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22. The known property of a circle, that "the angles in the same segment are equal to one another,” gives rise to an important anharmonic property of conic sections. The property of the circle may be expressed thus, that “if A, B be any two fixed points on the circumference of a circle, O any. moving point on it, the angle A OB is constant.” Project the circle into any conic, and let A', B', O' be the projections of A, B, O; H, K those of the circular points at infinity. Then, from the result of the last article, it follows that

{0'. A'B'HK} is constant. Or, the anharmonic ratio of the pencil, formed by joining any point of a conic to four fixed points on the curve, is constant.

Reciprocating this theorem, in accordance with Art, 13, Chap. vii, we see that if any tangent to a conic be cut by four fixed tangents, the anharmonic ratio of the range, formed by the points of section, is constant,

23. If P, Q, R be three points in a straight line, and p, q, r be their projections, and s the projection of the point

PO at infinity on the line PQR, then [pqrs]

QR'

[blocks in formation]

where S denotes the point at infinity on the line PQR.

PQ Also RS: QR in a ratio of equality, hence [pqrs] =

QR

24. If P, P, Q, Q, R, R'... be a system of points in involution, and p, p, q, q, r, 2"... their projections, then since by Art. 27, Chap. 1. (PQRS]=[P'Q'R'S®], and by Art. 20 of this Chapter [PQRS] =(pqrs], [POʻR'S'] = [p'g'r's'],

it follows that (pqrs] = [p'q'r's'], or p, p, q, d', ?, n... are a system of points in involution. Hence, any system of points in involution projects into a system in involution.

If P coincide with P, p will coincide with p', or the foci of one system project into the foci of the other. We may observe that the centre of one system will not, in general, project into the centre of the other.

25. Let a system of circles be described through two given points A, A', and let any circle of the system cut a given straight line in P, P'. Produce AA' to meet the given straight line in 0. Then

OP. OP= 0A.OA', or OP. OP is constant for all circles passing through A, A'. Hence, the system of points in which a system of circles, passing through two given points, cut a given straight line, are in involution. Project the system of circles into a system of conics, passing through four given points, and we learn that “a system of conics, passing through four given points, cut any straight line in a system of points in involution.”

Of this system of conics, one can be drawn so that one of its points of intersection with the given straight line shall be at an infinite distance,-in other words, so that one of its asymptotes shall be parallel to the given straight line. The other point, in which this conic cuts the given straight line, will be the centre of the system.

Again (see Art. 29, infra), two conics can be described, passing through the four given points, and touching the given straight line. The two points of contact of these conics will be the foci of the system of points in involution.

By reciprocating these propositions, we obtain analogous properties of the system of conics, inscribed in a given quadrilateral, whence, by projection, may be obtained those of a system of confocal conics.

26. When the vertex of the cone, used for purposes of projection, is infinitely distant, so that the cone itself becomes a cylinder, the projection is said to be orthogonal. In this

mode of projection, the line at infinity remains at an infinite distance, and any two parallel lines will therefore project into parallel lines. Also any area will bear to its projection a constant ratio; and the mutual distances of any three points in the same straight line will bear to one another the same ratios as the mutual distances of their projections. Two perpendicular diameters of a circle will, since each is parallel to the tangent at the extremity of the other, project into two conjugate diameters of an ellipse. By this method, many properties of conic sections, more especially those relating to conjugate diameters, may be readily deduced from those of the circle.

ON THE DETERMINATION OF A CONIC FROM FIVE GIVEN

GEOMETRICAL CONDITIONS.

27. If any five independent conditions be given, to which a conic is to be subject, each of these, expressed in algebraical language, will give an equation for the determination of the five arbitrary constants which the equation of the conic involves. Hence, five conditions suffice for the determination of the conic. It may, however, happen that some of the equations for the determination of the constants rise to a degree higher than the first, in such a case, the constants will have more than one value, and more than one conic may therefore be described, satisfying the required conditions, although the number will still be finite.

The geometrical conditions of most frequent occurrence are those of passing through given points and touching given straight lines, with such others as may be reduced to these. We proceed to consider how many conics may be described in each individual case.

28. Let five points be given.

In this case we have merely to substitute in the equation of the conic the co-ordinates of the several points for a, b, y; we shall thus obtain five simple equations for the determina

DETERMINATION OF A CONIC FROM FIVE CONDITIONS. 147

tion of the constants, and one conic only will satisfy the given conditions.

R18

+

V +

29. Let four points and one tangent be given.

Take three of the points as angular points of the triangle of reference. Let f, g, h be the co-ordinates of the fourth given point, la + inry = 0, the equation of the given tangent. Let the equation of the conic be

р

0. B

y Then for the determination of the ratios 1:4:v, we have the equations

λ
M

0,
9

h 1872 + p'm + v'n- 2uvmn - 2vinl 2rulm= 0. These equations will give two values for the ratios, and prove therefore that two conics can be described satisfying the required conditions.

+

30. Let three points and two tangents be given.

Take the three points as angular points of the triangle of reference. Let the two given tangents be represented by the equations

la + + ny= 0,

l'a+m'ß + n'y=0.
If then the conic be represented by the equation

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we have, for the determination of x:N:v, the equations

X?P + ?m? + v’n’ – 2uvmn 2vanl 2ulm = 0,

127"2 + pełm"? + v*n"– 2pvm'n' 2van'l' 21. l'm' =0, which, being both quadratics, give four values for each of the ratios, shewing that four conics may be described satisfying the given conditions.

31. Let two points and three tangents be given.

Take the three tangents as lines of reference, and let f, g, h; f", g', h' be the co-ordinates of the two given points. Then, if the equation of the conic be

2% + m*B? + very 2uvbry - 2v1ya - 21paß = 0, we shall get, writing f, g, h; f', g', h', successively for a,B,, two quadratics for the determination of the ratios λ:μ: ν, giving therefore four conics.

32. Let one point and four tangents be given.

Taking three of the tangents as lines of reference, the condition of touching the fourth given line gives a simple equation for the determination of the coefficients, and that of passing through the given point a quadratic. Hence, two conics may be described, satisfying the given conditions.

33. Let five tangents be given.

Taking three of the tangents as lines of reference, the condition of touching each of the others gives a simple equation for the determination of the constants, shewing that one conic only can be described satisfying these conditions.

The results of Arts. 31, 32, 33, may of course be deduced by the method of reciprocal polars, from those of Arts. 30, 29, 28.

34. Several other forms under which the data may be given, are reducible to a certain number of lines and points. Thus to have given a tangent and its point of contact is equivalent to having two points given, the points being indefinitely close together. Or, it may be regarded as equivalent to having two tangents given, these tangents being indefinitely nearly coincident. To have given that a conic is a parabola is equivalent to having a tangent given, since every, parabola touches the line at infinity. To have given that it is a circle is equivalent to having two points given, since all circles intersect the line at infinity in the same two points. And this explains the reason why four circles can

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