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of the «th degree, aud of the form (45), Art. 207; then each of the equations (131) is of n —1 dimensions, and therefore represents a curve of that degree; and if the points of intersection of these two curves are on the original curve, they axe the double points of that curve: hereby we have a limit to the number of double points which can be on a curve. The two curves (131) cannot intersect in more than (» —l)2 points, and therefore the number of double points cannot be greater than this; it may however be less because all the points of intersection may not be on the original curve: thus, in a conic for
example, (—jp = 0, and (~^) = 0, are the equations to two
diameters, and they intersect in the centre which is not on the curve, unless the conic can be broken up into two straight lines; in which case the centre is a double point. Thus the point of intersection of the two equations (131) in the case of a conic is not ordinarily a double point.
As two branches of a curve intersect at a double point, so a double point is two coincident points; and therefore if a straight line passes through a double point, this passage is equivalent to its passing through two points of a curve: hence a proper curve of the third degree cannot have more than one double point, because, if it had two, a straight line might be drawn through both, and would thus pass through four points on the curve of the third order, and this is impossible for a proper curve of that order, see Art. 210; for a proper curve, I say; because it is possible, if the curve can be broken up into a straight line and a conic, since then the two points where the line cuts the curve are double. Similarly a proper curve of the fourth order cannot have four double points; because if it had, through these and any fifth point on the curve a conic might be drawn, and as each double point is equivalent to two points, this conic would pass through nine points on the curve; and thus a conic and a curve of the fourth order would intersect in more than eight points, which is impossible, see Article 210. And in general a curve of
the nth degree cannot have more than ———— double points*; for suppose it to have one more: then through these
* See Salmon, Higher Plane Curves, page 31; and Annates de Mathematiques, Tome X, p. 95, Paris, 1851.
+ 1 double points and through n — 3 other points
of the curve, a curve of the («—2)th order may be drawn; and this must be considered to meet the original curve in (n — 1) (n—2) +2 +» — 3, that is, in n(ra—2)+l points, which is impossible if the given curve is a proper curve.
In Art. 223 it has been shown that generally n(n — 1) tangents can be drawn to a curve of the nth degree from a given point: in the proof of this theorem it is assumed that every line passing through two consecutive points on the curve is a tangent; a line however passing through a double point passes through two consecutive points on the curve and in two different ways, viz. through the point common to both branches and the consecutive point on each branch; and yet is not in either way a tangent in the meaning in which the word is commonly used; so that the preceding number of tangents must be diminished by 2 for every double point on the curve. If therefore a curve has 8 double points, the number of tangents which can be drawn to it from a given point is n(« —1)—28; and this is therefore the class of the curve.
252.] The number of cusps on a curve of the «th degree cannot be greater than 2ra (n—2); and all these are on a curve of the 2(«—2)th order. For the condition of a cusp on a curve is, see (127), the equation
which is manifestly of 2(»—2) dimensions; all cusps therefore are found on a curve of this order: and as the points of intersection of this curve with the original curve cannot be more than 2«(n —2), so the number of cusps on a curve of the »th order cannot be greater than 2n (n—2). Indeed, as a cusp is a
double point, the number cannot be greater than ———— » as is shewn in the preceding Article.
And for every cusp on a curve, the number of tangents which can be drawn to the curve from a given point must be diminished by 3. Because at a cusp three points of the curve become coincident; and as these may be taken in pairs in three different ways, so may three lines be drawn passing through these several pairs; and as these lines are not tangents in the
ordinary meaning of the word, the number of tangents which can be drawn to the curve from a given point is to be diminished by three for each cusp.
Hence if a curve has K cusps, the number of tangents which can be drawn to it from a given point, and therefore the class
of the curve, is . „ ,100,
«(» — 1) — 3 K. (133)
If therefore a curve has 5 double points, and K cusps, and if its class is m; then
m = «(w-1)-28-3k. (134)
253.] From (126) it appears that the equation which gives the directions of the tangents at a double point is
hence substituting -~ from (124), we have
which is the Hessian of the curve in terms of x and y, see Art. 246, and is the same equation as that which gives the points of inflexion; see Art. 244. Hence it appears that the Hessian passes through all the double points of a curve as well as through all its points of inflexion.
* Now the relation of a curve to its Hessian deserves closer consideration. In the first place, a double point on a curve is also a double point on the Hessian; and the tangents to the two are identical at the point. This may be shewn most easily if we take the double point to be the origin and the tangents at it to be the coordinate axes; in which case the equation (48), Art. 207, takes the form
b2xy + ua + ui + ...+«„ = v(x,y) = 0; (137)
and substituting in (136) and taking only the terms involving the lowest powers of x and y, the Hessian becomes
— 2b23xy + terms of three and higher dimensions — 0; (138)
which evidently passes through the origin, and at it touches the axes of x and y. Thus the original curve and its Hessian have the same tangents at the common double point. But when two
* See an Article by Mr. Cayley in Crelle's Journal, Vol. XXXIV, p. 43.
curves have a common double point, this is equivalent to four points of intersection being coincident; and if the tangents to the branches are also common, this is equivalent to two more points of intersection being coincident; and therefore a double point with tangents common to the curve and its Hessian is equivalent to six coincident points of intersection. Now the number of points of inflexion is 3ra (n—2), being the number of points of intersection at a finite distance of a curve and its Hessian; and by as many as these points are diminished, by so many is the number of the points of inflexion diminished. Therefore if a curve has 8 double points, the number of its points of inflexion cannot be more than
Also again, if the original curve has a cusp, three branches of the Hessian will pass through the point, the tangents of two of which will coincide with the tangent of the original curve at the cusp. Let us take the origin at the cusp, and let the tangent be the axis of y: in which case (48), Art. 207, takes the
orm' aix* + us + ui+ ... +un = F(3\y) = 0; (140)
and substituting in the Hessian, and retaining only the terms involving the lowest powers of the variables x and y, the Hessian becomes
^afx1 (~7^f) + terms of four and higher dimensions = 0. (141)
Hence the Hessian has three branches passing through the origin; to two of which the axis of y is the tangent, and to the
third the tangent is (^^) = 2c3a? + 6^y = 0. But when two
curves have a common point, through which two branches of one and three branches of the other pass, this point is equivalent to six coincident points of intersection; and as two branches of one have the same tangent as two branches of the other, two more points are common to the two curves; so that this cusp common to the curve and its Hessian is equivalent to eight coincident points of intersection. Hence if a curve has 8 double points, and K cusps, and if t is the number of its points of inflexion, i = 3n(»-2)-68-8(c. (142)
I must in conclusion observe, that M. Hesse has shewn, Crelle, Vol. XXVIII, that in curves of the third order, the points of inflexion on the original curve are also points of inflexion on the Hessian: but in other curves the points of inflexion are not generally points of inflexion of the Hessian.
254.] Lastly, let us consider the subject of double points iu curves whose equations are given in terms of three variables x, y, z, and in the form (49), Art. 208. Now the equation to the tangent, see (18), Art. 216, is
and therefore if at any point on the curve this equation is iden
tion of the tangent at that point is indeterminate; and to determine it we must pass to the next consecutive point on the curve: that is, in other words, we must take the x-, the y-, and the z-partial differentials of (143): hence we have the three equations (116), Art. 246; and hence arises the condition,
/d2F\ I d2F \2 id2F\ I d2F \2 id2F\ I d2F \2
which is the equation to the Hessian in terms of three variables. And this is the condition that a curve should have a double point. Let us apply it to a conic; that is, investigate the circumstances under which a conic has a double point. Let the equation to the conic be
AX2 + By2-\ cz2+%Eyz + 2azx + 2Hxy = 0; (145) then (144) becomes
which is the well-known condition that (145) should break up into two straight lines; and at the point of intersection of these two lines we have the characteristics of a double point. Therefore a proper curve of the second degree does not admit of a double point.
255.] Let us now return to equation (125), Art. 248, and