EXAMPLES.

Er. 1. To find the centre of gravity of the surface of the spherical segment DSE (fig. 196.)

Put SC=r, CD=y, the arc SD=z, and the radius VS=r; then, by the property of the circle, (see art. 46, Vince's Fluxions) ż: ::r: y; therefore, ży=rx; and rx being substituted for yż in the above general formula, we have _fxyż__ frxż fyż frx becomes a hemisphere, and the distance SO'r: that is, the centre of gravity of an orb, of evanescent thickness, is in the middle of the radius.

1. When a becomes r, the segment

Ex. 2. To find the centre of gravity of the surface of the right cone VBC (fig. 197.)

Put VS-a, BS=b, VI=x, and EI=y; then, by similar

bx

b

triangles, a : b ; : x : y===nx (putting -=n); therefore,

a

n2 x x + xx n2x+x

this being multiplied by nx2=xy, for y=nx),

; and

VE or z=√n2x2+x2, and ¿=·

√n2x2+x2 √n2+1

the fluent is n2 x + x

Xnx

√n2+1

which must be divided by the fluent of

2n2+1

; and we have VO=r, or, when a=a,

then, VO={a}VS.

The centres of gravity of the surface of a cylinder, of a cone, and of a conic frustum, are respectively at the same distances from the less ends, as the centres of gravity of the parallelogram, triangle, and trapezoid; which are vertical sections of those solids.

659. In this proposition, &c. the surfaces of the respective. bodies are supposed to be of evanescent thickness; when they are not so, the centre of gravity of the shell may be found by

art. 543; viz. by substituting the annular section at the distance of a from the vertex, in the general formula Axx, instead of the whole circular section A.

D=

CENTROBARYC METHOD.

660. The centrobaryc method consists in a very simple procedure, by which we can determine the surface and solidity of a body generated by the revolution of any curve about its axis; the equation of the curve, and the centre of gravity of the generating line or area being given."

The formula which are requisite for this purpose have been investigated in the preceding propositions; thus, if D denote the distance of the centre of gravity of the revolving line from

_fyż the axis of revolution, then, by art. 657, D= is the

distance of the centre of gravity of that line from the axis about which it revolves; and if the numerator and denominator

be multiplied equally by 2p, we shall have D-2pyż

2pz

This equation, by reduction, gives 2pDz=f2pyż; where 2pD is the circumference of which D is the radius, and is that which is described by the centre of gravity of the curve by revolving about its axis. The other quantity, that is f2pyż, is the expression for the area of the surface generated by the curve z during its rotation. Consequently, the surface generated by the rotation of any given curve about an axis, is equal to the product of the generating arc, into the circumference described by its centre of gravity.

In the same manner, by art. 655, D=~

fy
2fy

-denotes the

distance of the centre of gravity of any plane from the revolving

axis, and which, by thus revolving, generates a solid the capacity or content of which is required. Multiply both terms of the fraction by p, and the above formula becomes

Now, by reduction, pyx=f2pDyx; where Spyx is the solid generated by the area yx, and 2pD is the circumference described by the centre of gravity. Hence, it is plain that the solid generated by the rotation of any plane figure about an axis, has for its capacity, the product of the generating area into the circumference described by its centre of gravity. If, in the above formulæ, instead of p we introduce any

1

fraction np,

it will be equally obvious that the same method is applicable to the curve surfaces or capacities of figures generated by a partial revolution about a fixed axis.

From what has been observed, it is plain, that if any two of the three dimensions which enter into the computation, that is, the generating quantity, its centre of gravity and the quantity generated, be given, the third may always be found. EXAMPLES,

Ex. 1. To find the content of the solid generated by a parallelogram, whose sides are 6 and 4, revolving about its longer side as an axis.

Put m=6 and n=4; now the centre of gravity of the rectangle is at the distance of n from the revolving axe, and will describe, in a complete revolution, the circumference nx2p=np; and the area of the generating plane is mn; therefore, by the latter formula, mn×np=mn2p=301.5929, is the content of the solid.

Ex. 2. The base of a semi-parabola is b=4, and its axis a=6; to find the content of the paraboloid generated by a complete revolution of this curve about its axis.

The distance of the centre of gravity of the semi-parabola from the revolving axe is 3b, and 3b×2p=1bp, is the circumference described by that centre; also fab is the area of the generating plane; therefore, bp×}ab=}ab2p=1×6×43× 3.141593-150.79646, is the content of the paraboloid.

Ex. 3. Let a semicircle, whose radius is r and area ḍpr2, by revolving about its diameter generate a sphere the content of which is pr3; to find the distance of the centre of gravity of the semicircle from its centre.

In this case, pr3÷pr2=3r, is the circumference described

4r

by the centre of gravity; and fr÷2p==0.42441r is the

3p

distance of the centre of gravity of the semicircle from its centre (see ex. 4, art. 655.)

Ex. 4. Let the circle ADBE (fig. 198.) revolve about the line XY as an axis, at the distance BH; it is required to determine the content of the solid generated by the circle in making a complete revolution about XY,

Put the radius CB=r, and the distance CH=d; then, because C is the centre of gravity of the generating plane, the circumference described by that centre will be 2pd; also pr2 is the area of the circle ADBE; therefore, 2pdxpr2=2p2dr2 is the solidity, as required.

For more on this subject, see Stones' Dictionary, Hutton's Mensuration, or his Mathematical and Philosophical Dictionary, &c.

On variable motions in general.

661. When bodies in motion are acted upon by different forces in different successive times, their motions are variable.

This subject may be illustrated by the motion of a body falling towards the earth: thus, since the force of gravity is different at different distances from the earth's centre, it is plain that the falling body is acted upon by different forces, or at least by different degrees of the same force, at different distances from the earth's centre, and also in different successive instants of time. Several other circumstances might be adduced, by way of illustration, but the above is plain enough, and sufficient.

In the motion of bodies, treated of in the former part of this work, (art. 268. et sequel) the force by which the bodies were supposed to be urged, was constant; and in the case of bodies falling near the earth's surface, the force of gravity by which they are urged, may be esteemed as such; for the difference, in very small distances, is insensible.

The principles, necessary for determining the several circumstances relative to variable motions, are easily deduced from those already given, where the force was constant.

Thus let f, t, v, and s, be put to denote any standard force, time, velocity and space; and let F, T, V, and S, denote any other quantities of the same kind, compared respectively with the former; also put, as usual, g=16 feet.

V ft

and because in the small

1. Now, by art 268, element of time t, the force f may be looked upon as constant, let & be the velocity generated in that time, by the force ƒ;

then,

i f i

V

x or v∞ft; when V, F and T are each=1.

T'