# Calendar: 1875-1876 Page 541

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FHEAKE EXHIBITIONS 537 12 If HL be two consecutive convergents to the value of 'ןן continued fraction prove that g' p'q Express Λ4 as continued fraction and find the first four convergents to its value 13 Define prime י number and show that the number of primes is infinite Enunciate Fermat's theorem and prove that the 11th power of number is of one of the forms 23 or 23 14 Prove by geometrical constructions sin2 2sin sin sin cosB cos sini -f- cos 15 Prove that tan Λ tan2 £ --- tan2 Λ Find tan in terms 0ftan2 and explain the ambiguity of sign Apply your formula to find tan 15 16 If are the angles of triangle the expression sin cotA ----- smB sin will retain the same value if any two of the quantities are interchanged 17 Find an expression for the area of circle Two circles of equal radii intersect so that the common chord is equal to either radius find the area common to both circles 18 Show how to find the height and distance of an inaccessible object on horizontal plane person wishing to ascertain his distance from an inaccessible object finds three points in the horizontal plane at which the angular elevation of the summit of the object is the same Show how the distance may be found 19 Frove De Moine's theorem for fractional and negative indices Apply it to find the values of 20 Explain the advantages of the Napierian and of the common system of logrithms Reduce the negative logarithm 8753145

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