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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.

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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

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T

B

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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;

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R3+2
3(R+r)'

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

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then will

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w) ÷ 5, (x+m) + e +

R2

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W

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of P, or of w reduced to A. Mul-
for the sake of simplifying the pro-"

・rm

2R

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ánd n for p + (Mm), then

R2

r2 be a quantity which varies as the effect
·n+ W
R2

varies, and which, indeed, when multiplied by gt, dénotes
the effect itself. Putting the fluxion of this equal to nothing,
and reducing, we at length find

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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
BC; to find the length L of another inclined plane sc along
which a given weight w shall be raised from the horizontal
line AB to the point c, in the least time possible, by means of
another given weight P descending along the given plane CB :
the two weights being connected by an inextensible thread BCW
running always parallel to the two planes.
Here we must as a preliminary
to the solution of this proposition,
deduce expressions for the motion
of bodies connected by a thread, and
running upon double inclined planes.
Let the angle of elevation CAD be
E, while e is the elevation CBD.
Then at the end of the time t, P

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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.

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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
masses P, w, the respective velocities g sinctv,
-v, and
g
sin, the system will be in equilibrio. The quanti-
ties of motion being therefore equal, it will be

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.

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But in the triangle ABC it is AC: BC sin B sin A, that is,
L : 1 e:
:: sine sine; bence L=sin e, and l sin E;

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1

m

m

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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
PL W wl

= 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.

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.

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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.

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CD

P

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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 —

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
EC2.

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

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PROPOSITION IV.

Let the given weight e descend along CB, and by means of the
thread pcw (running parallel to the planes) draw a weight w
when
up the plane ac: it is required to find the values of w,
its momentum is a maximum, the lengths and positions of the
planes being given. (See the preceding fig.)

The general expression for the vel. in v—

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1

1

P sin e-w sin É

P+w

gt, which, by substitut. - L for sin e, and – 7 for sin E, becomes

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m

=

m 1

(PWL-W21)

m

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gt. This mul. into w, gives

which, by the prop. is to be a maximum. Or, striking out the

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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

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And the inertia which resists the com-
munication of motion to the point a will be the same as if the
§2π+r3w+R2P
R3

mass

were concentrated in a point a (Mechan.

prob. 50). If the former of these be divided by the latter,

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