# Calendar: 1875-1876 Page 578

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574 GENERAL LITERATURE AND SCIENCE opposite sides ab -f ac -f be Assuming De Moivre's Theorem for positive integral values of the index shew that it is true for negative and fractional values of the same Express sin" θ in terms of sines of multiples of Θ when η is an odd positive integer Obtain the three values of Shew that tan-' χ χ x3 -f &c and that li χ ץ V2 2נ T3 Find an expression for the cosine of an angle of spherical triangle in terms of the sides no side being greater than two right angles Prove that if two sides be supplementary the opposite angles are also supplementary or differ by an odd multiple of two right angles 10 Any great circle is drawn cutting the sides of an equilateral quadrantal tiiangle in angles β φ ψ shew that cos θ cos φ cos2 ψ 11 An arc of great circle AB of constant length β moves with its extremities on two fixed great circles OA OB at right angles to each other The locus of Ρ the foot of the perpen- dicular from on Β is given by sin sin 2Θ tan cos where OP and POA Θ 12 If AOD BOB COF be arcs of great circles drawn from the vertices perpendicular to the sides of any triangle ABC prove that tan AO tan OD tan BO tan OB tan CO tan OB cos OA cos cos OB cos cos OC cose 13 If point on sphere be defined by its distance φ from fixed point and the inclination θ of that distance to fixed

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