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GENERAL LITERATURE AND SCIENCE 553 10 right pyramid with square base stands on horizontal plane and when the sun has an altitude and the distance of the extreme point of its shadow from the angular points of the base nearest to it are Show that the height of the pyramid is tan sin φ cosec where and φ are given by the equations- b"- 72 tan 45- β -J c' 11 Prove the following relations between the parts of spherical triangle cosa cosA cose -f- sin sin cos י Β sin --- sm sin -ψ- cos 12 Define the polar triangle of given spherical triangle and establish the fundamental relation between the pair If any transversal DEF be drawn cutting the sides of spherical triangle ABC in prove that sin BD sin CE ainAF amBF sin CD sin AF and reciprocate this property into the corresponding one of tbe polar triangle 13 Find the area of spherical triangle It be the right angle prove that the spherical excess sin sin tan-' --ך-1 Vcose coso 14 If be the number of solid angles in any polyhedron tbe number of faces Ε the number of edges then S- -F Show that there are only five regular polyhedrons 15 If from an external point chord OAB be drawn cutting small circle in and and if OC be the tangent to the same circle from t&n OA tan 0B tan-£ OC VI -2h1alutical 0fo1nttrp AC are two given straight lines in which points Ρ and are taken so that AP AB AQ QC show that PQ passes through fixed point
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