# Calendar: 1899-1900 Page 759

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FACULTY OF SCIENCE CVll Through point in one edge of tetrahedron are drawn two planes whose intersections with the tetrahedron are parallelograms Shew that they cut the opposite edge at points equidistant from its extremities If ax2 4- by2 cz2 42 fyz 49 ZX 42 can be made perfect square for all values of ζ by the addition of px qy rz where are numerical coefficients prove that ρ2 q2 r2 bg2 42 gr 2chpq fo State and prove the exponential theorem and shew that it always leads to convergent series Hence prove that the limit of when Λ is log tf Prove that if θ be the circular measure of an angle sm --- and write down the 72th term of the series Expand θ sin -f cos powers of sin θ as far as sin 40 straight line is drawn from the origin to intersect 2x and χ 42 From each of these points of intersection perpendicular is drawn to the other line Find the locus of the intersection of these perpendiculars Transform y2 4tax x2 4ay 16 to polar co- ordinates the origin being at the pole and the initial line at an angle of 45 to the axis and solve the resulting equation for Obtain an expression for the length of the tangent drawn from fixed point to circle whose equation is given circle is drawn through fixed point so that the length of tangent drawn to it from another fixed point is constant find the locus of its centre Two ellipses have the same foci are points on one P' Q' on the other such that Ρ and P1 have the same eccentric angle and likewise and Q' prove that PQ' P'Q Also if Ρ and are conjugate prove that the difference between the squares on P'Q' and PQ is equal to half the dif- ference between the squares on the major axes State clearly the meaning of differential coefficient Β

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