Equilibrium of a particle. Angle of repose. (4) If it rest on a rough surface, friction will be called into play, resisting motion along the surface; and there will be equilibrium at any point within a certain boundary, determined by the condition that at it the friction is μ times the normal pressure on the surface, while within it the friction bears a less ratio to the normal pressure. When the only applied force is gravity, we have a very simple result, which is often practically useful. Let be the angle between the normal to the surface and the vertical at any point; the normal pressure on the surface is evidently W cose, where W is the weight of the particle; and the resolved part of the weight parallel to the surface, which must of course be balanced by the friction, is W sine. In the limiting position, when sliding is just about to cominence, the greatest possible amount of statical friction is called into play, and we have The value of thus found is called the Angle of Repose. Let (x, y, z)=0 be the surface: P, with direction-cosines 1, m, n, the resultant of the applied forces. The normal pressure is Attraction. аф do 2 аф αφ dz dy do do 2 dx P Hence, for the boundary of the portion of the surface within which equilibrium is possible, we have the additional equation 457. A most important case of the composition of forces acting at one point is furnished by the consideration of the attraction of a body of any form upon a material particle any where situated. Experiment has shown that the attraction Attraction. exerted by any portion of matter upon another is not modified by the proximity, or even by the interposition, of other matter; and thus the attraction of a body on a particle is the resultant of the attractions exerted by its several parts. To treatises on applied mathematics we must refer for the examination of the consequences, often very curious, of various laws of attraction; but, dealing with Natural Philosophy, we confine ourselves mainly, (and except where we give the mathematics of Laplace's beautiful and instructive and physically important, though unreal, theory of capillary attraction,) to the law of the inverse square of the distance which Newton discovered for gravitation. This, indeed, furnishes us with an ample supply of most interesting as well as useful results. law of 458. The law, which (as a property of matter) is to be care- Universal fully considered in the next proposed Division of this Treatise, attraction. may be thus enunciated. Every particle of matter in the universe attracts every other particle, with a force whose direction is that of the line joining the two, and whose magnitude is directly as the product of their masses, and inversely as the square of their distance from each other. Experiment shows (as will be seen further on) that the same law holds for electric and magnetic attractions under properly defined conditions. quantity 459. For the special applications of Statical principles to Special unit which we proceed, it will be convenient to use a special unit of of matter." mass, or quantity of matter, and corresponding units for the measurement of electricity and magnetism. Thus if, in accordance with the physical law enunciated in § 458, we take as the expression for the forces exerted on each other by masses M and m, at distance D, Mm it is obvious that our unit force is the mutual attraction of two units of mass placed at unit of distance from each other. Linear, surface, and volume, densities. 460. It is convenient for many applications to speak of the density of a distribution of matter, electricity, etc., along a line, over a surface, or through a volume. = quantity of matter per unit of length. Here line-density area. Electric and magnetic of quantity. volume. 461. In applying the succeeding investigations to electricity reckonings or magnetism, it is only necessary to premise that M and m stand for quantities of free electricity or magnetism, whatever these may be, and that here the idea of mass as depending on inertia will still repre negative masses ad mitted in abstract theory of is not necessarily involved. The formula Mm sent the mutual action, if we take as unit of imaginary electric or magnetic matter, such a quantity as exerts unit force on an Positive and equal quantity at unit distance. Here, however, one or both of M, m may be negative; and, as in these applications like kinds repel each other, the mutual action will be attraction or repulsion, according as its sign is negative or positive. With these provisos, the following theory is applicable to any of the above-mentioned classes of forces. We commence with a few simple cases which can be completely treated by means of elementary geometry. Uniform spherical 462. If the different points of a spherical surface attract shell. At equally with forces varying inversely as the squares of the disinternal tances, a particle placed within the surface is not attracted in any direction. traction on point. H Let HIKL be the spherical surface, and P the particle within it. Let two lines HK, IL, intercepting very small arcs HI, KL, be drawn through P; then, on account of the similar triangles HPI, KPL, those arcs will be proportional to the distances HP, LP; and any small elements of the spherical surface at HI and KL, each bounded all round by straight lines passing through P [and very nearly coinciding with HK], will be in the duplicate ratio of those lines. spherical traction on point. Hence the forces exercised by the matter of these elements Uniform on the particle P are equal; for they are as the quantities shell. Atof matter directly, and the squares of the distances, inversely; internal and these two ratios compounded give that of equality. The attractions therefore, being equal and opposite, balance one another and a similar proof shows that the attractions due to all parts of the whole spherical surface are balanced by contrary attractions. Hence the particle P is not urged in any direction by these attractions. on the divi faces into 463. The division of a spherical surface into infinitely small Digression elements will frequently occur in the investigations which sion of surfollow and Newton's method, described in the preceding de- elements. monstration, in which the division is effected in such a manner that all the parts may be taken together in pairs of opposite elements with reference to an internal point; besides other methods deduced from it, suitable to the special problems to be examined; will be repeatedly employed. The present digression, in which some definitions and elementary geometrical propositions regarding this subject are laid down, will simplify the subsequent demonstrations, both by enabling us, through the use of convenient terms, to avoid circumlocution, and by affording us convenient means of reference for elementary principles, regarding which repeated explanations might otherwise be necessary. tions and 464. If a straight line which constantly passes through a Explanafixed point be moved in any manner, it is said to describe, or definitions generate, a conical surface of which the fixed point is the cones. vertex. If the generating line be carried from a given position continuously through any series of positions, no two of which coincide, till it is brought back to the first, the entire line on the two sides of the fixed point will generate a complete conical surface, consisting of two sheets, which, are called vertical or opposite cones. Thus the elements HI and KL, described in Newton's demonstration given above, may be considered as being cut from the spherical surface by two opposite cones having P for their common vertex. regarding The solid angle of a 465. If any number of spheres be described from the vercone, or of tex of a cone as centre, the segments cut from the concentric a complete conical surface. Sum of all the solid angles spherical surfaces will be similar, and their areas will be as the squares of the radii. The quotient obtained by dividing the area of one of these segments by the square of the radius of the spherical surface from which it is cut, is taken as the measure of the solid angle of the cone. The segments of the same spherical surfaces made by the opposite cone, are respectively equal and similar to the former (but "perverted"). Hence the solid angles of two vertical or opposite cones are equal: either may be taken as the solid angle of the complete conical surface, of which the opposite cones are the two sheets. 466. Since the area of a spherical surface is equal to the square of its radius multiplied by 47, it follows that the sum of point=4. the solid angles of all the distinct cones which can be described with a given point as vertex, is equal to 4π. round a Sum of the solid angles of all the complete 467. The solid angles of vertical or opposite cones being equal, we may infer from what precedes that the sum of the conical sur- solid angles of all the complete conical surfaces which can be described without mutual intersection, with a given point as vertex, is equal to 2π. faces=2π. Solid angle subtended at a point by a 468. The solid angle subtended at a point by a superficial area of any kind, is the solid angle of the cone generated by a terminated straight line passing through the point, and carried entirely round the boundary of the area. surface. Orthogonal and oblique small cone. 469. A very small cone, that is, a cone such that any two sections of a positions of the generating line contain but a very small angle, is said to be cut at right angles, or orthogonally, by a spherical surface described from its vertex as centre, or by any surface, whether plane or curved, which touches the spherical surface at the part where the cone is cut by it. A very small cone is said to be cut obliquely, when the section is inclined at any finite angle to an orthogonal section; and this angle of inclination is called the obliquity of the section. The area of an orthogonal section of a very small cone is equal |