Calendar: 1873-1874 Page 553

GENERAL LITERATURE AND SCIENCE 553 13 Assuming that when η is odd sin η φ can be expressed as rational integral function of sin φ show that sin י -£- infin Find the value of ad infin 26 36 46 14 Prove that in any spherical triangle cos cos cos sin sin cos and deduce the value of tan Find the lengths of the arcs of great circles drawn through ABC perpendicular to the opposite sides 15 Prove that tan C0Sfff- tan- cosi And if are in arithmetical progression show that sin sin cos--sin 16 Explain Napier's rules for the solution of right-angled spherical triangles and prove the rules for the cases in which the complement of the hypotenuse is the middle part If be right angle and δ be the perpendicular drawn from to the hypotenuse AB show that cot δ Vcut2 cot24 17 Find the angular radius of the small circle inscribed in spherical triangle in terms of the sides Show that the angular radius of the circle described round the polar triangle is the complement of the angular radius of the small circle inscribed in the primitive triangle Hence find the radius of the small circle circumscribing spherical triangle in terms of the angles 18 Show that the area of sperical triangle is Er2 when χ C-n