same conic.* The theorem here proved is the reciprocal of that proved Art. 297, and may also be established by interpreting tangentially the equations there used. Thus, if P, P'; Q, Q' represent tangentially two pairs of corresponding points, P+λP', Q+λ represent any other pair of corresponding points; and the line joining them touches the curve represented by the tangential equation of the second order, PQ = P'Q.

Ex. Any transversal through a fixed point P meets two fixed lines OA, OA', in the points AA'; and portions of given length Aa, A'a' are taken on each of the given lines; to find the envelope of aa'. Here, if we give the transversal four positions, it is evident that {ABCD} = {A'B'C'D'}, and that {ABCD} = {abcd}, and {A'B'C'D'} = {a'b'c'd'}.

330. Generally when the envelope of a moveable line is found by this method to be a conic section, it is useful to take notice whether in any particular position the moveable line can be altogether at an infinite distance, for if it can, the envelope is a parabola (Art. 254). Thus, in the last example the line aa' cannot be at an infinite distance, unless in some position AA' can be at an infinite distance, that is, unless P is at an infinite distance. Hence we see that in the last example, if the transversal, instead of passing through a fixed point, were parallel to a given line, the envelope would be a parabola. In like manner, the nature of the locus of a moveable point is often at once perceived by observing particular positions of the moveable point, as we have illustrated in the last example of Art. 328.

331. If we are given any system of points on a right line we can form a homographic system on another line, and sueh that three points taken arbitrarily a', b', c' shall correspond to three given points a, b, c of the first line. For let the distances of the given points on the first line measured from any fixed

* In the same case if P, P' be two fixed points, it follows from the last article that the locus of the intersection of Pd, P'd' is a conic through P, P'. We saw (Art. 277) that if a, b, c, d, &c., a', b', c', d' be two homographic systems of points on a conic, that is to say, such that {abcd} always = {a'b'c'd'}, the envelope of dd' is a conic having double contact with the given one. In the same case, if P, P' be fixed points on the conic, the locus of the intersection of Pd, P'd' is a conic through P, P'. Again, two conics are cut by the tangents of any conic having double contact with both, in homographic systems of points, or such that {abcd} = {a'b'c'd'} (Art. 276); but it is not true conversely, that if we have two homographic systems of points on different conics, the lines joining corresponding points necessarily envelope a conic.

origin on the line be a, b, c, and let the distance of any variable point on the line measured from the same origin be x. Similarly let the distances of the points on the second line from any origin on that line be a', b', c', x', then, as in Art. 277, we have the equation

This equation enables us to find a point x' in the second line corresponding to any assumed point x on the first line, and such that (abcx}={a'b'c'x'}. If this relation be fulfilled, the line joining the points x, x' envelopes a conic touching the two given lines; and this conic will be a parabola if A= 0, since then x' is infinite when x is infinite.

The result at which we have arrived may be stated conversely thus: Two systems of points connected by any relation will be homographic, if to one point of either system always corresponds one, and but one, point of the other. For evidently an equation of the form

Axx+Bx+ Cx' + D=0

is the most general relation between x and x' that we can write down, which gives a simple equation whether we seek to determine x in terms of x' or vice versa. And when this relation is fulfilled, the anharmonic ratio of four points of the first system is equal to that of the four corresponding points of the (xy) (zw) second. is unaltered

For the anharmonic ratio.

(x − z) (y — w)

* M. Chasles states the matter thus: The points x, x' belong to homographic systems, if a, b, a', b' being fixed points, the ratios of the distances ax : bx, a'x': b'x', be connected by a linear relation, such as

Denoting, as above, the distances of the points from fixed origins, by a, b, x; a', b', x', this relation is

which, expanded, gives a relation between x and x' of the form

Axx' + Bx + Cx' + D = 0.

if instead of x we write

Bx+D
Ax+ C'

and make similar substitu

tions for y, z, w.

332. The distances from the origin of a pair of points A, B on the axis of x being given by the equation, ax2 + 2hx +b=0, and those of another pair of points A', B' by a'x2+2h'x+b′ = 0, to find the condition that the two pairs should be harmonically conjugate.

Let the distances from the origin of the first pair of points be a, B; and of the second a', '; then the condition is

which expanded may be written

(a + B) (a' + B′) = 2aß +2α'B'.

It is proved, similarly, that the same is the condition that the pairs of lines

aa2 + 2haß + bß3, a’a2 + 2h′aß +b′ß",

should be harmonically conjugate.

333. If a pair of points ax2+2hx+b, be harmonically conjugate with a pair a'x2+2h'x + b', and also with another pair a′′x2+2h′′x+b′′, it will be harmonically conjugate with every pair given by the equation

(a'x2 + 2h'x + b') +λ (a′′x2 + 2h′′x + b′′) = 0.

For evidently the condition

a (b′ + λb′′) + b (a' + λa′′) — 2h (h′ + λh′′) = 0,

will be fulfilled if we have separately

ab' + ba' - 2hh' = 0, ab" + ba′′ – 2hh" = 0.

*It can be proved that the anharmonic ratio of the system of four points will be given, if (ab' + a'b - 2hh')2 be in a given ratio to (ab — h2) (a'b' — h'2).

RR.

334. To find the locus of a point such that the tangents from it to two given conics may form a harmonic pencil.

If four lines form a harmonic pencil they will cut any of the lines of reference harmonically. Now take the second form (Art. 294) of the equation of a pair of tangents from a point to a curve given by the general trilinear equation, and make y=0 when we get

(CB” + By2 – 2Fß′y') a2 — 2 ( Ca′ß' – Fay' – GB'y' + Hy'2) aß + (Ca2 + Ay” − 2 Ga'y′) B2 = 0. We have a corresponding equation to determine the pair or points where the line y is met by the pair of tangents from aB'y' to a second conic. Applying then the condition of Art. 332 we find that the two pairs of points on y will form a harmonic system, provided that a'B'y satisfies the equation (CB2 + By3 - 2Fßy) (C'a2 + A'y2 – 2 Gay)

+ (Ca2 + Ay2 − 2 Gay) (C′ß2 + B′y3 – 2F′′ßy) =2(Caß - Fay - GBy+ Hy3) (C'aß - F'ay - G'By + H ́y3). On expansion the equation is found to be divisible by y2, and the equation of the locus is found to be

(BC′+B'C-2FF") a2+(CA'+C'A−2G G') B2+ (A.B'+A'B—2HH')y* +2(GH'+ G'H-AF' — A'F) By+2(HF'+H'F— BG'– BG) ya + 2 (FG' + F'G - CH' - C'H) aẞ=0;

a conic having important relations to the two conics, which will be treated of further on. If the anharmonic ratio of the four tangents be given, the locus is the curve of the fourth degree F=kSS', where S, S', F, denote the two given conics, and that now found.

335. To find the condition that the line λa + μẞ+vy should be cut harmonically by the two conics. Eliminating y between this equation and that of the first conic, the points of intersection are found to satisfy the equation

(cλ2 + av2 – 2gλv) a2 + 2 (cλμ − ƒ λv – gμv + hv3) aß

+ (cμ2 + bv2 - 2ƒμv) ß2 = 0. We have a similar equation satisfied for the points where the line meets the second conic; applying then the condition of

Art. 332, we find, precisely as in the last article, that the required condition is

(bc' + b'c - 2ff') λ" + (ca' + c'a - 2gg') μ2 + (ab' + a′b – 2hh') v*

+ 2 ( gh' + g′h − af'

-

− a′ƒ) μv + 2 (hƒ' + h'ƒ — bg′ — b′g) vλ

+2(fg' +f'g-ch' — c'h) Xμ = 0.

The line consequently envelopes a conic.*

INVOLUTION.

336. Two systems of points a, b, c, &c., a', b', c', &c., situated on the same right line, will be homographic (Art. 331) if the distances measured from any origin, of two corresponding points, be connected by a relation of the form

Axx+Bx+ Cx' + D= 0.

Now this equation not being symmetrical between x and x', the point which corresponds to any point of the line considered as belonging to the first system, will in general not be the same as that which corresponds to it considered as belonging to the second system. Thus, to a point at a distance x considered as belonging to the first system, corresponds a point at the disBx+D but considered as belonging to the second Ax+ α i C

tance

Two homographic systems situated on the same line are said to form a system in involution, when to any point of the line the same point corresponds whether it be considered as belonging to the first or second system. That this should be the case it is evidently necessary and sufficient that we should have B = C in the preceding equation, in order that the relation connecting x and x may be symmetrical. We shall find it

*If substituting in the equations of two conics U, V, for a, λa + ua', &c. we obtain results

then it is easy to see, as above, that UV'+U'V - 2PQ, represents the pair of lines which can be drawn through a'@'y', so as to be cut harmonically by the conics. In the same case (Art. 296), the equation of the system of four lines joining a'ß'y' to the intersections of the conics, is

(UV' + U'V − 2PQ)2 = 4 (UU' — P2) (VV' — Q2).

UU' - P2 and VV' - Q' denote the pairs of tangents from a'ß'y' to the conics.