And any number of forces acting on the particle m may [be resolved in the same manner, whatever their directions may be. If be employed to denote the sum of any number of finite quantities, represented by the same general symbol

is the sum of the partial forces urging the particle parallel to the axis

ox.

Likewise Σ.F.

; Σ.Γ

Fr
δε

; are the sums of the par

tial forces that urge the particle parallel to the axis oy and oz. Now if F, be the resulting force of all the forces F, F', F", &c. that act on the particle m, and if u be the straight line drawn from the origin of the resulting force to m, by what precedes

are the expressions of the resulting force F,, resolved in directions. parallel to the three co-ordinates; hence

or if the sums of the component forces parallel to the axis x, y, z, be represented by X, Y, Z, we shall have

F,

δι

δε

2.

F( ) = X; F, () = Y; F. (H) = z.

бу

If the first of these be multiplied by Sr, the second by dy, and the third by Sz, their sum will be

FSu X&x + Ydy + Zdz.

40. If the intensity of the force can be expressed in terms of the distance of its point of application from its origin, X, Y, and Z may be eliminated from this equation, and the resulting force will then be given in functions of the distance only. All the forces in nature are functions of the distance, gravity for example, which varies inversely as the square of the distance of its origin from the point of its application. Were that not the case, the preceding equation could be of

no use.

41. When the particle is in equilibrio, the resulting force is zero; consequently

Xộc + Yông + Zôi = 0

(3),

which is the general equation of the equilibrium of a free particle.

42. Thus, when a particle of matter urged by any forces whatever remains in equilibrio, the sum of the products of each force by the element of its direction is zero. As the equation is true, whatever be the values of dx, dy, dz, it is equivalent to the three partial equations in the direction of the axes of the co-ordinates, that is to X=0, Y = 0, Z = 0,

for it is evident that if the resulting force be zero, its component forces must also be zero.

On Pressure.

43. A pressure is a force opposed by another force, so that no motion takes place.

44. Equal and proportionate pressures are such as are produced by forces which would generate equal and proportionate motions in equal times.

45. Two contrary pressures will balance each other, when the motions which the forces would separately produce in contrary directions are equal; and one pressure will counterbalance two others, when it would produce a motion equal and contrary to the resultant of the motions which would be produced by the other forces.

46. It results from the comparison of motions, that if a body remain at rest, by means of three pressures, they must have the same ratio to one another, as the sides of a triangle parallel to the directions.

x and y being the co-ordinates of m, a and b those of N. If the point m be on a surface, or curve of double curvature, in which no two of its elements are in the same plane, then,

mN = √(x − a)2 + (y − b)2 + (≈ — c)2

x, y, z being the co-ordinates of m, and a, b, c those of N. The

centre of curvature N, which is the intersection of two consecutive normals mN, m'N, never varies in the circle and sphere, because the curvature is every where the same; but in all other curves and surfaces the position of N changes with every point in the curve or surface, and a, b, c, are only constant from one point to another. By this property, the equation of the radius of curvature is formed from the equation of the curve, or surface. If r be the radius of curvature, it is evident, that though it may vary from one point to another, it is constant for any one point m where dr = 0.

Equilibrium of a Particle on a curved Surface.

48. The equation (3) is sufficient for the equilibrium of a particle of matter, if it be free to move in any direction; but if it be constrained to remain on a curved surface, the resulting force of all the forces acting upon it must be perpendicular to the surface, otherwise it would slide along it; but as by experience it is found that re-action is equal and contrary to action, the perpendicular force will be resisted by the re-action of the surface, so that the re-action is equal, and contrary to the force destroyed; hence if R, be the resistance of the surface, the equation of equilibrium will be

Sr, dy, Szare arbitrary; these variations may therefore be assumed to take place in the direction of the curved surface on which the particle moves: then by the property of the normal, dr = 0; which reduces the preceding equation to

Xây + Tây + Z = 0.

But this equation is no longer equivalent to three equations, but to two only, since one of the elements dx, dy, dz, must be eliminated by the equation of the surface.

49. The same result may be obtained in another way. For if uO be the equation of the surface, then du = 0; but as the equation of the normal is derived from that of the surface, the equation Sr0 is connected with the preceding, so that dr Ndu. But

r = √(x− a)2 + (y−b)* + (z−c)2

whence

consequently,

2

{(c)*+ (^~^)*+ ((£)"} = 1,

on account of which, the equation

Nou gives N2

Νδι gives N3 {(d)*+ (Su)*+ (-)'} =

1,

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

Xdx + Ydy + Zdz + λdu = 0

(4),

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 N. 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

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.

a

m

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.

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, B,, &c. the angles above mentioned,

ma

MA cos a; ab AB cos ß; and be BC cos y,