COR. II.-When z coincides with the centre of the conic the polar xy passes to infinity, and the lines zp and zp' become conjugate diameters of the curve. Hence the pairs of conjugate diameters of a conic form pencils of rays in involution.

COR. III.-If a parallelogram ABCD be inscribed in a conic and a second parallelogram be circumscribed to the same curve by drawing tangents at the points A, B, C, D; the diagonals of the second parallelogram will bisect the pairs of sides AB and CD, AD and BC, forming two conjugate diameters parallel respectively to the two sides of the first parallelogram meeting in any of the four points A, B, C, or D. Hence the two chords joining any point on an ellipse or hyperbola with the ends of any diameter are parallel to two conjugate diameters of the curve.

24. Inversion. The inversion of a point-range may be defined as the permutation of the elements or letters composing it continuously in the same direction, beginning at any letter. Thus, if abcd be a harmonic point-range, bcda and badc are its two inversions branching from the letter b towards the right and left respectively. In general a pointrange abcd is projective of its left inversion from b, its right from c, and its left from d; i.e. of badc, cdab, and dcba. For, project abcd from any centre S upon a line traversing a in aefg; then, aefg from the centre d upon Sb in bexS;

x

and, finally, bexS from the centre f in bade. Thus abcd and bade, being the first and last of a sequence of point-ranges in perspective, are projective of each other (§ 3, Cor. IV.). The projectivity of the other two inversions can be proved in a similar way. By combining this deduction with Art. 2 it will be seen that a harmonic point-range is projective of all its inversions. Consider the involution MNaa' (Fig. 45), the points MaNa' of the first system corresponding to the points Ma'Na of the second, and the points M and N being double or common to both systems; so that the forms MaNa' and Ma'Na are projective. Project, therefore, MaNa' from any point or centre S upon a line traversing M. The form MrKt is projective of MaNa', and therefore also of Ma'Na. But these forms have the point M in common; hence they are also in perspective, or the lines ra', KN, and ta will meet in the same point Q, the second centre of perspective. We obtain, thus, the complete quadrangle QrSt, whose opposite sides tQ and rS meet in a and opposite sides rQ and tS meet in a', and whose diagonals traverse M and N respectively.

Fig. 46.

25. The three pairs of opposite sides of a complete quadrangle are cut by any line in the plane of the quadrangle which does not traverse any of the four apices of the latter in three pairs of points forming an involution. Let grst (Fig. 46) be a complete quadrangle whose pairs of opposite sides rt and qs, st and qr,

qt and rs are cut by the line xy respectively in a and a', b and b', c and c'; and let o be the point of intersection of qs and rt. The form ator is the projection of aca'b' from

the centre q, as well as the projection of aba'c' from the centres; so that aca'b' is projective of aba'c' and therefore, by inversion, of a'c'ab. Hence aa'. bb'. cc' forms an involu

tion (§§ 24, 20).

Similarly, it may be proved that the three pairs of opposite apices of a complete quadrilateral are projected from any point of their plane, not situate on any of the four sides, by three pairs of conjugate rays in involution.

COR. The theorem of the present article enables us to solve the following problem :-Given two pairs of points in involution situate upon the punctuated line xy to determine the point c' conjugate of any fifth point c, by simply constructing a complete quadrangle, of which one pair of opposite sides passes through a and a', another pair through b and b', and a fifth side through c; then the sixth side will pass through c'.

26. Any line which does not traverse any of the apices of a simple quadrangle inscribed in a conic will meet the curve and the two pairs of opposite sides of the quadrangle in three pairs of conjugate points in involution. Let xy (Fig. 47) be a line meeting the sides qt, rs, qr, ts of the quadrangle qrst respectively in aa', bb', and the curve in pp'. It will be seen that the two pencils q(prp't) and s(prp't) are projective of the same four points P,r,p',t on the curve; hence the forms pbp'a and pa'p'b' in which the line xy cuts these two pencils

x

b

Fig. 47.

are also projective. But by inversion the two forms pa'p'b' and p'b'pa' are projective; therefore pbp'a and p'b'pa' are also projective, and pp'. aa'. bb' forms an involution.

27. DOUBLE POINTS.-If a punctuated line has two double points, such as M and N (Fig. 48), in each of which two conjugate points coincide, we have (§ 19)

which expresses the fact that M and N are harmonically

gate points, such as a and a', of a punctuated line in involution, are situated on the same side of the centre O, when the line contains double points; but, when the line has no double points, the conjugate points a and a' are separated by the centre O.

Describe a series of circles on the segments aa', bb', cc' as diameters, and let r be the radius of any one of these circles such as that described on aa'; then, if d be the distance of its centre k from O, we have

centre O lies within the circle described upon any segment aa' (Fig. 49), and d is negative. Through the centre O draw pq perpendicular to aa'; then the half-chord po or 90 forms the side of a right-angled triangle of which dis the second side and r the hypothenuse. Since, in this

Thus all the circles pass through the points of intersection of any two, that is p and q, from each of which the system, aa', bb', cc', is projected by an involute pencil, whose conjugate rays are inclined at right angles to each other. A pencil of rays is in involution when two pairs of conjugate rays intersect at right angles.

If a series of right-angled triangles be inscribed in a conic, so arranged that the apices of the right angles fall upon the same point, the hypothenuses will all intersect in the same point; for the series of points are coupled in involution by the rectangular pencil issuing from a single point on the curve. The point in which the hypothenuses intersect is the pole of the line passing through the double points, if they exist.

28. If conjugate rays traversing pairs of points p and p' on an axis of a conic meet at right angles, the axis will form a punctuated line in involution. Let aa be an axis of the curve meeting two rectangular conjugate rays Sp and Sp' in p and p'. Then every ray of the brush or pencil p is per