or which conduct the power or work from the moving to the working points of the machine. The first class may be called RECEIVERS, the second OPERATORS, and the third coMMUNICATORS of work.

THE TRANSMISSION OF WORK BY MACHINES.

144. The moving power divides itself whilst it operates in a machine, first, Into that which overcomes the prejudicial resistances of the machine, or those which are opposed by friction and other causes uselessly absorbing the work in its transmission. Secondly, Into that which accelerates the motion of the various moving parts of the machine; so long as the work done by the moving power upon it exceeds that expended upon the various resistances opposed to the motion of the machine (Art. 129.). Thirdly, Into that which overcomes the useful resistances, or those which are opposed to the motion of the machine at the working point or points by the useful work which is to be done by it. Thus, then, the work done by the moving power upon the moving points of the machine (as distinguished from the working points) divides itself in the act of transmission, first, Into the work expended uselessly upon the friction and other prejudicial resistances opposed to its transmission. Secondly, Into that

accumulated in the various moving elements of the machine, and reproducible. Thirdly, Into the useful work, or that done by the operators, whence results immediately the useful products of the machine.

145. The aggregate number of units of useful work yielded by any machine at its working points is less than the number received upon the machine directly from the moving power, by the number of units expended upon the prejudicial resistances and by the number of units accumulated in the moving parts of the machine whilst the work is being done.

For, by the principle of vis viva (Art. 129.), if ΣU, represent the number of units of work received upon the machine immediately from the operation of the moving power, Zu the whole number of such units absorbed in overcoming the prejudicial resistances opposed to the working of the machine, EU, the whole useful work of the machine (or that done by its operators in producing the useful effect), and

1

2g

2

Σw(v,2 —v‚2) one half the aggregate difference of the vires vivæ of the various moving parts of the machine at the commencement and termination of the period during which the work is estimated, then, by the principle of VIS VIVA (equation 108),

EU1 =EU2+Σu+ — Σw(v,?—v‚?) . . . . .

2

1 2g

(112);

in which v1 and v, represent the velocities at the commencement and termination of the period, during which the work is estimated, of that moving element of the machine whose weight is w. Now one-half the aggregate difference of the vires vivæ of the moving elements represents the work accumulated in them during the period in respect to which the work is estimated (Art. 130.). Therefore, &c.

146. If the same velocity of every part of the machine return after any period of time, or if the motion be periodical, then is the whole work received upon it from the moving power during that time exactly equal to the sum of the useful work done, and the work expended upon the prejudicial resistances. For the velocity being in this case the same at the commencement and expiration of the period during which the work is estimated, Zw(v,2-v,2)=0, so that

Therefore, &c.

Συ, = Συ, + Σ« . . . . .

2

(113).

The converse of this proposition is evidently true.

2

147. If the prime mover in a machine be throughout the motion in equilibrium with the useful and the prejudicial resistances, then the motion of the machine is uniform. For in this case, by the principle of virtual velocities (Art. 127.), EU1=EU2+Σu; therefore (equation 112) Σw(v ̧2—v‚2)=0; whence it follows that (in the case supposed) the velocities v, and v, of any moving element of the machine are the same at the commencement and termination of any period of the motion however small, or that the motion of every such element is a uniform motion. Therefore, &c.

The converse of this proposition is evidently true.

THE MODULUS OF A MACHINE MOVING WITH A UNIFORM OR PERIODICAL MOTION.

148. The modulus of a machine, in the sense in which the term is used in this work, is the relation between the work constantly done upon it by the moving power, and that constantly yielded at the working points, when it has attained a state of uniform motion, if it admit of such a state