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CHAP. II.] ANHarmonic property of cubic cURVES.

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the notation (35) of anharmonics of pencils being retained. We obtain therefore thus the following Theorem :-" If the sides of any given plane* triangle ABC be cut (as in Fig. 21) by any given rectilinear transversal A"B"c", and if any two points P and P' in its plane be such as to satisfy the anharmonic rela


(A′′. PBP′B") . (B′′. PCP'C") . (C". PAP ́A") = 1,

then these two points P, P' are on one common cubic curve, which has the three collinear points A", B", C" for its three real points of inflexion, and has the sides BC, CA, AB of the triangle for its three tangents at those points;" a result which seems to offer a new geometrical generation for curves of the third order.

56. Whatever the order of a plane curve may be, or whatever may be the degree p of the function f in 46, we saw in 51 that the tangent to the curve at any point P = (x, y, z) is the right line

A = [l, m, n], if l=DÃ, m = D1f, n=D2ƒ;

expressions which, by the supposed homogeneity of f, give the relation, lx + my + nz = 0, and therefore enable us to establish the system of the two following differential equations,

ldx +mdy + ndz = 0, xdl + ydm + zdn = 0.

If then, by elimination of the ratios of x, y, z, we arrive at a new homogeneous equation of the form,

0 = F(DÃ, Dyf, Def),

as one that is true for all values of x, y, z which render the function f=0 (although it may require to be cleared of factors, introduced by this elimination), we shall have the equation

F(l, m, n) = 0,

* This Theorem may be extended, with scarcely any modification, from plane to spherical curves, of the third order.

as a condition that must be satisfied by the tangent A to the curve, in all the positions which can be assumed by that right line. And, by comparing the two differential equations,

dr (l, m, n) = 0, xdl + ydm + zdn = 0,

we see that we may write the proportion,

x: y: z = D/F : DmF: DF, and the symbol P = (D/F, DmF, DnF), if (x, y, z) be, as above, the point of contact P of the variable line [l, m, n], in any one of its positions, with the curve which is its envelope. Hence we can pass (or return) from the tangential equation F = 0, of a curve considered as the envelope of a right line A, to the local equation f=0, of the same curve considered (as in 46) as the locus of a point : since, if we obtain, by elimination of the ratios of l, m, n, an equation of the form

0 =ƒ(DF, DmF, D„F),

(cleared, if it be necessary, of foreign factors) as a consequence of the homogeneous equation F = 0, we have only to substitute for these partial derivatives, DF, &c., the anharmonic co-ordinates x, y, z, to which they are proportional. And when the functions fand F are not only homogeneous (as we shall always suppose them to be), but also rational and integral (which it is sometimes convenient not to assume them as being), then, while the degree of the function f, or of the local equation, marks (as before) the order of the curve, the degree of the other homogeneous function F, or of the tangential equation F = 0, is easily seen to denote, in this anharmonic method (as, from the analogy of other and older methods, it might have been expected to do), the class of the curve to which that equation belongs: or the number of tangents (distinct or coincident, and real or imaginary), which can be drawn to that curve, from an arbitrary point in its plane.

57. As an example (comp. 52), if we eliminate x, y, z between the equations,

1 = x - y-Z,

m=y-z - X,

n = z―x-y, lx + my + nz = 0,

where l, m, n are the co-ordinates of the tangent to the inscribed



conic of Art. 46, we are conducted to the following tangential equation of that conic, or curve of the second class,

F(1, m, n) = mn + nl + lm = 0;

with the verification that the sides [1, 0, 0], &c. (38), of the triangle ABC are among the lines which satisfy this equation. Conversely, if this tangential equation were given, we might (by 56) derive from it expressions for the co-ordinates of contact x, y, z, as follows:

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with the verification that the side [1, 0, 0] touches the conic, considered now as an envelope, in the point (0, 1, 1), or a', as before: and then, by eliminating 1, m, n, we should be brought back to the local equation, f= 0, of 46. In like manner, from the local equation f= yz + zx + xy = 0 of the exscribed conic (53), we can derive by differentiation the tangential co-ordinates,*

1 = D2f= y + z, m = 2 + x, n = x + y,

and so obtain by elimination the tangential equation, namely,

F (l, m, n) = l2 + m2 + n2 − 2mn − 2nl - 2lm = 0;

from which we could in turn deduce the local equation. And (comp. 40), the very simple formula

lx + my + nz = 0,

which we have so often had occasion to employ, as connecting two sets of anharmonic co-ordinates, may not only be considered (as in 37) as the local equation of a given right line A, along which a point p moves, but also as the tangential equation of a given point, round which a right line turns: according as we suppose the set l, m, n, or the set x, y, z, to be given. Thus, while the right line A"B"c", or [1, 1, 1], of Fig. 21, was

This name of " tangential co-ordinates" appears to have been first introduced by Dr. Booth in a Tract published in 1840, to which the author of the present Elements cannot now more particularly refer: but the system of Dr. Booth was entirely different from his own. See the reference in Salmon's Higher Plane Curves, note to page 16.

represented in 38 by the equation x + y + z = 0, the point o of the same figure, or the point (1, 1, 1), may be represented by the analogous equation,

l+m+ n = 0;

because the co-ordinates l, m, n of every line, which passes through this point o, must satisfy this equation of the first degree, as may be seen exemplified, in the same Art. 38, by the lines OA, OB, OC.

58. To give an instance or two of the use of forms, which, although homogeneous, are yet not rational and integral (56), we may write the local equation of the inscribed conic (46) as follows:

x2 + y + zł = 0;

and then (suppressing the common numerical factor), the partial derivatives are

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so that a form of the tangential equation for this conic is,

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which evidently, when cleared of fractions, agrees with the first form of the last Article: with the verification (48), that a ̄1 + b ̄1 + c-1 = 0 when the curve is a parabola; that is, when it is touched (50) by the line at infinity (38). For the exscribed conic (53), we may write the local equation thus,

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a form of the tangential equation which, when cleared of radicals, agrees again with 57. And it is evident that we could return, with equal ease, from these tangential to these local equations.

59. For the cubic curve with a conjugate point (54), the local equation may be thus written,*

*Compare Saimon's Higher Plane Curves, page 172.


x + y + z = 0;


we may therefore assume for its tangential co-ordinates the expressions,

1 = x}, m = y},

n = z};

and a form of its tangential equation is thus found to be,

}} + m2}+n} = 0.

Conversely, if this tangential form were given, we might return to the local equation, by making

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which would give x + y + z = 0, as before. The tangential equation just now found becomes, when it is cleared of radicals,

0 = 1 2 + m2 2 + n ̄2 — 2m ̄1n ̄1 — 2n ̄1l-1 — 21-1 m ̄1 ;

or, when it is also cleared of fractions,

0 = F = m2n2 + n2l2 + l2m2 − 2nl3m – 2lm2n — 2mn3l;

of which the biquadratic form shows (by 56) that this cubic is a curve of the fourth class, as indeed it is known to be. The inflexional character (54) of the points A", в", c" upon this curve is here recognised by the circumstance, that when we make m-n= 0, in order to find the four tangents from a” = (0, 1,− 1) (36), the resulting biquadratic, 0 = m2 – 4lm3, has three equal roots; so that the line [1, 0, 0], or the side BC, counts as three, and is therefore a tangent of inflexion: the fourth tangent from a" being the line [1, 4, 4], which touches the cubic at the point (− 8, 1, 1).

60. In general, the two equations (56),

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may be considered as expressing that the homogeneous equation,

f(nx, ny, - lx - my) = 0,

which is obtained by eliminating z with the help of the relation lx + my + nz = 0, from ƒ(x, y, z) = 0, and which we may

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