&c. and the sum of these partial forces will be MA cos a + AB cos B+ BC cos y + &c. = 0 by the general property of polygons, as will also be evident if we consider that dm, ma, ab lying towards o are to be taken positively, and bc, cd lying towards a negatively; and the latter making up the same whole bd as the former, their sums must be zero. Thus it is evident, that if any number of forces urge a particle of matter, the sum of these forces when estimated in any given direction, must be zero when the particle is in equilibrio; and vice versâ, when this condition holds, the equilibrium will take place. Hence, we see that a point will rest, if urged by forces represented by the sides of a polygon, taken in order. In this case also, the sum of the virtual velocities is zero; for, if M be removed from its place through an infinitely small space in any direction, since the position of or is arbitrary, it may represent that direction, and ma, ab, bc, cd, dm, will therefore represent the virtual velocities of M in directions of the several forces, whose sum, as above shown, is zero. 55. The principle of virtual velocities is the same, whether we consider a material particle, a body, or a system of bodies. Variations. 56. The symbol is appropriated to the calculus of variations, whose general object is to subject to analytical investigation the changes which quantities undergo when the relations which connect them are altered, and when the functions which are the objects of discussion undergo a change of form, and pass into other functions by the gradual variation of some of their elements, which had previously been regarded as constant. In this point of view, variations are only differentials on another hypothesis of constancy and variability, and are therefore subject to all the laws of the differential calculus. fig. 15. m 57. The variation of a function may be illustrated by problems of maxima and minima, of which there are two kinds, one not subject to the law of variations, and another that is. In the former case, the quantity whose maxi- M mum or minimum is required N C depends by known relations on some arbitrary independent variable; -for example, in a given curve MN, fig. 15, it is required to determine the point in which the ordinate p m is the greatest pos-➡ sible. In this case, the curve, or function expressing the curve, remains the same; but in the other case, the form of the function whose maximum or minimum is required, is variable; for, let M, N, fig. 16, be any two given points in space, and suppose it were fig. 16. M ds A N required, among the infinite number of curves that can be drawn between these two points, to deter mine that whose length is a minimum. If ds be the element of the curve, fds is the curve itself; now as the required curve must be a minimum, the variation of fds when made equal to zero, will give that curve, for when quantities are at their maxima or minima, their increments are zero. Thus the form of the function fds varies so as to fulfil the conditions of the problem, that is to say, in place of retaining its general form, it takes the form of that particular curve, subject to the conditions required. 58. It is evident from the nature of variations, that the variation of a quantity is independent of its differential, so that we may take the differential of a variation as d.dy, or the variation of a differential as .dy, and that d.dy.dy. 59. From what has been said, it appears that virtual velocities are real variations; for if a body be moving on a curve, the virtual velocity may be assumed either to be on the curve or not on the curve; it is consequently independent of the law by which the co-ordinates of the curve vary, unless when we choose to subject it to that law. CHAPTER II. VARIABLE MOTION. 60. WHEN the velocity of a moving body changes, the cause of that change is called an accelerating or retarding force; and when the increase or diminution of the velocity is uniform, its cause is called a continued, or uniformly accelerating or retarding force, the increments of space which would be described in a given time with the initial velocities being always equally increased or diminished. Gravitation is a uniformly accelerating force, for at the earth's surface a stone falls 16 feet nearly, during the first second of its motion, 48 during the second, 80 during the third, &c., falling every second 32 feet more than during the preceding second. 61. The action of a continued force is uninterrupted, so that the velocity is either gradually increased or diminished; but to facilitate mathematical investigation it is assumed to act by repeated impulses, separated by indefinitely small intervals of time, so that a particle of matter moving by the action of a continued force is assumed to describe indefinitely small but unequal spaces with a uniform motion, in indefinitely small and equal intervals of time. 62. In this hypothesis, whatever has been demonstrated regarding uniform motion is equally applicable to motion uniformly varied; and X, Y, Z, which have hitherto represented the components of an impulsive force, may now represent the components of a force acting uniformly. Central Force. 63. If the direction of the force be always the same, the motion will be in a straight line; but where the direction of a continued force is perpetually varying it will cause the particle to describe a curved line. Demonstration.-Suppose a particle impelled in the direction mA, fig. 17, and at the same time attracted by a continued force whose origin is in o, the force being supposed to act impulsively at equal successive infinitely small times. By the first impulse alone, in any given time the particle would move equably to A: but in the same time the action of the continued, or as it must now be considered the impulsive force alone, would cause it to move uniformly through ma; hence at the end of that time the particle would be found in fig. 17. B B, having described the diagonal mB. Were the particle now left to itself, it would move uniformly to C in the next equal interval of time; but the action of the second impulse of the attractive force would bring it equably to b in the same time. Thus at the end of the second interval it would be found in D, having described the diagonal BD, In this manner the particle would describe the polygon mBDE; but if the intervals between the successive impulses of the attractive force be indefinitely small, the diagonals mB, BD, DE, &c., will also be indefinitely small, and will coincide with the curve passing through the points m, B, D, E, &c. and so on. in 64. In this hypothesis, no error can arise from assuming that the particle describes the sides of a polygon with a uniform motion; for the polygon, when the number of its sides is indefinitely multiplied, coincides entirely with the curve. 65. The lines mA, BC, &c., fig. 17, are tangents to the curve in the points, m, B, &c.; it therefore follows that when a particle is moving in a curved line in consequence of any continued force, if the force should cease to act at any instant, the particle would move on in the tangent with an equable motion, and with a velocity equal to what it had acquired when the force ceased to act. 67. We shall consider the element or differential of time to be a constant quantity; the element of space to be the indefinitely small space moved over in an element of time, and the element of velocity to be the velocity that a particle would acquire, if acted on by a constant force during an element of time. Thus, if t, s and v be the time, space, and velocity, the elements of these quantities are dt, ds, and dv; and as each element is supposed to express an arbitrary unit of its kind, these heterogeneous quantities become capable of comparison. As a decrement only differs from an increment by its sign, any expressions regarding increasing quantities will apply to those that decrease by changing the signs of the differentials; and thus the theory of retarded motion is included in that of accelerated motion. 68. In uniformly accelerated motion, the force at any instant is directly proportional to the second element of the space, and inversely as the square of the element of the time. Demonstration.-Because in uniformly accelerated motion, the velocity is only assumed to be constant for an indefinitely small time, v= and as the element of the time is constant, the dif ds ferential of the velocity is dv= ing for an indefinitely small velocity, Fdt dv; hence F d's dt time, produces an indefinitely small des ; but since a constant force, act dt General Equations of the Motions of a Particle of Matter. 69. The general equation of the motion of a particle of matter, when acted on by any forces whatever, may be reduced to depend on the law of equilibrium. Demonstration.-Let m be a particle of matter perfectly free to obey any forces X, Y, Z, urging it in the direction of three rectangular co-ordinates x, y, z. Then regarding velocity as an effect of force, and as its measure, by the laws of motion these forces will produce in the instant dt, the velocities Xdt, Ydt, Zdt, proportional to the intensities of these forces, and in their directions. Hence when m is free, by article 68, for the forces X, Y, Z, being perpendicular to each other, each one is independent of the action of the other two, and may be regarded as |