If, therefore, the body be balanced at any two points of its surface, and verticals be drawn through the point, in these positions, their intersection will be the centre of gravity of the body.

It follows, from what has just been explained, that when a body is suspended by an axis, it can only come to a state of rest when the centre of gravity lies in a vertical plane passed through the axis.

The centre of gravity may lie above the axis, below the axis, or on the axis.

In the first case, if the body be slightly deranged, it will continue to revolve till the centre of gravity falls below the axis; in the second case, it will return to its primitive position; in the third case, it will remain in the position in which it is placed. These cases will be again referred to, under the head of Stability.

The preceding rules enable us to find the centres of gravity of all lines, surfaces, and solids; but, on account of the difficulty of applying them in certain cases, we shall annex an outline of some of the methods, by the Differential and Integral Calculus. Those magnitudes whose centres of gravity are most readily found by the calculus, are mathematical curves; areas bounded wholly, or in part, by these curves; curved surfaces; and volumes bound by curved surfaces.

Determination of the Centre of Gravity by means of the Calculus.

64. To place Formulas (13) under a suitable form for the application of the calculus, we have simply to substitute for the forces P, P, &c., the elementary volumes, or the differentials of the magnitudes, and to replace the sign of summation, Σ, by that of integration, s.

Making these changes, Formulas (13) become,

In which dm denotes the differential of the magnitude in question; x, y, and z, the co-ordinates of its centre of grav

ity, and x, y, and z1, the co-ordinates of the centre of gravity of the magnitude.

Application to plane curves.

65. The plane XY may be taken to coincide with that of the curve, in which case, z = 0 for every point of the curve; and, consequently, z1 = 0; dm becomes the differential of an arc of a plane curve, or dm = √de + dy3. Substituting in (32), we have,

66. Let ABC be the arc, O the origin of co-ordinates and centre of the circle, OX the axis of abscissas, perpendicular to the chord of the arc, and OY the axis of ordinates. Since the arc is symmetrically situated with respect to the axis of X, the centre of gravity is somewhere on this line (Art. 54); consequently, y1 = 0. To find 1, we have the equation of the circle,

Differentiating,

C

Fig. 42.

x2 + y2 = p2.

y

dy.

Substituting in the first of Formulas (33), and reducing,

That is, the centre of gravity of an arc of a circle is on the diameter which bisects its chord, and its distance from the centre is a fourth proportional to the arc, chord, and radius.

Application to Plane areas.

67. Let the plane XY be taken to coincide with that of the area. We shall have, as before, z1 = 0. In this case, we have dm = ydx; and, consequently, Formulas (32), reduce to

Sxydx
Sydx

Sy❜dx
Sydx

Centre of Gravity of a parabolic area.

68. Let AOB represent the area, having its chord at right angles to the axis. Let be the origin of co-ordinates, taken at the vertex, and let the axis of X coincide with the axis of the curve; the value of y, will, as before, be equal to 0. To find the value of x1, we have the equation of the parabola,

a

Fig. 43.

By substitution in the first of Formulas (34), we have,

That is, the centre of gravity of a segment of a parabola is on its axis, and at a distance from the vertex equal to three-fifths of the altitude of the segment.

Application to solids of revolution.

69. If we take the planes XY and XZ passing through the axis of revolution, the centre of gravity will lie in both these planes, therefore y, and z, will both be 0. In this case, the first of Formulas (32) will be sufficient.

Since the axis of X coincides with the axis of revolution, dm becomes equal to y'da. Substituting in the first of Formulas (32), we have,

Centre of Gravity of a semi-ellipsoid.

70. Let the semi-ellipse ACB, be revolved about the axis OC; it will generate a semi-ellipsoid whose axis coincides with the axis of X. being 0, it only remains to find the value

Both Yı

and 21

of 21.

(35.)

a

0

B

Fig. 44

The equation of the ellipse referred to its centre, is,

in which a and b are the semi-axes.

Substituting, in Equation (35), we have,

Integrating between the limits, x = 0, and x = a, we

That is, the centre of gravity of a semi-prolate spheroid of revolution is on its axis of revolution, and at a distance from the centre equal to three-sixteenths of the major axis of the generating ellipse.

The examples above given are enough to indicate the method of applying the calculus to the determination of the centre of gravity.

Centre of Gravity of a system of bodies.

71. When we have several bodies, and it is required to find their common centre of gravity, it will, in general, be found most convenient to employ the principle of moments.