and this is the expression for the cosine of the angle of a plane triangle in terms of the sides.

Again, in Art. 110, from the formula

that is, in a plane triangle the sides are as the sines of the opposite angles.

EXAMPLES.

1. In a spherical triangle, if C and c remain constant while a and b receive the small increments da and Sb respectively, shew

2. If C and c remain constant, and a small change be made in a, find the consequent changes in the other parts of the triangle. Find also the change in the area.

3. Supposing A and c to remain constant, prove the following equations, connecting the small variations of pairs of the other elements,

4. Supposing b and c to remain constant, prove the following equations connecting the small variations of pairs of the other elements,

5. Supposing B and C to remain constant, prove the following equations connecting the small variations of pairs of the

duce the formula for the area of a plane triangle, namely

7. If A and C are constant, and b be increased by a small quantity, shew that a will be increased or diminished according as c is less or greater than a quadrant.

8. Two spherical triangles ABC, abc, equal in all respects, differ slightly in position; shew that

cos ABb cos BCc cos CAa + cos ACc cos CBb cos BA a

=

0.

9. What formulæ in Plane Trigonometry are deducible from Napier's Analogies; and what from Gauss's Theorems?

deduce the area of a plane triangle in terms of the sides and one

of the angles.

11. What result is obtained from example 7 to Chap. VI. by supposing the radius of the sphere infinite?

T. S. T.

H

XII. POLYHEDRONS.

132. A polyhedron is a solid bounded by any number of plane rectilineal figures which are called its faces. A polyhedron is said to be regular when its faces are similar and equal regular polygons, and its solid angles equal to one another.

133. If S be the number of solid angles in any polyhedron, F the number of its faces, E the number of its edges, then S+FE+ 2.

Take any point within the polyhedron as centre, and describe a sphere of radius r, and draw lines from the centre to each of the angular points of the polyhedron; let the points at which these lines meet the surface of the sphere be joined by arcs of great circles, so that the surface of the sphere is divided into as many polygons as the polyhedron has faces.

Let s denote the sum of the angles of any one of these polygons, m the number of its sides; then the area of the polygon is 2 {8-(m-2) π} by Art. 99. The sum of the areas of all the polygons is the surface of the sphere, that is, 42. Hence since the number of the polygons is F, we obtain

Απ = Σ8 - Σm + 2Γπ.

Now Σs denotes the sum of all the angles of the polygons, and is therefore equal to 27 × the number of solid angles, that is, to 2TS; and Σm is equal to the number of all the sides of all the polygons, that is, to 2E, since every edge gives rise to an arc which is common to two polygons. Therefore

therefore

4π = 2πS - 2πE + 2Fπ ;

S+ F = E + 2.

134. There can be only five regular polyhedrons.

Let m be the number of sides in each face of a regular polyhedron, n the number of plane angles in each solid angle; then the entire number of plane angles is expressed by mF, or nS, or 2E; thus

mF= nS = 2E, and S+F=E + 2 ;

from these equations we obtain

These expressions must be positive integers, we must therefore have 2 (m + n) greater than mn; therefore

1

n

and therefore must be greater than ; and as m must be an

m

integer and cannot be less than 3, the only admissible values of m are 3, 4, 5. It will be found on trial that the only values of m and n which satisfy all the necessary conditions are the following; each regular polyhedron derives its name from the number of its plane faces.

It will be seen that the demonstration establishes something more than the enunciation states; for it is not assumed that the faces are equilateral and equiangular and all equal.

It is in fact

demonstrated that, there cannot be more than five solids each of which has all its faces with the same number of sides, and all its solid angles formed with the same number of plane angles.

135.

The sum of all the plane angles which form the solid angles of any polyhedron is 2 (S′ − 2) π.

For if m denote the number of sides in any face of the polyhedron, the sum of the interior angles of that face is (m − 2) – by Euclid 1. 32, Cor. 1. Hence the sum of all the interior angles of all the faces is (m-2) π, that is mπ-2Fπ, that is

2 (E-F), that is 2 (S-2) π.

136. To find the inclination of two adjacent faces of a regular polyhedron.

Let AB be the edge common to the two adjacent faces, C and D the centres of the faces; bisect AB at E, and join CE and DE; CE and DE will be perpendicular to AB, and the angle CED is the angle of inclination of the two adjacent faces; we shall denote it by I. In the plane containing CE and DE draw CO and DO at right angles to CE and DE respectively, and meeting at 0: about O as centre describe a sphere meeting OA, OC, OE at a, c, e respectively, so that cae forms a spherical triangle. Since AB is perpendicular to CE and DE, it is perpendicular to the plane CED, therefore the plane AOB which contains AB is perpendicular to the plane CED; hence the angle cea of the spherical triangle is a right angle. Let m be the number of sides in each face of the