This integral is to be taken from y = 0,

which makes ≈ = 0; or from y = 0, to y

to that value of y

b

=

Va2x2; and

· (a2 x − } x2 + C).

This integral is to be taken from x = 0, to x = a; and therefore

Again, to find the value of faЛy (xx) we observe that

which, taken between the same limits as before, viz. x = 0,

Ex. 2. To find the centre of gravity of a portion of a paraboloid, comprehended between two planes passing through its axis at right angles to each other.

If a be its length, and b the radius of its base, the coordinates of its centre of gravity will be

179. To find the centre of gravity of a surface of revo

lution.

Employing the notation and figure of Art. 177, let u be the moment of the surface generated by the arc AP, and therefore du the moment of that generated by PQ; let S denote the former, and SS the latter of these surfaces so generated. Then the moment of SS about Oy is greater than if it were all collected in the circumference of the circle described by P, and less than if collected in the circumference of that described by Q, that is,

Su is greater than x. SS, and less than (x + dx). SS;

u = the moment of S about Oy=xS= x. 2π [(yds) ;

..π.2πf(yds) = 2π f(xyds) ;

·· xf(yds) = f(xyds).

And it is evident, from the symmetrical form of the surface, that = 0.

y

Ex. 1. To find the centre of gravity of the surface of a cone.

If a be the altitude and b the radius of the base of the cone, the equation of the line by which the surface is generated is

which, taken between the limits = 0, and a = a, gives

[(yds) = ba2 + b2.

Ex. 2. To find the centre of gravity of the surface generated by the revolution of an arc of a circle about a diameter.

The centre of gravity bisects the axis of the zone.

Ex. 3. To find the centre of gravity of the surface generated by the revolution of a semi-cycloid about its axis,

Ex. 4. To find the centre of gravity of the surface of a paraboloid.

Taking the focus as origin of co-ordinates, we find the distance of the centre of gravity from the directrix

Ex. 5. To find the centre of gravity of the surface generated by the revolution of a node of the Lemniscate.

180. To find the centre of gravity of a surface of any form.

If, in Art. 178, we use A to denote the elementary surface PS instead of the prism Ps, we shall have

the limit of

A

Sx Sy

=

√1 + (d ̧ð)2 + (d,≈)2;

and by proceeding exactly as in that Article, we shall find x • Sx Sy√ 1 + (d ̧ ≈)2 + (d,≈)2= fxfy {x\/1 + (d ̧≈)2 + (d ̧≈)2}, ÿ • SxSy√1 + (d ̧~)2 + (dy≈)2= SxSy {Y√1 + (d ̧≈)2 + (d„≈)2}, ≈ • SxSy√1 + (d ̧ñ)2 + (d,≈)2= fxfy {≈√/1+ (d ̧~)2 + (d ̧≈)2}.

181. To find the centre of gravity of a curve of double

curvature.

If we use S for the length of the curve line, and SS for the length of a very small portion of it, we shall have the

d2S′ = √1 + (d2y)2 + (d ̧≈)2, and it will be

xS = √2x√/1 + (d2y)2 + (d„≈)3,

7S = Sy√1+ (d ̧y)2 + (d ̧≈)*,

zS = √2≈√/1 + (d ̧y)2 + (d ̧≈)2.

182. We shall now add a few examples of finding the centre of gravity when the density is variable. Questions of this kind depend upon the formulæ of Art. 138, viz.—

183. To find the centre of gravity of a physical line, the density of which at any point varies as the nth power of its distance from a given point in the line produced.