-1

neglecting powers of (+) above the first, since in all

cases its value is less than unity. Integrating this quantity between the limits 0 and 2 the second term disappears, so that

2

1

1

+ =U2 +

2

;

since 2α, is the space through which the point of application of the constant pressure P2 is made to move in each revolution. Therefore by equation (187), in the case in which P, is constant,

179. If the pressure P2 be supplied by the tension of a rope winding upon a drum whose radius is a, (as in the capstan), then is the effect of the rigidity of the rope (Art. 142.) the same as though P, were increased by it so as to become

2

D+EP2, or E

P2+ or (1+)P,+D

2

Now, assuming P, to be constant, and observing that U2=2P2a2, we have, by equation (187),

Substituting in this equation the above value for P2,

Performing the actual multiplication of these factors, ob

serving that is exceedingly small, and omitting the term

D. a2

E p +2xD.

U1 = P12 (1+) { 2x + P sin, ©/Ldv}

Whence performing the integration as before, we obtain 1 1

U1 = U. (1+) { 1 + (+)'p sin. e } +2«D.

απ

If this equation be multiplied by n, and if instead of U1 and U, representing the work done during one complete revolution, they be taken to represent the work done through n such revolutions, then

1

U.=U,(1+) { 1+ (+) p sin. p}

which is the MODULUS.

+2nяD... (189),

180. If the quantity (+2) be not so small that terms

a

of the binomial expansion involving powers of that quantity above the first may be neglected, the value of the definite integral

2′′

de may be determined as follows: —

2π

:

a12+2a ̧a‚cos. §+a22)*d☺= / {(a1 + a2)2 — 2a ̧ a2 (1 — cos. 0)} * dł

2

— d0

= 2(a1 + a») [(1 − k2 sin. 20)' dê * = 2(a1+-a,)E1(k), where E ̧ (4)

*See Encyc. Met. art. DEF. INT. theorem 2.

represents the complete elliptic function of the second order, whose modulus is k.* The value of this function is given for all values of k in a table which will be found at the end of this work.

Substituting in equation (187),

U1=U2+o sin.$.2(a1+a2). E‚(k)†· P ̧=U2+(

а

1/1 1

π

.. U1 = U, { 1 + ¦ (1 + 4)p sin. }..

π а1

P

THE CAPSTAN.

181. The capstan, as used on shipboard, is represented in

the accompanying figure. It consists of a solid timber CC, pierced through the greater part of its length by an aperture AD, which receives the upper portion of a solid shaft AB of great strength, whose lower extremity is prolonged, and strongly fixed into the timber framing of the

ship. The piece CC, into the upper portion of which are fitted the moveable arms of the capstan, turns upon the shaft AB, resting its weight upon the crown of the shaft, coiling the cable round its central portion CC, and sustaining the tension of the cable by the lateral resistance of the shaft. Thus the capstan combines the resistances of the pivot and the axis, so that the whole resistance to its motion is equal to the sum of the resistances

с

See Encyc. Met. art. DEF. INT. theorem 2.

An approximate value of E,(k) is given when k is small by the 2/k

formula E,(k)=(1+K−1), where K=1 (See Encyc. Met. art. DEF. INT. equation (W'), 14.)

due separately to the axis and the pivot, and the whole work expended in turning it equal to the whole work which would be expended in turning it upon its pivot were there no tension of the cable upon it, added to the whole work necessary to turn it upon its axis under the tension of the cable were there no friction of the pivot. Now, if U, represent the work to be done upon the cable in n complete revolutions, the work which must be done upon the capstan to yield this work upon the cable is represented (equation 189.) by

1

2

(1+) { 1+ (+) p sin. p. } U,

α

p sin. 4. }U2+2nπD,

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. 175.), the work due to the friction of the

4

pivot in n complete revolutions is represented by rp, W.

On the whole, therefore, it appears that the work U1 expended upon n complete revolutions of the capstan is represented by the formula

E

U1 = (1 + 5) {1 + (4; +4)*psin.e}U, +2n={D+p,/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 P1, 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, per-
pendicular to the same straight line passing through the
centre, and at equal distances from it.*

2

Suppose that they may thus be 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 P2 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, they would be in equilibrium about that point (Art. 8.). But when so removed, R, and R2 will act in the same straight line and in opposite directions. Moreover, they are equal to one another; R1 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 equilibrium; whence it follows, that R is equal to P, or the whole pressure upon the axis, equal in this case to the whole tension P2 upon the cable. So that the friction of the axis is represented in every position of the capstan by P, tan. 4 (tan. 4 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

2

2

the angle by Papo tan. 4, or by Pa() tan. 9, or by U.(1) tan. ; so that, on the whole, introducing the cor

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

2

2

U1=U, (1+) { 1+ (£) tan. 4 } +20%

{

This is manifestly the least possible value of the modulus,

*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.