L; then the circle described through the points T, L, and touching the conic at T, is the circle of curvature.

This construction shows that the osculating circle at either vertex has a contact of the third degree.

Ex. 1. Using the notation of the eccentric angle, find the condition that four points a, ß, y, d should lie on the same circle (Joachimstal, Crelle, xxxvi. 95).

The chord joining two of them must make the same angle with one side of the axis as the chord joining the other two does with the other; and the chords being

we have tan (a + B) + tan} (y + d) = 0 ; a + ß+y+d=0; or =

=2mπ.

Ex. 2. Find the co-ordinates of the point where the osculating circle meets the conic

Ex. 3. There are three points on a conic whose osculating circles pass through a given point on the curve; these lie on a circle passing through the point, and form a triangle of which the centre of the curve is the intersection of bisectors of sides (Steiner, Crelle, xxxii. 300; Joachimstal, Crelle, xxxvi. 95).

Here we are given d, the point where the circle meets the curve again, and from the last Example the point of contact is a = - But since the sine and cosine of d would

δ

3

δ

not alter if d were increased by 360°, we might also have a=-= +120°, or =—. 3

+240o, and from Ex. 1, these three points lie on a circle passing through d. If in the last Example we suppose XY given, since the cubics which determine x' and y' want the second terms, the sums of the three values of x and, of y are respectively equal to cipher: and therefore (Ex. 4, p. 5) the origin is the intersection of the bisectors of sides of the triangle formed by the three points. It is easy to see that the normals at these points are the three perpendiculars of this triangle, and therefore that they meet in a point. 248. To find the radius of curvature of the parabola. The equation, referred to any diameter and tangent (y2 = px), is transferred to the tangent and normal by the same substitution as in Art. 246, and we find

The construction, therefore, used in the last Article, applies also to the case of the parabola.

Ex. 1. In all the conic sections the radius of curvature is equal to the cube of the normal divided by the square of the semi-parameter.

Ex. 2. Express the radius of curvature of an ellipse in terms of the angle which the normal makes with the axis.

Ex. 3. Find the lengths of the chords of the circle of curvature which pass through the centre or the focus of a central conic section.

26'2

Ans.

and

26'2 a

a''

Ex. 4. The focal chord of curvature of any conic is equal to a focal chord of the conic drawn parallel to the tangent at the point.

Ex. 5. In the parabola the focal chord of curvature is equal to the parameter of the diameter passing through the point.

249. To find the co-ordinates of the centre of curvature of a

central conic.

These are evidently found by subtracting from the co-ordinates of the point on the conic the projections of the radius of curvature upon each axis. Now it is plain that this radius is to its projection on y as the normal to the ordinate y. We find the projection, therefore, of the radius of curvature on the axis of

We should have got the same values by making a = in Ex. 7, Art. 236.

B = Y

Or again, the centre of the circle circumscribing a triangle is the intersection of perpendiculars to the sides at their middle points; and when the triangle is formed by three consecutive points on a curve, two sides are consecutive tangents to the curve, and the perpendiculars to them are the corresponding normals, and the centre of curvature of any curve is the intersection of two consecutive normals. Now if we make x' x" = X,

=

y = y' = Y, in Ex. 4, p. 161, we obtain again the same values as those just determined.

250. To find the co-ordinates of the centre of curvature of a parabola.

The projection of the radius on the axis of y is found in like manner by multiplying the radius of curvature

The same values may be found from Ex. 9, p. 194.

251. The evolute of a curve is the locus of the centres of curvature of its different points. If it were required to find the evolute of a central conic, we should solve for x'y' in terms of the x and y of the centre of curvature, and, substituting in the equa

tion of the curve, should have (writing

c?

a

=

c2
A, = B),
Ђ

In like manner the equation of the evolute of a parabola is found to be

27py2 = 16 (x - p)3,

which represents a curve called the semicubical parabola.

*CHAPTER XIV.

METHODS OF ABRIDGED NOTATION.

252. We have proved (Art. 15) that we obtain an equation of the mnth degree to determine the co-ordinates of the points of intersection of two curves of the mth and nth degrees; and since an equation of the mnth degree has always mn roots, real or imaginary, we infer (as in Art. 69) that a curve of the mth degree will always intersect a curve of the nth degree in mn points, real or imaginary. Two conic sections, therefore, S = 0, S' = 0, always intersect each other in four points, real or imaginary;

and (Art. 36) S+kS' = 0 is the equation of another conic through these four points of intersection.

253. This will, of course, still be true if either or both the quantities S, S' be resolvable into factors. Thus, let S' be resolvable into factors, and represent the pair of right lines a, ẞ; then S+kaẞ = 0, which is evidently satisfied by the co-ordinates of the points where either a or ẞ meets S, will represent a conic passing through the four points where S is met by this pair of right lines. It is, therefore, the equation of a conic having a and B for its chords of intersection with S. If either a or ẞ do not meet S in real points, it must still be considered as a chord of imaginary intersection, and will preserve many important properties in relation to the two curves, as we have already seen in the case of the circle (Art. 111).

If both S and S' break up into factors, the equation ay + kßd = 0 represents the conic circumscribing the quadrilateral (aßyd), as we have already seen, p. 97.

It is obvious that in what precedes a need not denote a line. whose equation has been reduced to the form x cos a + y sin a = p, but that S LM = 0 (see convention, Art. 52) will in like manner represent a conic passing through the points where L and M meet S, &c.

Ex. 1. What is the equation of a conic passing through the points where a given conic S meets the axes?

=

Here the axes x = 0, y 0 are the chords of intersection, and the equation must be of the form S + kxy = 0, where ✯ is indeterminate. Compare Ex. 1, p. 137.

Ex. 2. Find the equation of the conic passing through five given points. Having formed the equations of a, ß, y, d, the sides of the quadrilateral formed by four of the given points, we know that the equation must be of the form ay kßd; and, substituting in this equation the co-ordinates of the fifth point, we are able to determine k.

=

Ex. 3. Form the equation of the conic which passes through the points (1, 2), (3, 5) (− 1, 4), (− 3, − 1), (− 4, 3).

Considering the quadrilateral formed by the first four points, we see that the equation must be of the form

(3x - 2y + 1) (5x − 2y + 13) = k (x − 4y + 17) (3x − 4y + 5). Substituting in this the co-ordinates - 4, 3, which must satisfy it, we obtain k

Substituting this value, and reducing the equation, it becomes

221

19

254. We have seen that the equation S+ kaß = 0 represents a conic passing through the

and Q to q. Suppose then that the lines a and ẞ coincide, then the points P, p; Q, q coincide, and the second conic will touch the first at the points P, Q. We learn then that the equation Ska2 = 0 represents a conic having double contact with S, and whose chord of contact is a. In like manner ay + kß2 = 0 represents a conic, to which a and y are tangents, while ẞ is their chord of contact, as we have already seen (Art. 104). Similarly S+L2 = 0 represents a conic having double contact with S, L being the chord of contact; and LN M2 denotes a conic to which L and N are tangents, while M is their chord of contact.

=

If the line a were a tangent to S,the two points P and Q would coincide, and the conic S + ka2 would have four consecutive points common with S, and would therefore have with it a contact of the third degree. Thus, for instance, we have seen (Art. 244) that the equations of two conics which have contact of the third order at a point on the axis of x are of the form

S = 0 and S+ ky2 = 0.

255. The forms given in the preceding articles receive important modifications, if any of the lines which they involve be at an infinite distance. It was proved (Art. 64) that when a line is removed to an infinite distance, its equation is reduced to the constant term. If, then, in any of the preceding equations, we substitute a constant for any of the quantities a, ß, &c., we shall have the form which that equation will assume when the line a, B, &c., is at an infinite distance.

=

Thus we know that the lines L, N touch the conic LN = M2 at the points where they meet M; if, then, we substitute for M a constant m, we see that the conic LN m2 is touched by the lines L, N at the points at infinity on those lines: in other words, that the lines L, N are asymptotes to this conic. If we suppose