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then R,or becomes Adu, and the equation of the equilibrium of a particle m, on a curved line or surface, is

Xdx + Ydy + Zdz + λdu = 0


where du is a function of the elements dx, dy, dz: and as this equation exists whatever these elements may be, each of them may be made zero, which will divide it into three equations; but they will be reduced to two by the elimination of X. And these two, with the equation of the surface u = 0, will suffice to determine x, y, z, the co-ordinates of m in its position of equilibrium. These found, N and consequently λ become known. And since R, is the resistance

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is the pressure, which is equal and contrary to the resistance, and is therefore determined.

50. Thus if a particle of matter, either free or obliged to remain on a curved line or surface, be urged by any number of forces, it will continue in equilibrio, if the sum of the products of each force by the element of its direction be zero.

Virtual Velocities.

51. This principle, discovered by John Bernouilli, and called the principle of virtual velocities, is perfectly general, and may be expressed thus:

If a particle of matter be arbitrarily moved from its position through an indefinitely small space, so that it always remains on the curve or surface, which it ought to follow, if not entirely free, the sum of the forces which urge it, each multiplied by the element of its direction, will be zero in the case of equilibrium.

On this general law of equilibrium, the whole theory of statics depends.



fig. 13.



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52. An idea of what virtual velocity is, may be formed by supposing that a particle of matter m is urged in the direction mA by a force applied to m. If m be arbitrarily moved to any place n indefinitely near to m, then mn will be the virtual velocity of m.

53. Let na be drawn at right angles to mA, then ma is the virtual velocity of m resolved in the direction of the force mA: it is also the projection of mn on mA; for

mn: ma :: 1: cos nma and mamn cos nma.

54. Again, imagine a polygon ABCDM of any number of sides, either in the same plane or not, and suppose the sides MA, AB, &c., fig. 14.



a c m

to represent, both in magnitude and direction, any forces applied to a particle at M. Let these forces be resolved in the direction of the axis or, so that ma, ab, bc, &c. may be the projections

of the sides of the polygon, or the cosines of the angles made by the sides of the polygon with or to the several radii MA, AB, &c., then will the segments ma, ab, bc, &c. of the axis represent the resolved portions of the forces estimated in that single direction, and calling a, ẞ, y, &c. the angles above mentioned,

ma= MA cos a; ab AB cos ẞ; and be BC cos y,

&c. and the sum of these partial forces will be

MA cos a + AB cos ß + 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.


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.


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

fig. 15.




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 possible. 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 required, among the infinite number of curves that can be drawn between these two points, to deter

fig. 16.






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 Sds 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 8.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.



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

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