Calendar: 1873-1874 Page 544

544 GENERAL LITERATURE AND SCIENCE If η be positive integer show that ο r' χ Given and show that esiny sin du du fdu MS £ wind is blowing directly off straight shore find what course vessel should take so as to reach the shore soonest if the speed vary as cos β being the angle between the course of the vessel and the direction of the wind 10 Prove the following properties of plane curve -- where ss dP ds normal ν dx Show that the normal is the longest or shortest straight line that can be drawn from given point to the curve 11 Find the area of loop of the lemniscate r5 a2 cos 20 If circles be described on its radii vectores show that their envelope is the curve cos φ and trace it 12 The evolute of curve may be regarded as the en- velope of consecutive normals From this definition deduce the general formulae for determin- ing the evolute Find the evolute of the curve and trace it