where a, represents the length of the arm, and a, the radius of that portion of the capstan on which the cable is winding. Moreover (Art. 176.), the work due to the friction of the

4

pivot in ʼn complete revolutions is represented by p1f W. On the whole, therefore, it appears that the work U, expended upon n complete revolutions of the capstan is represented by the formula

E

1

U.= (1+4); 1 + (+)', sin. • } U,+

2

2n= {D+W... (191).

which is the MODULUS of the capstan.

A single pressure P, applied to a single arm has been supposed to give motion to the capstan; in reality, a number of such pressures are applied to its different arms when it is used to raise the anchor of a ship. These pressures, however, have in all cases,-except in one particular case about to be described,--a single resultant. It is that single resultant which is to be considered as represented by P, and the distance of its point of application from the axis by a,, when more than one pressure is applied to move the capstan.

The particular case spoken of above, in which the pressures applied to move the capstan have no resultant, or cannot be replaced by any single pressure, is that in which they may be divided into two sets of pressure, each set having a resultant, and in which these two resultants are equal, act in opposite directions, on opposite sides of the centre, perpendicular to the same straight line passing through the centre, and at equal distances from it.*

Suppose that they may be thus compounded into the equal pressures R, and R, and let them be replaced by these. The capstan will then be acted upon by four pressures, the tension P, of the cable, the resistance R of the shaft or axis, and the pressures R, and R. Now these pressures are in equilibrium. If moved, therefore, parallel to their present directions, so as to be applied to a single point, Two equal pressures thus placed constitute a STATICAL COUPLE. The properties of such couples have been fully discussed by M. Poinsot, and by Mr. Pritchard in his Treatise on Statical Couples; some account of them will be found in the Appendix to this work.

they would be in equilibrium about that point (Art. 8.). But when so removed, R, and R, will act in the same straight line and in opposite directions. Moreover, they are equal to one another; R, and R, will therefore separately be in equilibrium with one another when applied to that point; and therefore P, and R will separately be in equili brium; whence it follows, that R is equal to P, or the whole pressure upon the axis, equal in this case to the whole tension P, upon the cable. So that the friction of the axis is represented in every position of the capstan by P, tan. © (tan. ¢ being equal to the co-efficient of friction (Art. 138.)), and the work expended on the friction of the axis, whilst the capstan revolves through the angle by Pp tan. 9, or by

Po, (2) tan. 9, or by U,(,) tan. ; so that, on the whole,

introducing the correction for rigidity and for the friction of the pivot, the modulus (equation 191) becomes in this case

2

2nx {D+W}.
3r. ƒ W } . . . . (192).

This is manifestly the least possible value of the modulus, being very nearly that given (equation 191) by the value infinity of a,.*

Thus, then, it appears generally from equation (191), that the loss by friction is less as a, is greater, or as P, is applied at a greater distance from the axis; but that it is least of all when the pressures are so distributed round the capstan as to be reducible to a COUPLE, that case corresponding to the value infinity of a,. This case, in which the moving pressures upon the capstan are reducible to a couple, manifestly occurs when they are arranged round it in any number of pairs, the two pressures of each pair being equal to one another, acting on opposite sides of the centre, and perpendi cular to the same line passing through it. This symmetrical distribution of the pressures about the axis of the capstan is therefore the most favourable to the working of it, as well as to the stability of the shaft which sustains the pressure upon it.

being exceedingly small, tan. is very nearly equal to sin.

182. THE MODULUS OF A SYSTEM OF THREE PRESSURES APPLIED TO A BODY MOVEABLE ABOUT A CYLINDRICAL AXIS, TWO OF THESE PRESSURES BEING GIVEN IN DIRECTION AND PARALLEL TO ONE ANOTHER, AND THE DIRECTION OF THE THIRC CONTINUALLY REVOLVING ABOUT THE AXIS AT THE SAME PERPENDICULAR DISTANCE FROM IT.

Let P, and P, represent the parallel pressures of the system, and P, the revolving pressure. From the centre of the axis C, let fall the perpendiculars CA,, CA,, CA, upon the directions of the pressures, and let represent the inclination of CA, to CA, at any period of the revolution of P1. Let P, be the preponderating pressure, and let P, act to turn the system in the same direction as P., and P, in the opposite direction; also let R represent the resultant of P, and P,, and the perpendicular distance CA of its direction from C. Suppose the pressures P, and P, to be replaced by R; the conditions of the equilibrium of P, throughout its revolution, and therefore the work of P, will remain unaltered by this change, and the system will now be a system of two pressures P, and R instead of three; of which pressures R is given in direction. The modulus of this system is therefore represented (equation 187) by the formula

where U, represents the work of R, and L represents the distance AA, between the feet of the perpendiculars and a,, so that L'a,'-2a,r cos. +=(a,-r cos. )+ sin.;

.. R'L'=(Ra, — Rr cos. 4)2+ R22 sin.3ë.

..R'L'= {(P ̧+P,)a,—(P,a,-P,a,)cos.}'+ (P,a,—P,ɑ,)'sin., [Now if the relations of a, to a, are such that

{(P,+P,)a,—(P,a,—P,a,) cos. *>(P,a,-P,a,)'sin.

then the value of R'L' will be represented by the sum of the

squares of two quantities the first of which is greater thar the second. ED.] Therefore, extracting the square root by Poncelet's theorem, (see Appendix B.)

RL=a {(P,+P,)a, -(P,a,-P,a,) cos. } +(P,a,-P,a,) sin. very nearly; or,

RL=aa,(P,+P1)-(P,a,—P,a,)(a cos. ê—ß sin. 4).

(a cos. - sin. )d3. . . . (195).

If P, and P, be constant, the integral in the second member of this equation becomes (P,a,-P2a,) (a sin. + cos. ); Pa-Pa U‚—U whence observing that P,a,-P,a,

=

also, that U,Rr=¿P‚a‚—¿P‚a‚=U,-U,, and substituting in equation (193), we have

(U) (« sin. 6+8 cos. 6)}.... (196);

(a

(a

for complete revolution making =2, we have

U. U

U1 =U‚—U,+p sin. 9 { a (U' + 1) - () };

reducing,

2

α

Σπα,

U={1+p sin. (2) U-1

2a,

which is the modulus of the system where a and 6 are to be determined, as in Note B, (Appendix.)

183. If the pressure P, be supplied by the tension of a cord which winds upon a cylinder or drum at the point A,, then allowance must be made for the rigidity of the cord, and a correction introduced into the preceding equation for that purpose. To make this correction let it be observed (Art. 142.) that the effect of the rigidity of the cord at A, is the same as though it increased the tension there from

or (multiplying both sides of this inequality by a,, and integrating in respect to d,) as though it increased

Thus the effect of the rigidity of the rope to which P, is applied upon the work U, of that force is to increase it to

(1+) U,+2-D. Substituting this value for U, in equa

tion (197), and neglecting terms which involve products of E p sin. p sin. ❤ the exceedingly small quantities

U,= { 1+2+p sin. 9 (~,——) } U

α B

-P

Σπα,

{1-p sin.c(+)}U,+2 D... . (198).

, and D,

To determine the modulus for n revolutions we must sub. stitute in this expression n for .

THE CHINESE CAPSTAN.

184. This capstan is represented in the accompanying