« PreviousContinue »
under the form,
a + b
we see that the conic is an ellipse,
parabola, or hyperbola, according as c,c'< or = or > AB; the arrangement being still, in other respects, that which is represented in Fig. 26. Or, to express the same thing more symmetrically, if we complete the parallelogram CABD, then according as the point D falls, Ist, beyond the chord B'c', with respect to the point a; or IInd, on that chord; or IIIrd, within the triangle AB'C', the general arrangement of the same Figure being retained, the curve is elliptic, or parabolic, or hyperbolic. In that other arrangement or configuration, which answers to the system of inequalities, b> 0, c> 0, a + b + c < 0, the point a' is still upon the side BC itself, but o is on the line A'A prolonged through a; and then the inequality,
a (b + c) + bc < − (b2 + bc + c2) < 0,
shows that the conic is necessarily an hyperbola; whereof it is easily seen that one branch is touched by the side BC at A', while the other branch is touched in B' and c', by the sides CA and BA prolonged through A. The curve is also hyperbolic, if either a + b or a + c be negative, while b and c are positive as before.
50. When the quadratic (48) has its roots real and unequal, so that the conic is an hyperbola, then the directions of the asymptotes may be found, by substituting those roots, or the values of t, u, v which correspond to them (or any scalars proportional thereto), in the numerator of the expression (46) for p; and similarly we can find the direction of the axis of the parabola, for the case when the roots are real but equal for we shall thus obtain the directions, or direction, in which a right line or must be drawn from o, so as to meet the conic at infinity. And the same conditions as before, for distinguishing the species of the conic, may be otherwise obtained by combining the anharmonic equation, f= 0 (46), of that conic, with the corresponding equation ax + by + cz = 0 (38) of the line at infinity; so as to inquire (on known principles of modern geometry) whether that line meets that curve in two
CHAP. 11.] DIFFERENTIALS-TANGENTS—POLARS.
imaginary points, or touches it, or cuts it, in points which (although infinitely distant) are here to be considered as real.
51. In general, if f(x, y, z) = 0 be the anharmonic equation (46) of any plane curve, considered as the locus of a variable point P; and if the differential* of this equation be thus denoted,
0 =df(x, y, z) = Xdr + Ydy + Zdz;
then because, by the supposed homogeneity (46) of the function f, we have the relation
Xx + Yy + Zz = 0,
we shall have also this other but analogous relation,
x' - x: y-y: z' - z = dx : dy: dz;
that is (by the principles of Art. 37), if p' = (x', y', z) be any point upon the tangent to the curve, drawn at the point P= (x, y, z), and regarded as the limit of a secant. The symbol (37) of this tangent at P may therefore be thus written, [X, Y, Z], or [D1ƒ, Dyƒ, Dzƒ]; where Dr, Dy, D are known characteristics of partial deriva
52. For example, when ƒ has the form assigned in 46, as answering to the conic lately considered, we have Drf=2(x−y−z), &c.; whence the tangent at any point (x, y, z) of this curve may be denoted by the symbol,
in which, as usual, the co-ordinates of the line may be replaced by any others proportional to them. Thus at the point a', or (by 36) at (0, 1, 1), which is evidently (by the form off) a point upon the curve, the tangent is the line [-2, 0, 0], or [1, 0, 0]; that is (by 38), the side BC of the given triangle, as
In the theory of quaternions, as distinguished from (although including) that of vectors, it will be found necessary to introduce a new definition of differentials, on account of the non-commutative property of quaternion-multiplication: but, for the present, the usual significations of the signs d and D are sufficient.
was otherwise found before (46). And in general it is easy to see that the recent symbol denotes the right line, which is (in a well known sense) the polar of the point (x, y, z), with respect to the same given conic; or that the line [X', Y', Z'] is the polar of the point (x, y, z): because the equation
Xx' + Yy + Zz' = 0,
which for a conic may be written as X'x + Y'y + Z'z = 0, expresses (by 51) the condition requisite, in order that a point (x, y, z) of the curve should belong to a tangent which passes through the point (x', y', z'). Conversely, the point (x, y, z) is (in the same well-known sense) the pole of the line [X, Y, Z]; so that the centre of the conic, which is (by known principles) the pole of the line at infinity (38), is the point which satisfies the conditions aX=b1Y= c1Z; it is therefore, for the present conic, the point K = (b+c, c + a, a + b), of which the vector OK is easily reduced, by the help of the linear equation, aa +bB+cy = 0 (27), to the form,
with the verification that the denominator vanishes, by 48, when the conic is a parabola. In the more general case, when this denominator is different from zero, it can be shown that every chord of the curve, which is drawn through the extremity K of the vector κ, is bisected at that point K: which point would therefore in this way be seen again to be the centre.
53. Instead of the inscribed conic (46), which has been the subject of recent articles, we may, as another example, consider that exscribed (or circumscribed) conic, which passes through the three corners A, B, C of the given triangle, and touches there the lines AA", BB", Cc" of Fig. 21. The anharmonic equation of this new conic is easily seen to be,
yz + zx + xy = 0;
*If the curve f=0 were of a degree higher than the second, then the two equations above written would represent what are called the first polar, and the last or the line-polar, of the point (x, y, z), with respect to the given curve.
VECTOR OF A CUBIC CURVE.
the vector of a variable point p of the curve may therefore be expressed as follows,
and it is an ellipse, a parabola, or an hyperbola, according as the denominator of this last expression is negative, or null, or positive. And because these two recent vectors, κ, k', bear a scalar ratio to each other, it follows (by 19) that the three points o, K, K' are collinear; or in other words, that the line of centres KK', of the two conics here considered, passes through the point of concourse o of the three lines AA', BB', cc'. More generally, if L be the pole of any given right line ▲ = [1, m, n] (37), with respect to the inscribed conic (46), and if L' be the pole of the same line A with respect to the exscribed conic of the present article, it can be shown that the vectors OL, OL', or A, A', of these two poles are of the forms,
λ=k(laa+mbẞ + ncy), X' = k' (laa + mbẞ +ncy),
where k and k' are scalars; the three points o, L, L' are therefore ranged on one right line.
54. As an example of a vector-expression for a curve of an order higher than the second, the following may be taken :
with t + u+v=0, as before. Making x = 13, y = u3, z = v3, we find here by elimination of t, u, v the anharmonic equation,
the locus of the point P is therefore, in this example, a curve of the third order, or briefly a cubic curve. The mechanism (41)
of calculations with anharmonic co-ordinates is so much the same as that of the known trilinear method, that it may suffice to remark briefly here that the sides of the given triangle ABC are the three (real) tangents of inflexion; the points of inflexion being those which are marked as A", B′′, c′′ in Fig. 21; and the origin of vectors o being a conjugate point.* If a=b=c, in which case (by 29) this origin o becomes (as in Fig. 19) the mean point of the trian
branches, inscribed within the angles vertically opposite to those of the given triangle ABC, of which the sides are the three asymptotes.
55. It would be improper to enter here into any details of discussion of such cubic curves, for which the reader will naturally turn to other works. But it may be remarked, in passing, that because the general cubic may be represented, on the present plan, by combining the general expression of Art. 34 or 36 for the vector p, with the scalar equation
83=27kxyz, where s=x+y+z;
k denoting an arbitrary constant, which becomes equal to unity, when the origin is (as in 54) a conjugate point; it follows that if P = (x, y, z) and p' = (x, y, z) be any two points of the curve, and if we make s' = x + y + z', we shall have the relation,
Answering to the values t=1, u= 0, v= = 02, where 0 is one of the imaginary cube-roots of unity; which values of t, u, v give x = y = z, and p = 0.
Especially the excellent Treatise on Higher Plane Curves, by the Rev. George
Salmon, F. T. C. D., &c. Dublin, 1852.