Ex. 2. Every diameter of a conic bisects the area enclosed by the curve.

For if we suppose a number of ordinates drawn to this diameter, since the diameter bisects them all, it also bisects the trapezium formed by joining the extremities of any two adjacent ordinates, and by supposing the number of these trapezia increased without limit, we see that the diameter bisects the area.

395. Ex. 1. The area of the sector of a hyperbola made by joining any two points of it to the centre, is equal to the area of the segment made by drawing parallels from them to the asymptotes.

For since the triangle PKC=QLC, the area PQC=PQKL.
Ex. 2. Any two segments PQLK, RSNM, are equal, if
PK: QL:: RM: SN.

and therefore QR is parallel to PS. We can now easily prove that the sectors PCQ, RCS are equal, since the diameter bisecting PS, QR will bisect both the hyperbolic area PQRS, and also the triangles PCS, QCR.

If we suppose the points Q, R to coincide, we see that we can bisect any area PKNS by drawing an ordinate QL, a geometric mean between the ordinates at its extremities.

Again, if a number of ordinates be taken, forming a continued geometric progression, the area between any two is constant.

396. The tangent to the interior of two similar, similarly placed, and concentric conics cuts off a constant area from the

exterior conic.

For we proved (Art. 236, Ex. 4) that this tangent is always bisected at the point of contact; now if we draw any two tangents, the angle AQA' will be equal to BQB' and the nearer we suppose the point Q to P, the more nearly will the sides AQ, A'Q approach to equality with the sides BQ, B'Q; if, therefore, the two

P P

B

B

tangents be taken indefinitely near, the triangle AQA' will be equal to BQB, and the space AVB will be equal to A'VB'; since, therefore, this space remains constant as we pass from any tangent to the consecutive tangent, it will be constant whatever tangent we draw.

COR. It can be proved, in like manner, that if a tangent to one curve always cuts off a constant area from another, it will be bisected at the point of contact; and, conversely, that if it be always bisected it cuts off a constant area.

Hence we can draw through a given point a line to cut off from a given conic the minimum area. If it were required to cut off a given area, it would be only necessary to draw a tangent through the point to some similar and concentric conic, and the greater the given area, the greater will be the distance between the two conics. The area will, therefore, evidently be least when this last conic passes through the given point; and since the tangent at the point must be bisected, the line through a given point which cuts off the minimum area is bisected at that point.

In like manner, the chord drawn through a given point which cuts off the minimum or maximum area from any curve is bisected at that point. In like manner can be proved the following two theorems, due to the late Professor Mac Cullagh.

Ex. 1. If a tangent AB to one curve cut off a constant arc from another, it is divided at the point of contact, so that AP : PB inversely as the tangents to the outer curve at A and B.

Ex. 2. If the tangent AB be of a constant length, and if the perpendicular let fall on AB from the intersection of the tangents at A and B meet AB in M, then AP will = MB.

397. To find the radius of curvature at any point on an ellipse. The centre of the circle circumscribing any triangle is the intersection of perpendiculars erected at the middle points of the sides of that triangle; it follows, therefore, that the centre of the circle passing through three consecutive points on the curve is the intersection of two consecutive normals to the curve.

Now, given any two triangles FPF", FP'F', and PN, P'N, the two bisectors of their vertical angles, it is easily proved by elementary geometry, that twice the angle PNP'= PFP+ PF'P. (See figure, Art. 392, Ex. 1).

Now, since the arc of any circle is proportional to the angle it subtends at the centre (Euc. VI. 33), and also to the radius (Art. 391), if we consider PP' as the arc of a circle, whose centre

is N, the angle PNP' is measured by

taking FR=FP, PFP' is measured by

PP'

PN

In like manner,

therefore, denoting this angle by 0, PN by R, FP, F'P, by p, p', we have

2
R sin 0

=

1 1
+
ρ

Hence it may be inferred, that the focal chord of curvature is double b

the harmonic mean between the focal radii. Substituting for ō

sine, 2a for p + p', and b" for pp', we obtain the known value

The radius of curvature of the hyperbola or parabola can be investigated by an exactly similar process. In the case of the parabola we have p' infinite, and the formula becomes

I owe to Mr. Townsend the following investigation, by a different method, of the length of the focal chord of curvature: Draw any parallel QR to the tangent at P, and describe a circle through PQR meeting the focal

P

S

M

C

RN

points S and P coincide, therefore PC: PL:: MN: PL; or, the

focal chord of curvature is equal to the focal chord of the conic drawn parallel to the tangent at the point (p. 219, Ex. 4).

398. The radius of curvature of a central conic may otherwise be found thus:

Let Q be an indefinitely near point on the curve, QR a parallel to the tangent, meeting the

normal in S; now, if a circle be described passing through P, Q, and touching PT at P, since QS is a perpendicular let fall from Q on the diameter of this circle, we have PQ-PS multiplied by the diameter;

PQ* 2PS'

T

or the radius of curvature = Now, since QR is always drawn parallel to the tangent, and since PQ must ultimately coincide with the tangent, we have PQ ultimately equal to QR; but, by the property of the ellipse (if we denote CP and its conjugate by a', b′),

b": a" :: QR: PR.RP' (= 2a′. PR),

small PR, PS are taken, we have, by similar triangles, their

PR CP

α'

ratio

It is not difficult to prove that at the intersection of two confocal conics the centre of curvature of either is the pole with respect to the other of the tangent to the former at the intersection.

398 (a). If we consider the circle circumscribing the triangle formed by two tangents to a curve and their chord, it is evident geometrically, that its diameter is the line joining the intersection of tangents to the intersection of the corresponding normals. Hence, in the limit, the diameter of the circle circumscribing the triangle formed by two consecutive tangents and their chord is the radius of curvature; that is to say, the radius of the circle here considered is half the radius of curvature (Compare Art. 262, Ex. 4).

399. If two tangents be drawn to an ellipse from any point of a confocal ellipse, the excess of the sum of these two tangents over the arc intercepted between them is constant.*

For, take an indefinitely near point 7", and let fall the perpendiculars TR, T'S, then (see fig.)

PT=PR=PP' + P'R

(for P'R may be considered as the continuation of the line PP') in like manner

TT'

Hence (PT+ TQ) − (P'T'+ T' Q') = PP' − QQ' = PQ — P' Q'.

COR. The same theorem will be true of any two curves which possess the property that two tangents TP, TQ to the inner one always make equal angles with the tangent TT'' to the outer.

400. If two tangents be drawn to an ellipse from any point of a confocal hyperbola, the difference of the arcs PK, QK is equal to the difference of the tangents TP, TQ.†

*This beautiful theorem was discovered by Bishop Graves. See his Translation of Chasles's Memoirs on Cones and Spherical Conics, p. 77.

†This extension of the preceding theorem was discovered by Mr. Mac Cullagh, Dublin Exam. Papers, 1841, p 41; 1842, pp. 68, 83. M. Chasles afterwards independently noticed the same extension of Bishop Graves's theorem. Comptes Rendus, October, 1813, tom. XVII. p. 838.

CCC.