Calendar: 1860-1861 Page 479

480 GENERAL LITERATURE AND SCIENCE Prove that if un ex xY -J 12Un η tl Un-2- Show that the greatest rectangle that can be drawn with its sides passing through the angles of the square ABCD is another square whose side is equal to the diagonal AD Draw an asymptote to the curve defined by the equation yl -x 3axy find also the maximum ordinate and show that the portion of the curve which lies in the first quadrant is circumscribed by square whose area a2 ζ curve is convex or concave to the axis of according as and x2 have the same or different signs Ex y3 x2 Prove the equation in spirals ρ -f- Ex Find the equation to the Lemniscata to rectangular coordinates deduce the polar equation and by the preceding formula find that between ρ and 10 Prove that in any curve to rectangular coordinates ds 2ע -ךv -0 and find when dx x' χ α Θ sin ff cos Θ 11 Find the radius of curvature at the origin of the curve x3 x2 and if ma curve dx y2 ί2 ay2 -ךi-s- then radius of curvature y1 i2 12 Integrate the differentials χ tan χ χ jSi' 21 2' tan he ea sin χ 13 Find the equation to and area of the cissoid 7r sin d6 between the limits of β and ad xx χ between the limits of and