Calendar: 1899-1900 Page 770

CXV111 FACULTY OF SCIENCE Shew that dn dfc r2 a2 where wr lh 17 Ira If is developable in series proceeding by powers of determine the coefficients Develope tan tan2 as far as Shew that the maximum triangles which can be in- scribed in an ellipse are the projections of equilateral triangles inscribed in circle Shew how the maximum triangle inscribed in any oval curve may be found If curve is given by χ cos nt c' cos n't sin nt c' sin n't shew that the radius vector is perpendicular to the tangent at angular intervals of 2π η η' Sketch the curve for 2c' 71' 3n Evaluate the integrals dx dx ex -f a2 dx χ β If cos Θ cos φ shew that e2 'n οοβφ -1 cos Hence evaluate a2 cos2 -j- sin2 Prove that dx יי Shew that the arithmetic mean of the harmonic means between and when their number is indefinitely great is 10ge2 Shew that the area swept out by any given straight line in small displacement is r8s9 where is the length of the line and 8s the displacement of the middle of the line per- pendicular to the length Describe Amsler's Planimeter and prove its theory