is even simpler when dealing with the ellipse or the hyperbola. The tangents at the vertices of the principal axis cut any third tangent in two points P and Q, which are projected from every point of the principal axis by a couple of conjugate rays (compare § 17). Thus it is only necessary to find the two points on the principal axis from which the segment PQ is projected by two rectangular conjugate rays, by describing upon PQ as a diameter a circle whose points of intersection with the principal axis are the required foci.

Or again, by Art. 29, if ab be any segment cut off upon the minor axis of a central conic by two conjugate rays, the intercept ab will subtend a right angle at each focus. Hence a circle described on ab as a diameter will intersect the major axis in the required foci.

36. The variable segment cut off from any tangent to a conic by two fixed tangents is projected from one of the foci under a constant angle; or, in other terms, subtends a constant angle at the focus (Fig. 54). Let TA

and TB be the fixed tangents,
and A'B' the segment cut off
by the fixed tangents from
any third tangent; then, just
as in Fig. 51 the angle AFB T
is bisected by FT, so here FB'
bisects BFC; and, therefore,

B'FC B'FB=BFC.

Similarly

C

F

B'

B

Fig. 54.

A'FC A'FA AFC;

and, by addition,

that is

B'FC+A'FC={(BFC+AFC);

B'FA' BFT=AFT;

inasmuch as BFC + AFC is equal to the supplement of BFA, which is bisected by FT.

COR. I.-When A' passes to infinity on TA and B' to

G

infinity on BT, we have FA parallel to TA and FB parallel to TB, wherefore

B'FA'= a constant=180° - BTA,

and the angles A'TB' and A'FB' are together equal to two right angles; so that a circle may be described about the quadrangle A'TB'F. Now A'B' passes to infinity in the case of the parabola; therefore the circle circumscribed to any triangle formed by three tangents of a parabola traverses the focus.

COR. II.—When, in the case of the hyperbola, the fixed tangents TA and TB coincide with the asymptotes, and have therefore their points of contact situate at an infinite distance, the lines TB and FB are parallel; and the angle B'FA', being equal to BFT, is also equal to one of the angles which the asymptote TB makes with the principal axis. Similarly the second angle formed by TB and TF is equal to the angle A'F'B' which the segment A'B' subtends at N

Fig. 55.

M

The

the second focus. Thus the angles A'FB' and A'F'B' are supplementary. Hence the foci of a hyperbola and the points of intersection of any tangent with the asymptotes are four points situate upon the same circumference. centre of this circle and the same two points of intersection determine a second circumference which passes through the centre of the hyperbola.

37. To determine the double points in two projective pencils

A)A'B'C' and A')ABC (Fig. 55). Let A'C and AC' meet in c; AB' and A'B in b; join bc, and produce it to meet the conic in the double points M and N. If the pencils be in involution, the lines AA', BB', CC' drawn through the corresponding points in the two systems meet in the pole z of MN.

COR. I.-If the two given elementary projective forms be two superposed punctuated lines xy and x'y' (Fig. 56), we can reduce this to the preceding case by projecting the lines from any point S upon the circumference of a circle traversing S. Thus let ABC of the line ay be projected C'N B' B

x C

MA

y

الا

a

S

Fig. 56.

upon the circle in abc, and the correlative points A'B'C' upon the same circle in a'b'c'. Join ba' and b'a, bc' and b'c, meeting in a, and c, and produce ac。 to intersect the circle in m and n, the projections of the double points. Lastly, project m and n from S upon xy in M and N, the required double points of the line of projection. Since the systems ABC and A'B'C' are in projection, any points on the projective pencils S)ABC and S)A'B'C', which lie upon the same line or conic, are likewise in projection. The double series ABC and A'B'C' are not necessarily in involution, unless the similar letters, such as A and A', are interchangeable in the two systems; that is, unless the point A of the first system, when assumed to be a point of the second

system, correspond to the point A' of the second system, regarded as a point of the first system.

COR. II. The centre z of a series of pairs of points in involution on a conic is also the centre of perspective of pairs of conjugate homologous points, such as AA', BB', CC' (Fig. 55). If, therefore, be the centre of a second involute series aa', bb', cc' on the same curve, and the line of centres zz meet the conic in X, X'; the pair of conjugate homologous points XX' will be common to both involute systems.

COR. III. The conjugate diameters of a conic form an involute pencil at the centre of the curve. Hence, if a circle be described about any point O so as to pass through the centre S of a hyperbola and two given pairs of conjugate diameters meet this circle in B,B' and C, C'respectively; the lines BB' and CC' will meet in the centre z of the involute system. Moreover, if the line zO meet the circle in X, X'; SX and SX' will be the rectangular conjugate diameters or axes of the hyperbola.

38. If from the two foci S and H (Fig. 57) two lines be drawn to any point V, of which one, HV, is equal to the major axis, and if from the point of bisection T of the other focal ray SV a perpendicular TP be drawn, TP will be a tangent to the conic at P. For, in that case, the triangles SPT and VPT being equal,

SP=PV,

SP+PH=HV= major axis.

Moreover, owing to the similarity of the triangles SPT and VPT, the line PT makes equal angles with the focal rays SP and HP (§ 29).

39. Given the focus and major axis, to describe a conic passing through two points or touching two lines.

CASE I.-If the two points P and p be given (Fig. 57), describe from P and p as centres with the respective lengths

major axis SP and major axis Sp

two circular arcs meeting in the required second focus H. The upper or minus sign refers to the case of the ellipse, the lower or plus sign to that of the hyperbola.

H

=

Fig. 57.

CASE II. When the tangent lines TP and tp are given, draw ST perpendicular to TP and St to tp, making TV = ST and tv St. Then from V and v as centres, with the given major axis as a radius, describe circular arcs intersecting in H, the required second focus.

40. Given the focus S of a parabola, a tangent TR, and a point P on the curve, to describe the parabola (Fig. 58). From P with radius SP describe a circular arc F; then draw ST perpendicular to RT, making TV ST. From V draw a line VF tangent to the circular arc at F and from S let fall

=