Now, if we draw lines from the point L to each of these points, we form two pencils which have the three legs, CL, DE, AB, common, therefore the fourth legs, NL, LM, must form one right line. In like manner, Brianchon's theorem is derived from the anharmonic property of the tangents. Ex. 12. Given four points on a conic, ADFB, and two fixed lines through any one of them, DC, DE, to find the envelope of the line CE joining the points where those fixed lines again meet the curve. The vertices of the triangle CEM move on the fixed lines DC, DE, NL, and two of its sides pass through the fixed points, B, F; therefore, the third side envelopes a conic section touching DC, DE (by the reciprocal of Mac Laurin's mode of generation). Ex. 13. Given four points on a conic ABDE, and two fixed lines, AF, CD, passing each through a different one of the fixed points, the line CF joining the points where the fixed lines again meet the curve will pass through a fixed point. For the triangle CFM has two sides passing through the fixed points B, E, and the vertices move on the fixed lines AF, CD, NL, which fixed lines meet in a point, therefore (p. 268) CF passes through a fixed point. The reader will find in the Chapter on Projection how the last two theorems are suggested by other well-known theorems. (See Ex. 3 and 4, Art. 355). Ex. 14. The anharmonic ratio of any four diameters of a conic is equal to that of their four conjugates. This is a particular case of Ex. 2, p. 260 that the anharmonic ratio of four points on a line is the same as that of their four polars. We might also prove it directly, from the consideration that the anharmonic ratio of four chords proceeding from any point of the curve is equal to that of the supplemental chords (Art. 179). Ex. 15. A conic circumscribes a given quadrangle, to find the locus of its centre. (Ex. 3, p. 148). Draw diameters of the conic bisecting the sides of the quadrangle, their anharmonic ratio is equal to that of their four conjugates, but this last ratio is given, since the conjugates are parallel to the four given lines; hence the locus is a conic passing through the middle points of the given sides. If we take the cases where the conic breaks up into two right lines, we see that the intersections of the diagonals, and also those of the opposite sides, are points in the locus, and, therefore, that these points lie on a conic passing through the middle points of the sides and of the diagonals. 329. We think it unnecessary to go through the theorems, which are only the polar reciprocals of those investigated in the last examples; but we recommend the student to form the polar reciprocal of each of these theorems, and then to prove it directly by the help of the anharmonic property of the tangents of a conic. Almost all are embraced in the following theorem: If there be any number of points a, b, c, d, &c. on a right line, and a homographic system a', b', c', d', &c. on another line, the lines joining corresponding points will envelope a conic. For if we construct the conic touched by the two given lines and by three lines aa', bb', cc', then, by the anharmonic property of the tangents of a conic, any other of the lines dd' must touch the 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+XP', Q+λQ' 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 such 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 (ab) (c-x) which expanded is of the form = Axx' + Bx +Сx' + D= 0.* 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' +Вx+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 (x − y) (z — w) second. For the anharmonic ratio (x − z) (y-w) is unaltered * 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 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'x" + 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', B'; then the condition is which expanded may be written (a + B) (a' + B') = 2aß + 2a'ß'. It is proved, similarly, that the same is the condition that the pairs of lines aa2+2haß +bß2, 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). 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 (given 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 (C3'2 + By'2 — 2FB'y') a2 — 2 (Ca'B' — Fa'y' — GB'y' + Hy'2) aß + (Ca” + Ay” − 2 Ga'y') B2 = 0. We have a corresponding equation to determine the pair of points where the line y is met by the pair of tangents from a'B'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 + By2 – 2Fßy) ( C'a2 + A'y2 − 2 G'ay) = + ( Ca2 + Ay2 − 2 Gay) (C'ß2 + B'y2 — 2F′′ßy) · 2 (Caß – Fay — Gßy + 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−2 GG')ẞ2+(AB'+A'B−2HH')Ÿ” +2(GH' + G'H-AF" — A'F) By+2(HF"+H'F− BG' – B'G) 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, F2=kSS', where S, S', F, denote the two given conics, and that now found. 335. To find the condition that the line λa+uß+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 − fλv — gμv + hv2) aß +(cμ2 + bv2 − 2fμv) B2 = 0. We have a similar equation satisfied for the points where the line meets the second conic applying then the condition of |