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 P1 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 perpendicular 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. 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 THIRD CONTINUALLY REVOLVING ABOUT THE AXIS AT THE SAME PERPENDICULAR DISTANCE FROM IT. 2 3 Let P, and P, represent the parallel pressures of the system, and P1 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 P2 act to turn the system in the same direction * 2 being exceedingly small, tan. & is very nearly equal to sin. . 2 as P1, and P, in the opposite direction; also let R represent the resultant of P, and P, and r 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 r and a,, so that L2=a2-2a,r cos. + r2=(a1—r cos. 6)2+r2 sin.3 0; ... R2L2= (Ra ̧ — Rr cos. 9)2 + R22 sin.2 0. Now, R=P2+P2, Rr=P ̧a ̧-P,ɑ2; 3 ... R3L2={(P2+P2)α, -(P ̧α-Pa2) cos. 0}+(Pα-P12) sin.20. Now, if a, be>a,, then a1(P ̧+P2)>(P ̧a ̧—Р2α2), 3 .'. a,(P ̧+P2)>(P ̧a ̧—Р2a1⁄2) (sin. + cos. 4); for sin. follows, that 3 cos. is never greater than unity. Whence it a1(P3+P2) −(P ̧a ̧—Р ̧α) cos. ◊ >(P ̧a ̧− P2ɑ2) sin. §. The value of R2L2 is therefore represented by the sum of the squares of two quantities, of which the first is in all cases greater than the second. Therefore, extracting the square root by Poncelet's theorem, (see Appendix B.) 3 RL=a{(P ̧+P2)α-(P,α-Pα) cos. } +6(P ̧ ̧-Pa2) sin. ◊ very nearly; or, - RL=aa,(P ̧+P2) — (P ̧ɑ ̧ — P2ɑ2)(a cos. ◊ — ß sin. §). . . . . (194). .... If P, and P, be constant, the integral in the second member of this equation becomes (Pa ̧ — Pɑ1⁄2) (« sin. + ß cos. 0); Pa,0-Pa. U2-U, whence observing that P,a,-Pɑ3= 3 2 ; also, that U,=0Rr=0P ̧a ̧―0P2a1⁄2=U,- U2, and substituting in equation (193), we have U3 U‚=U.—U2+psin.ø { a (~;+)-(UU) (a sin. 0+ẞ cos. 0)} ·· (196) ; for a complete revolution making 0=2′′, we have a a B Σπαι U.={1+psin.(-2)}U,-{1-psin (+-)}U.... (197), 2απ which is the modulus of the system where a and ß are to be determined, as in Note B, (Appendix.) 3 183. 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 6,) 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 Ε (1+) U2+2-D. Substituting this value for U ̧ in equa tion (197), and neglecting terms which involve products of E psin. psin., and D, the exceedingly small quantities a3 a3 we have E U1 = {1++psin. ( α Σπα, a αι )}U.—{1—psin. (2+)}U2+2xD... (198). To determine the modulus for n revolutions we must substitute in this expression në for лë. cord is coiled upon one of these, and the other, in an opposite direction, upon the other; so that when the axle is turned, and the cord is wound upon one of these two parts of the drum, it is, at the same time, wound off the other, and the intervening cord is shortened or lengthened, at each revolution, by as much as the circumference of the one cylinder exceeds that of the other. In thus passing from one part * A figure of the capstan with a double axle was seen by Dr. O. Gregory among some Chinese drawings more than a century old. It appears to have been invented under the particular form shown in the above figure by Mr. G. Eckhardt and by Mr. M'Lean of Philadelphia. (See Professor Robison's Mech. Phil. vol. ii. p. 255.) of the drum to the other, the cord is made to pass round a To determine the modulus of this machine, let us and us Then, since in this case we have a body moveable about a cylindrical axis, and acted upon by three pressures, two of which are parallel and constant, viz. the tensions of the two parts of the cord; and the point of application of the third is made to revolve about the axis, remaining always at the same perpendicular distance from it; it follows, (by equation 198), that, for n revolutions of the axis, where U1=Au2-Bu2+2nD . . . . . β A={1 2na, (199); a β and B = { 1 – psin. 4 (2 a, and a representing the radii of the two parts of the drum, Pl Also, since the two parts of the cord pass over a pulley, if S represents the length of cord which passes over the pulley, where E 201 :{ a t2=A1t2+B1 ; 2 W cos. +2P1 sin.p}, and B, =1++), sin. e{t α D{1 P1 a representing the radius of the pulley, p1 the radius of its |