Calendar: 1850-1851 Page 315

GENERAL LITERATURE AND SCIENCE 311 If be the length of curve defined by χ then ן y2 and deduce from it the formula for polar ux dx2 co-ordinates Find the length of the semi-cubical parabola ay2 1x and of Archimedes' spiral αθ body revolves in the circumference of circle round an eccentric point find the law of force If Ρ be drawn parallel to the base of the triangle ABC and Β Ρ be joined intersecting in find the locus of as Ρ moves from Β to χ dn If χ prove strictly that -j- and if χ tan φ cos tan Θ prove that φ Θ tan 2- sin 2Θ tan 4" sin θ &c 10 semi-cycloid revolves round its axis find the centre of gravity of the surface generated 11 An uniform beam rests with one end against rough wall its other being supported by string attached to fixed point in the wall determine the position of the beam its pressure against the wall and the tension of the string 12 Find the relation between the power and the weight in the screw 13 parabola rolls on straight line find the locus of its focus χ 14 Find --- being odd from χ to χ a2 x2 cos θ άθ and -- יcos ϊ 15 Describe the greatest isosceles triangle about given ellipse 16 If Ρ and QK be normals drawn from the extremities of two conjugate diameters to meet the axis major the sum of their squares is constant 17 Trace the curve χ 10 χ χ and find its area 18 If and be the foci and Ρ point in an ellipse find the locus of the centre of the circle touching the sides of SΗ