tude is EF.

And that this may be equal to the whole

convex surface we must have Σ(b⋅ SV) = 0.

But as all the elements have the same width, b is constant, and SV: SE is constant;

.. (ES)=0;

or, CD passes through the centre of figure of the upper base; and hence AB passes through the centre of figure of the lower base.

Therefore, the area of the convex surface of a cylindroid is the circumference of a right section multiplied by the distance between the centres of figure of the

bases.

B

175. Let the plane figure X, invariable in form and dimensions, move with centre of figure on the path OPQR, and so that the direction of the path is at all points normal to the plane of the figure, and let GH and JK be two near positions, which at the limit come into coincidence.

X

HK

C

A

The surface of the elementary cylindroid GK is the circumference of Xx PQ; and the area of the surface of the figure generated by the motion of X is

(PQ) × circum. of X.

But (PQ) is the path OPQR., and circumference of X is constant.

Therefore, the area of the surface described by a plane figure, invariable in form and magnitude, which moves so that its direction of motion is at each point normal to

its plane, is the circumference of the generating figure multiplied by the length of path described by its centre of figure.

Cor. When a plane figure revolves about a complanar line as axis, the direction of motion is necessarily normal to the plane of the figure, and the surface described has for its area the circumference of the figure multiplied by the circumference traced by its centre of figure.

Ex. To find the surface of an anchor ring. The centre of figure is the centre of the generating circle, and the circumference traced is 2πR.

.. area of surface = 2πr · 2πR=4π2Rr.

176. The two theorems which go under the name of Guldin's theorems, but which were discovered by Pappus, express relations of the highest importance in mathematics both pure and applied. They enable us to find the centroid of a generating figure when the volume of the generated figure is known, or the centre of figure of a generating figure when the area of the surface of the generated figure is known, and vice versa.

Thus knowing the volume of a sphere, we can readily find the centroid of a semicircle, and knowing the surface of a sphere enables us to find the centre of figure of a semicircular arc.

EXERCISES M.

1. The circle describing an anchor ring is divided by a diameter parallel to the axis. Show that the difference between the surfaces described by the outer and the inner part is eight times the area of the generating circle.

2. The convex surface of a cone is πrs; and the entire surface is r(r+s); where s is the slant height, and r is the radius of the base.

3. The entire surface of a conical frustum is

π {s (r + r1) + p2 + r22}.

4. The areas of the surfaces of the regular polyhedra are as follows:

Tetrahedron, e2√3; Cube, 6e2; Octahedron, e22√3; Dodecahedron, e215√(1+ √5); Icosahedron, e25 √3.

5. The distance between the centre of a circle and the centre 2 cr of figure of any arc of the circle is where is the length of

the arc, and c and r denote as usual.

6. The area of the surface of a circular spindle is

7. The convex surface of an ungula of a right circular cylinder is

PART IV.

PROJECTIONS AND SECTIONS.

SECTION 1.

PERSPECTIVE PROJECTION.

177. Def. Let P be a variable point on a plane figure X; let O be a fixed point not complanar with X; and let L be the line OP.

When P describes the figure X, L describes the perspective projection of X in space, or the spatial projection of X, and O is the centre of the projection.

If the spatial projection be cut by a plane V, the figure of section is a plane figure called the perspective projection of X on V for the centre O.

When O goes to infinity, L has a fixed direction, and is parallel to a fixed line for all positions of P, and the projection becomes parallel projection.

If the direction of O at infinity is normal to V, the projection on V becomes orthogonal or orthographic projection, which is thus a special case of perspective projection.

If the direction of O at infinity is perpendicular to

the plane of X, but oblique to V, the projection may be called the ant-orthogonal projection of X on V.

In what follows in this section, projection will mean projection with O finite unless otherwise stated, and the projection will mean the figure of section.

178. We observe that in a way projection and section are reciprocal processes, as by projecting a plane figure we get a spatial one, and by cutting the spatial one by a plane we return to a plane figure.

And this passing from one plane figure to another through a spatial figure may be repeated as often as we please.

179. Since the generator L is unlimited, the spatial figure extends to infinity on both sides of the centre, and admits of section on either side or section on both sides by the same plane, examples of which will occur hereafter.

180. The following theorems are fundamental:

1. A line projects into a line.

For the spatial projection of a line is a plane, and every plane section of a plane is a line.

2. The point of intersection of two lines projects into the point of intersection of the projections of the lines. Hence the projection of a plane rectilinear figure is a plane rectilinear figure having the same number of sides and vertices as the projected figure.

3. A curve projects into a curve, and a tangent to the curve into a tangent to its projection.