obtain a maximum effect on the wheel and axle, (the weight of the machinery not being considered), we may adopt the conclusions of cors. 6 and 7 of this prop. And in the extreme case where the wheel and axle becomes a pulley, the expression in cor. 8 may be adopted. The like conclusions may be applied to machines in general, if R and r represent the distances of the impelled and working points from the axis of motion; and if the various kinds of resistance arising from friction, stiffness of ropes, &c. be properly reduced to their equivalents at the working points, so as to be comprehended in the character w for resistance overcome. PROPOSITION II. Given R and r, the arms of a straight lever, s and m their respective weights, and r, the power acting at the extremity of the arm R; to find the weight raised at the extremity of the other arm when the effect is a maximum. In this case im is the weight of the shorter end reduced to B, and T B applied at A, would balance the shorter end; therefore M +w, would sustain both the shorter end and the weight 2r R w in equilibrio. Put + is the power really acting at the longer end of the lever; consequently mr x + 1 m - (2 +w), is the absolute moving power. Now 2r R the distance of the centre of gyration of the beam from * * The distance of R, the centre of gyration, from c the centre or axis of motion, in some of the most useful cases, is as below; R3+2 is = √ prop. 50) R2 which let be denoted by e; then (Mechán. (M+m) will represent the mass equivalent to the beam or lever when reduced to the point a; while the weight equivalent to w, when referred to that point, p2 will be w. Hence, proceeding as in the last prop. we R2 then will w) ÷ 5, (x+m) + e + R2 W of P, or of w reduced to A. Mul- ・rm 2R ánd n for p + (Mm), then R2 r2 be a quantity which varies as the effect varies, and which, indeed, when multiplied by gt, dénotes Cor. When R=r, and м=m, if we restore the values of n and 9, the expression will become w⇒✓✓(2p2+2mr+‡m2)—(P+m). PROPOSITION III. Given the length l and angle e of elevation of an inclined plane i will have a velocity v; and gravity would impress upon it in the instant i following, a new velocityg sin e. t, provided the weight p were then entirely free but, by the disposition of the system, will be the velocity which obtains in reality. Then, estimating the spaces in the direction cr, as the body w moves with an equal velocity but in a contrary sense, it is obvious that by applying the 3d Law of Motion, the decomposition may be made as follows. At the end of the time 1+ we have, for the velocity in ressed on, * . . . v†g sín e. t, where Sv + v effective veloc. from c towards B. g sin e.t v,.... velocity destroyed. ข effective veloc. from c towards A.. ŵ. ~~ v4 g sín E. t, where+g sin E. t,.. velocity destroyed. If, therefore, gravity impresses, during the time upon the E pg sin e.t Po wg sin E i + w v Whence the effective accelerating force is found, i. e. è sin e -w sin E PW X g. Thus it appears that the motion is uniformly varied, and we readily, find the equations for the velocity and space from which the conditions of the motion are determined: viz. But in the triangle ABC it is AC: BC sin B sin A, that is, 1 m m any quantity, as t, is a minimum, its square is manifestly a minimum: so that substituting for its equal , and striking out the constant factors, we have 2LL(PL — Wl—PL2 L (PLWl) L3 = a min. or its fluxion =0. Here, as in all similar cases, since the fraction vanishes, its numerator must be equal to 0; consequently 2PL2-2wll—PL2=0, PL=2wl, or L :/:: 2w : r. Cor. 1. Since neither sin e nor sin E enters the final equation, it follows, that if the elevation of the plane Bc is not given, the problem is unlimited. Cor. ३ Cor. 2. When sin e=1, Bc coincides with the perpendicu lar CD, and the power p acts with all its intensity upon the weight w, This is the case of the present problem which has commonly been considered. This proposition admits of a neat geometrical demonstration. Thus, let ce be the plane upon which, if w were placed, it would be sustained in equilibrio by the power e on the plane CB, or the power r hanging freely in the vertical co; then (Mechan. prop. 23), BC: CD: CE:P:P: W. CD P But w is to the force with which it tends to descend along the plane ca, as ca to CD; consequently, the weight p is to the same force in the same ratio; because either of these weight in their respective positions would sustain w on CE, Therefore the excess of r above that force (which excess is the power accelerating the motions of r and w) is to P, as CA - CE to ca; or taking CH=CA, as En to CA. Now, the motion being uniformly accelerated, we have s¤ FT2, or т2 α = ; consequently, the square of the time in which ac' is described by w, will be as ac directly, and as — ∞ Ꭶ F SCA2 EN AC inversely; and will be least when is a minimum; that is CE2 EH CE2 EH EH when +En+2CE, or (because 2ce is invariable) when Ex EH is a minimum. Now, as, when the sum of two quantities is given, their product is a maximum when they are equal to each other; so it is manifest that when their product is given, their sum must be a minimum when they are equal. But the product of and EH is CE2, and consequently given ; therefore the sum of CE2 EH EH and EH is least when those parts are equal; that is when EH-CE, or CA=2CE. So that the length of the plane ca is double the length of that on which the weight w would be kept in equilibrio by r acting along ce. P When co and Cв coincide, the case becomes the same as that considered by Maclaurin, in his View of Newton's Philosophical Discoveries, pa. 183, 8vo. edit. PROPOSITION ་ PROPOSITION IV. Let the given weight e descend along CB, and by means of the The general expression for the vel. in v— 1 1 P sin e-w sin É P+w gt, which, by substitut. - L for sin e, and – 7 for sin E, becomes m = m 1 (PWL-W21) m gt. This mul. into w, gives which, by the prop. is to be a maximum. Or, striking out the a max. Putting this into fluxions, and reducing, we have PL-2pwl –w3l=0, or w=p√/ (~+1)—p. Cor. When the inclinations of the planes are equal, L, and 7 are equal, and w P/2-P-X(2-1)=4142P agreeing with the conclusion of the lever of equal arms, or the extreme ease of the wheel and axle, i. e. the pulley. PROPOSITION V. Given the radius R of a wheel, and the radius r of its axle, the weight of both, w, and the distance of the centre of gyration from the axis of motion e; also a given power P acting at the circumference of the wheel; to find the weight w raised by a cord folding about the axle, so that its momentum shall be a And the inertia which resists the com- mass were concentrated in a point a (Mechan. prob. 50). If the former of these be divided by the latter, |