EXAMPLES m 1. If I denote the inclination of two adjacent faces of a regular polyhedron, shew that cos I= } in the tetrahedron, = 0 in the cube, =- } in the octahedron, =-15 in the dodecahedron, and=-}5 in the icosahedron. 2. With the notation of Art. 153, shew that the radius of the sphere which touches one face of a regular polyhedron and all the adjacent faces produced is a cotcot 1 1. 3. A sphere touches one face of a regular tetrahedron and the other three faces produced : find its radius. 4. If a and b are the radii of the spheres inscribed in and described about a regular tetrahedron, shew that b= 3a. 5. If a is the radius of a sphere inscribed in a regular tetrahedron, and R the radius of the sphere which touches the edges, shew that R2 = 3a". 6. If a is the radius of a sphere inscribed in a regular tetrahedron, and R' the radius of the sphere which touches one face and the others produced, shew that R' = 2a. 7. If a cube and an octahedron be described about a given sphere, the sphere described about these polyhedrons will be the same; and conversely. 8. If a dodecahedron and an icosahedron be described about a given sphere, the sphere described about these polyhedrons will be the same; and conversely. 9. A regular tetrahedron and a regular octahedron are inscribed in the same sphere : compare the radii of the spheres which can be inscribed in the two solids. 10. The sum of the squares of the four diagonals of a parallelepiped is equal to four times the sum of the squares of the edges. 11. If with all the angular points of any parallelepiped as centres equal spheres be described, the sum of the intercepted portions of the parallelepiped will be equal in volume to one of the spheres. 12. A regular octahedron is inscribed in a cube so that the corners of the octahedron are at the centres of the faces of the cube: shew that the volume of the cube is six times that of the octahedron. 13. It is not possible to fill any given space with a number of regular polyhedrons of the same kind, except cubes; but this may be done by means of tetrahedrons and octahedrons which have equal faces, by using twice as many of the former as of the latter, 1 3 14. A spherical triangle is formed on the surface of a sphere of radius p; its angular points are joined, forming thus a pyramid with the straight lines joining them with the centre: shew that the volume of the pyramid is špo *p J(tan r tanr, tanr, tan r.), where r, ?, ?, ?, are the radii of the inscribed and escribed circles of the triangle. 15. The angular points of a regular tetrahedron inscribed in a sphere of radius r being taken as poles, four equal small circles of the sphere are described, so that each circle touches the other three. Shew that the area of the surface bounded by 3 16. If O be any point within a spherical triangle ABC, the product of the sines of any two sides and the sine of the included angle sin AO sin BO sin CO cot AO sin BOC vo {cot XIV, ARCS DRAWN TO FIXED POINTS ON THE SURFACE OF A SPHERE. 164. In the present Chapter we shall demonstrate various propositions relating to the arcs drawn from any point on the surface of a sphere to certain fixed points on the surface. 165. ABC is a spherical triangle having all its sides quadrants, and therefore all its angles right angles; T is any point on the surface of the sphere: to shew that cos' TA + cos2 TB + cos TC = 1. Similarly cos TC= sin TB cos TBC = sin TB sin TBA. Square and add ; thus cos' TA + cos' TC = sin TB=1 - cos' TB; therefore cos' TA + cos TB + cos’TC=1. 166. ABC is a spherical triangle having all its sides quadrants, and therefore all its angles right angles; Tand U are any points on the surface of the sphere: to shew that cos TU=cos TA cos UA + cos TB cos UB + cos TC cos UC. cos TU By Art. 37 we have = cos TA cos UA + sin T'A sin UA cos TAU, and cos TAU=cos (BAU – BAT) = cos BAU cos BAT + sin BAU sin BAT = cos BAU cos BAT + cos CAU cos CAT; therefore = cos TA cos UA + sin TA sin UA (cos BAU cos BAT + cos CAU cos CAT'); and cos TB = sin TA cos BAT, cos UC = sin UA cos CAU; therefore cos TU: = cos TA cos UA + cos TB cos UB + cos TC cos UC. 167. We leave to the student the exercise of shewing that the formulæ of the two preceding Articles are perfectly general for all positions of T and U, outside or inside the triangle ABC : the demonstrations will remain essentially the same for all modifications of the diagrams. The formulæ are of constant application in Analytical Geometry of three dimensions, and are demonstrated in works on that subject; we have given them here as they may be of service in Spherical Trigonometry, and will in fact now be used in obtaining some important results. 1 168. Let there be any number of fixed points on the surface of a sphere; denote them by H., H., H..... Let T be any point on the surface of the sphere. We shall now investigate an expression for the sum of the cosines of the arcs which join T with the fixed points. Denote the sum by E; so that == = cos TH, + cos TH, + cos THz + ... Take on the surface of the sphere a fixed spherical triangle ABC, having all its sides quadrants, and therefore all its angles right angles. Let is My v be the cosines of the arcs which join T with A, B, C respectively; let l, m, n, be the cosines of the arcs which join with A, B, C respectively; and let a similar notation be used with respect to H,, H .... Then, by Art. 166, =l,+ m, + n,v + led + moje + nyv + .. =PX + Qu + Rv; where P stands for 1, +1, +1, + ..., with corresponding meanings for Q and R. 22 + 169. It will be seen that P is the value which takes when T coincides with A, that Q is the value which takes when I coincides with B, and that R is the value which takes when T coincides with C. Hence the result expresses the general value of in terms of the cosines of the arcs which join T to the fixed points A, B, C, and the particular values of which correspond to these three points. |