dy" dx' at the point (x, y) of the curve, we shall have for the determination of a, ß, p, the three equations (x − a)2 + (y − ẞ)2 = p2, 1 + (y − ẞ) dx d'y dy2 + dx2 dx2 0, = 0. Now the equations (1) and (2) of Art. (136), supposing de to be zero, or x to be the independent variable, coincide with the last two of these equations. We see therefore that the coordinates of the centre of the circle, which has a contact of the second order with any proposed curve at any proposed point, coincide with those of the centre of the osculating circle at the same point; or that the osculating circle is identical with the circle which has a contact of the second order. Ex. 2. To determine the parabola which has a contact of the second order with an ellipse at an extremity of the latus rectum of the ellipse; the equation to the ellipse being and the axis of the parabola being parallel to the major axis of the ellipse. Let the equation to the parabola be Now the coordinates of an extremity of the latus rectum of the When the Radius of Curvature is a Maximum or Minimum, the Contact is of the third order. 144. Let the equation to the curve be y" = p (x''), and the equation to the circle of curvature, at a point (x, y) 906 2 log p = 3 log (1 + p'2) - 2 log q', and therefore, differentiating again, p being invariable as we pass from one point of the circle to another, Again, putting p, q, r, for the values of the point (x, y), we know that 3 log (1 + p2) - 2 log q. Now, by the nature of the contact between a curve and its circle of curvature, the values of p', q', at the point (x, y), are P, q; hence, by (2) and (3), we see that the values of r' at the point (x, y) is the same as that of r. Thus the contact between the circle and the curve must be of the third order. CHAPTER VIII. ENVELOPS. Case of a single Parameter. 145. LET the equation to a family of curves be x', y', being the coordinates of any point in any one of the curves, and a being a parameter, the particular values of which determine the individual curves of the family. Suppose that a becomes a + da, da being an indefinitely small increment of a. Then the equation (1) becomes Let x, y, be the values of x', y', at the intersection of the curves (1) and (2); that is, of any two consecutive individuals of the family of curves. Then and therefore, when Sa is diminished without limit, we have ultimately df ( x, y, a) Between the two equations (3) and (4) we may eliminate the parameter a, and we shall thus obtain an equation expressing the relation between the coordinates of the point of intersection of any and every two consecutive individuals of the family of curves: this equation will therefore represent a curve, which is the locus of such consecutive intersections. It is easy to see that the curve (5) touches each of the individuals of the family (1). In fact, differentiating the equation (1), we get Again, since, by virtue of (3) and (4), a as well as y is a function of x, we have, differentiating (3), hence, from (6) and (7), we see that the ratio of dy' to de' is the same in the curve (1) as that of dy to dx in the curve (5) at their common point (x, y). The locus of the consecutive intersections of the individuals of a family of curves has been called their envelop in consequence of this property. Ex. 1. To find the nature of the curve which shall touch all the curves represented by the equation y = ax – a3, whatever be the value of a. Differentiating with regard to a, we have |