mined accordingly. We may therefore obtain the equation to an individual of the family of curves denoted by the equation (1) which shall have a contact of the (n - 1)th order with any proposed curve at any point (x, y), by assigning to the quantities (3) the values of the corresponding quantities in the proposed curve, and obtaining the values of the n parameters accordingly. Thus suppose that u' = F(x', y', a,,a1⁄2, αz,.... α„) and that f(x) is the ordinate in the proposed curve at the point of contact: then v1, 2, 3,.... v, denoting certain functions of x, we shall have

will represent an individual of the family of curves represented by the equation (1), which shall have a contact of the (n - 1)th order with the proposed curve at the point (x, y).

Ex. 1. The general equation to a circle is

(x' − a)2 + (y' - ß)2 = p2,

a, ß, p, being disposable constants, upon the particular magnitudes of which the dimensions and position of the circle depend.

Let it be proposed to determine the values of a, ß, p, that the circle may have a contact of the second order with any proposed curve y′′ = 4 (x") at a point (x, y) of the curve.

Differentiating the equation to the circle twice with regard to

dy"

dx"2

dx2

are supposed to denote the values of dy" and

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,

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) of the curve,

(x' − a)2 + (y' – B)2 = p2

(1).

2

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,

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

Between the two equations (3) and (4) we may eliminate the parameter a, and we shall thus obtain an equation