U Its form will be greatly simplified if we assume cos. 4=1, since is small, suppose the co-efficient of friction at each axis to be the same, so that P1=P2=43=91, and divide by the coefficient of U,, omitting terms above the first dimension sin.4, &c.; whence we obtain by reduction. in 258. The best position of the axis of the beam. (321). Let a be taken to represent the length of the beam, and r the distance aC of the centre of its axis from the extremity to which the driving pressure is applied. Let the influence of the position of the axis on the economy of the work necessary to open the valves, to work the air-pump, and to overcome the friction produced by the weight of the axis, be neglected; and let it be assumed to be that, by which a given amount of work U2 may be yielded per stroke upon the crank rod, by the least possible amount U, of work done upon the piston rod. If then, in equation (321), we assume the three last terms of the second member to be represented by A, and observe that a, in that equation is represented by x, and a, by a-x, we shall obtain In practice the angle 0 never exceeds 20°, so that cos. O never differs from unity by more than 060307. The error, resulting from which difference, in the friction, estimated as above, must in all cases be inconsiderable. The best position of the axis is determined by that value of a which renders this function a minimum; which value of x is represented by the equation If p2>P1 then(+2) *> (322). > 1 and x <a; in this case, there fore, the axis is to be placed nearer to the driving than to the working end of the beam. If p2 <P1, the axis is to be fixed nearer to the working than to the driving end of the beam. 259. It has already been shown (Art. 168.), that a machine working, like the beam of a steam engine, under two given pressures about a fixed axis, is worked with the greatest economy of power when both these pressures are applied on the same side of the axis. This principle is manifestly violated in the beam engine; it is observed in the engine worked by Crowther's parallel motion*, and in the marine engines recently introduced by Messrs. Seaward, and known as the Gorgon engines. It is difficult indeed to defend the use of the beam on any other legitimate ground than this; that in some degree it aids the fly-wheel to equalise the revolution of the crank arm †, an explanation which does not extend to its use in pumping engines, where nevertheless it retains its place; adding to the expense of construction, and, by its weight, greatly increasing the prejudicial resistances opposed to the motion of the engine. * As used in the mining districts of the north of England. †The reader is referred to an admirable discussion of the equalising power of the beam, by M. Coriolis, contained in the thirteenth volume of the Journal de l'E'cole Polytechnique. BB THE CRANK. 260. The modulus of the crank, the direction of the resistance being parallel to that of the driving pressures. D E Let CD represent the arm of the crank, and AD the con necting rod. And to simplify the investigation, let the connecting rod be supposed always to retain its vertical position. * Suppose the weight of the crank arm CD, acting through its centre of gravity, to be resolved into two other weights (Art. 16.), one of which W, is applied at the centre C of its axis, and the other at the centre c of the axis which unites it with the connecting rod. Let this latter weight, when added to the weight of the connecting rod, be represented by W1. Let P2 represent a pressure opposed 2 to the revolution of the crank, which would at any instant be just sufficient to balance the driving pressure P1 transmitted through the connecting rod; and to simplify the investigation, let us suppose the direction of the pressure P2 to be vertical and downwards. 2 2 Let Cc-a, CA1=a1, CA2=a2, cCW2=0, radii of axes C and c=P1, P2, lim. ▲s of resistance =1, 2, W=whole weight of crank arm and connecting rod W1+ W2 = 1 Since the crank arm is in the state bordering upon motion, the perpendicular distance of the direction of the resistance upon its axis C from the centre of that axis, is represented by Pi sin. (Art. 153.). The resistance is also equal to * Any error resulting from this hypothesis affecting the conditions of the question only in as far as the friction is concerned, and being of two dimensions at least in terms of the coefficient of friction and the small angular deviation of the connecting rod from the vertical. P1±(P2+W); P, being supposed greater than P2+W, and cording as the crank arm is sures in equilibrium. There fore, by the principle of the equality of moments, observing that the driving pressure is (P1+W1)a1 = Pa2+ {P1±(P2+W)} p1 sin. 1. 2 Since moreover the axis c, which unites the connecting rod and the crank arm, is upon the point of turning upon its bearings, the direction of the pressure P, is not through the centre of that axis, but distant from it by a quantity represented by f2 sin. 2, which distance is to be measured on that side of the centre c which is nearest to C, since the friction diminishes the effect of P1 to turn the crank arm. P2 1 Substituting this value of a, in the preceding equation, (P1±W1)(asin.8—f2sin.42)=P2a2+ {P,±(P2+W)} p, sin.ø, . . .(324). Transposing and reducing Pasin. 0-pg sin. 4-p, sin.,}=P, {a,±p, sin. 4,}+Wp, sin., FW, (a sin. 6-p, sin.); 2 which is the relation between P, and P, in their state border- 2 ment AU, of the work yielded by the crank whilst that small P12 {a sin. 0—p2 sin. 2—p; sin. 41} A0= {a2±p1 sin. Q1} AU1⁄2±Wa1⁄2 sin. 140 ‡ W ̧a1⁄2(a sin. 0—p2 sin. ø2)A0 . whence passing to the limit, integrating from 0=✪ to @= π-, and dividing by a, P1 {2a cos. (325). − (x−2®) (pg sin. Ø2+pi | sin. 41)} = {1±(1 sin. 41} U2+W(x−2®)p; sin. ¡FW, {2a cos. ✪—¿1⁄2(~—2 O) sin. 2} . (32) Now, let it be observed that 2a cos. O represents the projection of the path of the point c upon the vertical direction of P1, whilst the arm revolves between the positions and T-; so that P,2a cos. represents (Art. 52.) the work U1 done by P1 upon the crank whilst the arm passes from one of these positions to the other, or whilst the work U2 is yielded by the crank. Whence it follows that P1=sec. . Substituting this value of P1, dividing by a, and reducing, we obtain U 2a ... 29 W(-20) p, sin., W, {2a cos. —P2(π-20) sin. 1} . . . . (327). By which equation is determined the modulus of the crank in respect to the ascending or descending stroke, according as we take the upper or lower signs of the ambiguous terms. 1 Adding these two values of the modulus together, and representing by U1 the whole work of P1, and by U, the whole work of P2, whilst the crank arm makes a complete revolution, also by u, the work of P, in the down stroke, and ug in the up stroke, we obtain 2 © (P2sin. 4, + sin. 4,) } = U2+ (u, —u,)lisin. 4, ... (3:28). Θ a which is the modulus of the crank in respect to a vertical |