the two directions according to which the same power will sustain a given weight upon the same plane, are equally inclined. with respect to a perpendicular to this plane, and consequently with respect to this plane itself; and they both fall on the side of a perpendicular to this plane, opposite to that in which the gravity of the body is directed.

206. In the same proportion,

qp: sin HFp: sin HFq,

if, instead of the angle HF p, we put the inclination ABG of the piane, which is equal to this angle, and instead of sin HFq, its Geom. equal cos A'Fq, FA' being drawn parallel to BA, we shall 209.

Therefore, the inclination of the plane and the weight remaining the same, the power q must be so much the smaller, as the cosine of its inclination to the plane is greater; accordingly, as the greatest of all the cosines is that of 0°, we say that the direction Fig.113; in which a power acts to the greatest advantage, in sustaining a weight upon an inclined plane, is that which is parallel to this plane.

207. In this case the proportion

q:p :: sin ABG : cos A'Fq

becomes

qp sin ABG 1 or radius.

Now if, from the point A, we let fall the perpendicular AL upon the horizontal line BG, we shall have in the right-angled triangle

Fig 113

ᎪᏞᏴ,

sin ABG: 1 :: AL: AB;

Trig. 30

therefore

q : p :: ᎯᏞ : ᎯᏴ ;

that is, when the power acts in a direction parallel to the plane; it is to the weight as the height of the plane is to its length.

Mech.

16

Fig.114.

Trig. 32.

208. If the direction of the power be horizontal, the angle A'Fq, being the complement of BAL, the proportion becomes q : p :: sin ABG cos A'Fq,

that is, when the direction of the power is parallel to the base of the inclined plane, the power is to the weight as the height of the plane to its base.

From the proportion

qp: sin ABG cos A'Fq,

we infer, as a general conclusion, that so much less power is required according as the inclination of the plane is less, and according also as the inclination of the power to the plane is less.

We have said nothing of the point where the direction of the power is to be applied to the body. This point is determined only by the condition, that the direction of the power meet the ver tical drawn through the centre of gravity of the body in a point from which a perpendicular let fall upon the plane has the conditions mentioned, article 196, &c.

We hence see that a homogeneous sphere cannot be sustained upon an inclined plane, except when the direction of the sustaining force passes through the centre of the figure, which is at the same time the centre of gravity.

209. If several powers, instead of one, are opposed to the action of the weight, what we have said respecting the power q, is to be understood of the resultant of these several powers. If the Fig.115. body p, for example, is supported upon an inclined plane by the

combined action of a power q, and of the resistance of a fixed point B, to which is attached the cord HD q, passing round the body; through the point of meeting S, of the two cords BH, q D, suppose a line SF drawn so as to bisect the angle formed by the cords. If this line cut a vertical line passing through the centre of gravity in a point F, from which a perpendicular can be let fall upon the plane that shall pass through the point of contact

H, the equilibrium will be possible; and the ratio of the weight p to the effort in the direction SF will be determined by the foregoing rules. The ratio of the effort, in the direction SF, to the power q, will be the same as in the moveable pulley. Thus if the power q is exerted in a direction parallel to the plane, the 207. weight p will be to the power 9, as the length of the plane to half its height; that is, the power will be only one half of what would be necessary without the aid of the pulley, or fixed point B.

210. With respect to the whole pressure exerted upon the plane, it will be easily determined by the ratios above establish ed. As to the particular pressure, however, that takes place upon each of the points where the body rests upon the plane, it is absolutely indeterminate, except in the case where the body touches only in two points; and in this case the whole pressure is divided between these two points in the inverse ratio of the 161. distances of its direction from these points. In every other case there are no other conditions for determining the several pressures except (1.) That the sum of them must be equal to the whole pressure. (2.) That the sum of their moments, taken with respect to an axis perpendicular to the direction of the whole pressure, is zero; the same will be true of the sum of the moments with respect to another axis perpendicular to the first. These two axes, moreover, pass through a point in the direction of the whole pressure. Thus, when a body rests upon a plane by means of a plane surface, there is no reason for supposing that all the points upon which it rests should experience equal pressures, except when it has the figure of a right prism or a right cylinder.

211. With respect to bodies which rest upon several planes at once, either in virtue of a single force, or of several forces, in which we comprehend their gravity, the general law of equilibrium is, (1.) That the resultant of all these forces must admit of being decomposed into as many forces as there are points on which the body rests; (2.) That these must be perpendicular to the plane touching the body at this point.

Let a heavy body KGI be placed in equilibrium upon two inclined planes; this state can continue only while the weight of Fig.116. the body is destroyed by the resistance of the planes; if there

48.

fore the body is in contact with each of the planes only in a single point, and perpendiculars IO, KO, be drawn through these points, they must meet in some common point O, of the vertical passing through the centre of gravity G, in order that the weight of the body may admit of being decomposed into two other forces having directions perpendicular to these planes. The components IO, KO, will represent the pressures exerted upon the planes. It hence results, that the plane which passes through the points of support and the centre of gravity must be vertical, or perpendicular to the inclined planes, or to their common intersection, which will consequently be horizontal.

What is here said is not peculiar to the case of a body urged by gravity simply. Whatever be the forces acting at I, K, their resultant must conform to what we have said of the vertical passing through the centre of gravity.

Let XZ be a horizontal plane passing through the intersection B of the inclined planes; and through the point K, draw KH also horizontal; and let the weight of the body KIG be represented by g, and the pressures exerted upon the two planes AB, BC, by p, q, respectively. In order to obtain these pressures, we must suppose the weight o of the body to be a vertical force applied at O; thus regarded, it may be decomposed into two others, directed according to OI, OK; we have accordingly the following proportions,

pq sin IOK: sin GOK: sin IOG,

Geom.

80.

Trig. 13.

Trig. 32.

or, since the angle CBZ = GOK, and ABX
angles IBK, IOK, are supplements of each other,

=

IOG, and the

pqsin ABC sin CBZ: sin ABX,
:: sin HBK: sin BKH: sin KHB,
HK : HB : BK.

212. These principles are sufficient for determining, under all circumstances, the conditions of equilibrium, where planes are concerned. By means of them we are enabled to explain the strength of arches, and in general why hollow bodies, whose exterior surface is convex, are better fitted, on this account, to resist a compressing force. If, for example, a body is composed

of four parts ABCD, CDFE, FEGH, ABGH, the exterior and Fig.116. interior surfaces of which are circular and concentric, and the same force be applied to the centre of gravity of each part, and be directed toward the common centre of the whole, no separation can take place among the parts, however great the force employed, provided the material itself be sufficiently hard. For it will be seen, that the force belonging to each part may be considered as decomposed into two others perpendicular respectively to the two plane faces of this part, and that consequently between each pair of contiguous planes there will be two equal and directly opposite forces; so that the several forces will mutually destroy each other, and a general equilibrium will be the result. The parts ABCD, &c., are called voussoirs. In a regular arch, the upper voussoir is distinguished by the name of key-stone. The surfaces which separate the voussoirs are technically termed joints. The interior curve of the arch is called the intrados, and the exterior, or that which limits all the voussoirs, when they are in equilibrium, is called the extrados; the masses of masonry at each end, that support the arch, are the abutments. The beginning of the arch is called the spring, the middle the crown, and the parts between the spring and the crown, the haunches of the arch. The part of the abutment from which the arch springs, is termed the impost; and the distance between the imposts the span of the

arch.

Of the Screw.

118.

213. The screw AB, is a solid cylinder having a protuberance Fig.117. or thread raised upon its convex surface, and carried round obliquely, and continually with the same inclination to the axis.

The nut is a hollow cylinder with a spiral groove cut upon the concave surface, and fitted to receive the thread of the screw. The former is sometimes called the external, and the latter the internal screw.

Sometimes the nut is fixed, and the screw in turning has all its threads carried successively through it; sometimes the screw is fixed, and the nut in turning passes the whole length of the screw. In each case, while the power is applied at the same distance from the axis of the screw, there is always the same