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."

Note (8) Ed. App.

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 works 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 prejudi cial 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 ≤U, 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

Zw(v,'—v‚3) 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),

£U1 =£U,+£u+ __Σw(v,'—v‚') . . . . . (112) ;
1 . . .

2g

in which v, 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 repect 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, £w(v,'—v‚2)=0, so that

*Note (t) Ed. App.

Therefore, &c.

£U ̧=ΣU2+£u . . . . . (113).

The converse of this proposition is evidently true.

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.), U=U,+; therefore (equation 112) Σw(v,'—v‚3)=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 of motion; or if the nature of its motion be periodical, then is its modulus the relation between the work done at its moving and at its working points in the interval of time which it occupies in passing from any given velocity to the same velocity again.

The modulus is thus, in respect to any machine, the parti cular form applicable to that machine of equation (113), and being dependent for its amount upon the amount of work Eu expended upon the friction and other prejudicial resistances opposed to the motion of the various elements of the machine, it measures in respect to each such machine the loss of work due to these causes, and therefore constitutes a true standard for comparing the expenditure of moving power necessary to the production of the same effects by different ma

chines: it is thus a measure of the working qualities of machines.*

Whilst the particular modulus of every differently con structed machine is thus different, there is nevertheless a general algebraical type or formula to which the moduli of machines are (for the most part and with certain modifications) referable. That form is the following,

where U, is the work done at the moving point of the machine through the space S, U, the work yielded at the working points, and A and B constants dependent for their value upon the construction of the machine: that is to say, upon the dimensions and the combinations of its parts, their weights, and the co-efficients of friction at their various rubbing surfaces.

It would not be difficult to establish generally this form of the modulus under certain assumed conditions. As the modulus of each particular machine must however, in this work, be discussed and determined independently, it will be better to refer the reader to the particular moduli investigated in the following pages. He will observe that they are for the most part comprised under the form above assumed; subject to certain modifications which arise out of the discussion of each individual case, and which are treated at length.

149. There is, however, one important exception to this general form of the modulus: it occurs in the case of machines, some of whose parts move immersed in fluids. It is. only when the resistances opposed to the motion of the parts of the machine upon one another are, like those of friction, proportional to the pressures, or when they are constant resistances, that this form of the modulus obtains. If there be resistances which, like those of fluids in which the moving parts are immersed (the air, for instance), vary with the velocity of the motion, and these resistances be considerable, then must other terms be added to the modulus. This subject will be further discussed when the resistances of fluids are treated of. It may here, however, be observed, that if the machine move uniformly subject to the resistance of a fluid during a given time T, and the resistance of the fluid.

* The properties of the modulus of a machine are here, for the first time, discussed.