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 MacCullagh.

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 FFF", 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 the first figure, p. 352).

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 In like manner,

is N, the angle PNP' is measured by

PP'

PN

PR

taking FR=FP, PFP' is measured by and we have

FP'

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

2

=

1 1
+

R sin 0 ρ ρ

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

b

the harmonic mean between the focal radii. Substituting for b'

sin 0, 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 chord PL of the conic at C. Then, by the circle PS.SC=QS. SR, and by the conic (Ex. 2, p. 179)

PS. SL: QS. SR:: PL: MN; therefore, whatever be the circle,

SC: SL:: MN: PL;

but for the circle of curvature the

P

R N

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

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;

or the radius of curvature =

PQ2 2PS

P'

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'),

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

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.

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 figure next page)

PT=PR=PP' + P'R

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

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

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.*

For it appears, precisely as before, that the excess of T'P'-PK over TP-PK=T"R,

and that the excess of TQ-Q'K over TQ-QK is T'S, which is equal to T'R, since (Art. 189) TT" bisects the angle RT'S. The difference, therefore, between the excess of TP over PK, and that of TQ over QK, is constant; but in the particular case where T coincides with K, both these excesses, and consequently their dif

A

P

K

F

R

S

F' Α'

ference, vanish; in every case, therefore, TP— PK=TQ — QK.

COR. Fagnani's theorem, "That an elliptic quadrant can be so divided, that the difference of its parts may be equal to the difference of the semi-axes," follows immediately from this Article, since we have only to draw tangents at the extremities of the axes, and through their intersection to draw a hyperbola

* 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 Dr. Graves's theorem. Comptes Rendus, October, 1843, tom. XVII., p. 838.

confocal with the given ellipse. The co-ordinates of the points where it meets the ellipse are found to be

401. If a polygon circumscribe a conic, and if all the vertices but one move on confocal conics, the locus of the remaining vertex will be a confocal conic.

In the first place, we assert that if the vertex T of an angle PTQ circumscribing a conic, move on a confocal conic (see fig., Art. 399); and if we denote by a, b, the diameters parallel to TP, TQ; and by a, ß, the angles TPT', TQ' T", made by each of the sides of the angle with its consecutive position, then aa=bB. For (Art. 399) TR=T'S; but TR=TP.a; T'S=T' Q'.ß, and (Art. 149) TP and TQ are proportional to the diameters to which they are parallel.

Conversely, if aa=b8, T moves on a confocal conic. For by reversing the steps of the proof we prove that TR=T'S; hence that TT" makes equal angles with TP, TQ, and therefore coincides with the tangent to the confocal conic through T; and therefore that T" lies on that conic.

If then the diameters parallel to the sides of the polygon be a, b, c, &c., that parallel to the last side being d, we have aa=bß, because the first vertex moves on a confocal conic; in like manner bß=cy, and so on until we find aa=dd, which shows that the last vertex moves on a confocal conic.*

* This proof is taken from a paper by Dr. Hart; Cambridge and Dublin Mathematical Journal, Vol. IV., 193.