Calendar: 1859-1860 Page 447

448 GENERAL LITERATURE AND SCIENCE 16 No real value of J1 lies between and 2xl 2x 11 17 If Sn be the sum of η terms of the geometric series first term and common ratio show that Sn Sn-l- rSn-2 18 Sum the series- ar a6 r2 ab ab2 &c to infinity 1' &c to terms 19 are points arranged in straight line and heing fixed and the distances in harmonic pro- gression prove that if move towards that will also move towards 20 Find xyz when 17χ 23y 3ζ 200 II £ucliU To bisect given finite straight line that is to divide it into two equal parts When the same straight line bisects the angle and opposite side of triangle this triangle is isosceles prove this If one side of triangle be produced the exterior angle is greater than either of the interior opposite angles Define parallelogram The opposite sides and angles of parallelograms are equal to one another and the diameter bisects them that is divides them into two equal parts The quadrilateral figure whose diameters bisect each other is parallelogram prove this In any right-angled triangle the square which is de- scribed upon the side subtending the right angle is equal to the squares described upon the sides which contain the right angle In the figure of the above proposition corresponding diagonals of the smallest squares are in the same direction If straight line be divided into any two parts the square of the whole line is equal to the squares of the two parts together with twice the rectangle contained by the parts