x

Ex. 4. If y be the co-ordinates of the centre of gravity of the area cut off from a parabola (y2 = 4mx) by a focal chord inclined to the axis at an angle a,

Ex. 5. To find the centre of gravity of the area of the quadrant of a circle, whose equation is x2 + y2 = a2

Ex. 6. To find the centre of gravity of the node of the lemniscate, whose equation is a cos 20.

177. To find the centre of gravity of a solid of revolution.

Let AB (fig. 33) be the curve by the revolution of which round Or the given solid is generated. Make the same construction and notation as before. Let V denote the volume of the solid generated by the revolution of AMP, and that generated by PMNQ; u the moment of V round Oy, and Su that of 8V about Oy.

The moment of V about Oy is greater than it would be if V were all collected in the circular plane generated by PM, that is,

Su> i. SV;

and it is less than it would be if SV were all collected in the circular plane generated by QN, that is,

But, y being the co-ordinates of the centre of gravity of V,

Now dVyda, by the Differential Calculus; and, therefore, Vfy'da; consequently

Ex. 1. To find the centre of gravity of a hemisphere. A hemisphere is generated by the revolution of a quadrant whose equation is

which gives, for the whole hemisphere, by writing a for x,

Ex. 2.

Given the altitude (c) and the radii (a, b) of

the ends of a parabolic frustum, to find its centre of

being measured along the axis from the smaller end whose radius is a.

Ex. 3. In a cone, generated by the revolution of a right-angled triangle about one of its sides,

Ex. 4. In the solid formed by the revolution of a semi-cycloid about its axis,

Ex. 5.

In the paraboloid, formed by the revolution of the parabola, whose equation is y"+"a"x",

x =

m + 3n 00

m + 2n 2

178. To find the centre of gravity of a solid of any form.

Let Ox, Oy, Oz (fig. 36) be the rectangular co-ordinates to which the solid is referred by its equation. Let ABPC be a portion of the surface of the solid, comprehended between the co-ordinate planes Ox, yox, and the planes PpNC, PpMB respectively parallel to them. Through the point S very near to P draw planes Ssnc, Ssmb parallel to the former. Let xyz be the co-ordinates of P, and x + dx, y + dy, ≈ + dx those of S. Then, denoting the volume of the parallelopiped Ps by A, its moment about the axis Ox is greater than if it were all collected in the plane Pq, and less than if collected in the plane Rs; that is, the moment of A is

greater than y▲,

and less than (y+dy) A.

But now if u be the moment of the solid PO about Or, the moment of SBm Pn about Or will be (by Taylor's theorem applied to two variables x, y).

and by the same theorem, applied to the variable a, the moment of the solid BmP about Ox is

du. S x + 1 d2 u. (8x)2 + ...

and, similarly, the moment of the solid Cn P, is du. Sy + du. (dy)2 + ...

Subtracting both these from the former, we find the moment of the parallelopiped Ps to be equal to dedu. Sady +...; consequently, this quantity always lies between y and (y+dy) A; and, therefore, dd,u+ always lies between

...

διδυ

tends to ≈ as its limit, and consequently the

equality with yz; and dd,u + ... which always lies between them, tends to dd, u as its limit; the three limits. are therefore equal; consequently,

dxdyu = yz;

... U = Safy (y).

Now the volume of PO is equal to ffy, and its moment about Ox is

By a similar investigation, we should find

SxSy≈ = Sef y (≈≈).........(1).

And observing that the centre of gravity of the parallelopiped A is ultimately in its middle point, we should find

z⋅ Sx Sy≈ = & Sx Sy (≈2)......................... (3).

REMARK. It is evident, that by taking an elementary parallelopiped, at right angles to the plane Oz, we might

also obtain

and if the elementary parallelopiped were at right angles to the plane yox, we should find

These formulæ are in fact, often more convenient than those first given; and which are the most convenient in a given example is to be determined by the form of the body and its situation with respect to the co-ordinate planes; the choice must, however, be left to the skill of the reader, as no general rule can be laid down. In every case, the greatest care is requisite to take the integrals between proper limits.

All the three sets of formulæ are comprehended in the following:

which may be readily investigated after the manner of Art. 175.

Ex. 1. To find the centre of gravity of the eighth part of an ellipsoid.

The equation of the surface of the ellipsoid is