58. Show that the equation of the involute of the catenary which begins at the point where x = 0, y = c, is the Tractrix 59. If a straight line be drawn through the pole perpendicular to the radius vector of a point on the equiangular spiral r=ae cota to meet the corresponding tangent, show that the distance between the point of intersection and the point of contact of the tangent is equal to the arc of the curve measured from the pole to the point of contact. Hence prove that the locus of this point of intersection is one of the involutes of the spiral, and show that it is an equal equiangular spiral. 60. An equiangular spiral has contact of the second order with a given curve at a given point; prove that its pole lies on a certain circle, and that, if the contact be the closest possible, the distance of the pole from the point of contact is 61. If the tangent and normal to a curve at any point be taken as the axes of x and y respectively, and if s be the distance, measured along the arc, of a point very near to the origin, show that the Cartesian co-ordinates of that point are approximately x= 8 $2 y = the values of p, $3 6p2 х 84 dp 8p ds 83 dp dp d'p ds' and being those at the origin. 62. If a line be drawn parallel to the common tangent of a curve and its circle of curvature, and so near to it as to intercept on the curve a small arc of length s measured from the point of contact, of the first order of small quantities, show that the distance between the two points on the same side of the common normal in which the line cuts the curve and the circle i.e., is of the second order of small of curvature is 82 dp 6p ds' quantities, the values of p and dp ds being those at the point of contact; and again, if a line be drawn parallel to the common normal, the distance between the points of intersection with the curve and the circle is 83 dp 6p2 ds and is of the third order of small quantities. 63. Prove that the circle √2(x2 + y2+2)=3(x+y) has contact of the third order with the conic very near the origin the shape of the evolute is approximately given by 1225x3y2 = 16a3. 65. A line is drawn through the origin meeting the cardioide r = a(1 - cos 0) in the points P, Q, and the normals at P and Q meet in C. Show that the radii of curvature at P and Q are proportional to PC and QC. 66. If PQ be an are not containing a point of maximum or minimum curvature, the circles of curvature at P, Q will lie one entirely within the other. 67. If in the plane curve p(x, y) = 0, we have at any point branches of the curve which passes through that point is 68. If be the angle between the normal at any point P of a plane curve (x, y) =0, and the line drawn from P to the centre of the chord parallel and indefinitely near to the tangent at P, prove cos that bp2 - 2hpq + aq2 √p*+q* √' {(b2 +h2)p2 − 2 (a+b)hpq+(a2 +h2)p2}' дф дф 22& 724 a = h= = джду and b Əy2 69. A curve is such that any two corresponding points of its evolute and an involute are at a constant distance. Prove that the line joining the two points is also constant in direction. CHAPTER XI. ENVELOPES. 302. Families of Curves. If in the equation p(x, y, c)=0 we give any arbitrary numerical values to the constant c, we obtain a number of equations representing a certain family of curves; and any member of the family may be specified by the particular value assigned to the constant c. The quantity c, which is constant for the same curve but different for different curves, is called the parameter of the family. 303. Envelope. Definition. Let all the members of the family of curves p(x, y, c) = 0 be drawn which correspond to a system of infinitesimally close values of the parameter, supposed arranged in order of magnitude. We shall designate as consecutive curves any two curves which correspond to two consecutive values of c from the list. Then the locus of the ultimate points of intersection of consecutive members of this family of curves is called the ENVELOPE of the family. 304. The Envelope touches each of the Intersecting Members of the Family. It is easy to show that the envelope touches every B, C represent three curve of the system. For, let A, C P B A Fig. 65. Now, by definition, P and Q are points on the envelope. Thus the curve B and the envelope have two contiguous points common, and therefore have ultimately a common tangent, and therefore touch each other. Similarly, the envelope may be shown to touch any other curve of the system. 305. To find the Equation of an Envelope. To find the equation of the envelope of the family of curves of which p(x, y, c)=0 is the typical equation. Let p(x, y, c)=0, 1 } p(x, y, c+dc)=0, Š be two consecutive members of the family. the latter we have Ә p(x, y, c)+dc_~_p(x, y, c) + ... = 0. .(A) Expanding Hence in the limit, when dc is infinitesimally small, we obtain a Sep(x, y, c) = 0 as the equation of a curve passing through the ultimate point of intersection of the curves (A). If we eliminate c between the equations. |