Calendar: 1848-1849 Page 259

APPLIED SCIENCES 257 14 If ί when its value may in general be found by the diff calculus isa u- -x log 1-x when κ 15 The chord arc and tangent are ultimately equal Lemma 1G Find the locus of the vertices of the parabolas which have given focus and which touch given line 17 weight Ρ is attached to the extremity of the string CP which passes through ring at the end of string BC find position of equilibrium and tension of and Β being in horizontal line 18 Find the equation to the uniform catenary and the co-ordinates of the centre of gravity of the semi-arc 19 uniform rod rests with one end against vertical line and the other on curve find the equation to the curve that the rod may rest in any position 20 Integrate sin θ -3 e- x2 'l esin 20 II $L1tI cmnttc5' Euclid Book xi prop The difference between the second and fourth powers of prime number is divisible by 12 The sum of the squares of the diagonals of parallelo- pipedon the sum of the squares of the 12 edges In the expansion of if η be integral every co-efficient is integral The sides of triangle are in arithmetical proportion and the distance between the centres of the inscribed and circum- scribed circles is mean proportional between the greatest and least find the sides If Ρ be the sum of the series formed by taking the 1st and every plh term of an infinite geometric series first term unity the sum by taking the 1st and every q'h term prove that pq £ QP