As the equation of a straight line is of the first degree, so does a straight line meet a curve of the nth degree in ra points which may he either real or imaginary. Thus a straight line may cut a curve of the second degree in two points, a curve of the third degree in three points, and so on: and even if the straight line is at an infinite distance, yet it intersects a curve of the rath degree just as truly in ra points. The following theorem due to Newton and called now by his name, is such a simple application of this property, that I must insert it. Let us take the general equation (45), or its abridged form (48); let us replace x and y by Ir and mr; where r is the radius vector of a point on the curve from the origin, and I and in are projective coefficients: then we have an equation of the rath degree in terms of r, the roots of which are the n distances from the origin of the ra points where the curve is met by the straight Hue mx — ly = 0. Through the origin let two transversals be drawn, of which let the equations be m\X—l\y = 0 and mix—hy = 0; and let them meet the curve in the points Pi, p2,... p„, Qi, Qa,... Q«; then by the theory of equations Opi.op2 ... Op„ = ± hnan + hn-lmibn+_+knmi», and therefore by division QPi.QPg ,..op, _ hnan + hn~lmil>n+ ••• +knm2n oo.i.oo.2 ... Oq„ — hnan + h"-lm1bn + ...+knmf' and as both the numerator and the denominator of this ratio are unaltered by a change of origin, provided that the directions of the lines Op and Oq are the same, so we infer that if from a point in the plane of a curve two transversals are drawn in given directions, the ratio of the product of the segments of the one taken from the given point to the product of the segments of the other is the same whatever is the place of the point. We have also another important theorem first given by Cotes in his Harmonia Mensurarum. Let each radius vector Op drawn from o cut the curve in the points vu p2,... p„; and let r be taken such that _L_ J_ 1 then the locus of p will be a straight line. For substituting Ir and tnr for x and y in (45), and dividing through by r" we have, «o ail + h m ^ + rn-\ + — U' 1 1 1 ai / + bx m opj + Op2 + "' + Op„ — flo' and therefore by the given condition, if Op = r, and r refers to the new locus, »,»._' r ~ «o .-. a\X + biy + n«0 = 0, which is the equation to a straight line. 211.] As a curve of the nth degree has n ^ determinable constants in its equation, so may these be determined if a sufficient number of independent conditions is given. Hereafter we shall meet with many different ways in which they may be determined. It is evident however that they may generally be found, if the coordinates of - ^n^~^ points through which the curve is to pass are given, because in this case we may replace successively x and y in equation (45) by the given coordinates, and we shall have n linear equations, by means of which the arbitrary constants may be determined. This is clearly a definite and generally an unique problem: and thus only one curve of the nth degree can be drawn passing through the assigned points. Hence a curve of the second degree may pass through five points; a curve of the third degree through nine points; a curve of the fourth degree through fourteen points; and so on. If however the points have certain relative positions, it may be that a proper curve of the nth degree cannot be drawn through them. That is, although we have shewn that the coefficients of the algebraical equation of the nth degree may be found in terms of the coordinates of the given points, yet the resulting equation may be susceptible of resolution into factors of lower degrees; so that the result may be the combination of two or more curves of lower degrees, the sum of which is equal to n; and thus not be the equation of a proper curve of the nth degree. Thus if n = 2, the number of points which is sufficient for the determination of the constants is 5; but if three of these are in a straight line, no conic can be drawn which shall pass through them, because a conic cannot cut a straight line in more points than two; and thus the only equation of the second degree which can satisfy such a system of points is that composed of two straight lines passing respectively through the three and the other two points. determination of the curve; that is, suppose a curve of the nth Generally of the number of points which are sufficient for the determination of a proper curve of the nth degree not more than nm can be on a curve of the mth degree; because two curves of the nth and mth degrees respectively cannot intersect in more than nm points. And generally too the number of points which may be on a curve of the mth degree, so that a proper curve of the nth degree may pass through them, is less than win. For let us, for instance, suppose the curve of the nth degree to be made up of two curves of the mth and (n — m)th degrees respectively. Under any circumstances the possibility of such a resolution requires one condition amongst the constants, so that the number of coefficients yet remaining to be determined is - — — 1. For the determination of the curve may manifestly be on the curve of the mth degree. Besides this case, wherein the resulting equation admits of resolution into other equations of lower degrees, it may be that the assigned number of points will not yield a definite result. For if the values of a certain number of unknown quantities are determined by means of an equal number of linear equations of the form (45), each will be expressed by a fraction; and the constants of the equation may be such as to make both the numerator and the denominator of any one or more to vanish, in which case the quantities are indeterminate; and thus the curve is not determined; and the number of curves passing through the given points may be infinite. Thus, suppose the number of points through which a curve is to pass to be less by one than that required for the complete 71 (tl 4- 3) degree to pass through = 1 different points; let u = 0 and v = 0, be two equations of the »th degree of the form (45), which represent two curves passing through these - ^M_^ ^ — 1 points; then, if k is an undetermined constant, u + kv = 0 is the equation to a curve of the nth degree passing through all the points of intersection of u = 0 and of v = 0, and therefore clearly passing through the - ^n^^ _ l given points; and as k is undetermined, the number of curves of the nth degree passing through these is also indeterminate. Now let us suppose the curve u + kv = 0 to pass through another point, so that it becomes completely determined; then if u and v' are the values of u and v, when x and y are replaced by the coordinates of this last point, we have u + kv = 0, whence we know k, and the curve is completely determined. There is however one case in which k will not have the required determinate value; and that is when the last point, through which the curve is to pass, and by means of which k is determined, is the point of intersection of M = 0 and v = 0, because in this case u = 0 and v = 0, and k takes an indeterminate form. This is also manifest from the equation u + kv = 0; for this curve miinifestly passes through all the points of intersection of u = 0 and v = 0; that is, through n2 points, and yet is not determinate, because its equation contains an undetermined constant k. Hence we conclude that all curves of the nth degree, which pass through U ^ — 1 given points, also pass til through as many others as n2 is in excess of this number; that is, pass through ———— other fixed points. And of this equation we have the following particular cases. All curves of the third degree which pass through the same eight points also pass through a common ninth point. And thus if eight points are given, through which a curve of the third degree is to pass, the curve is not determined if the ninth point is the other of the intersection of the two curves u = 0, v = 0, of the third degree. All curves of the fourth degree which pass through the same Price, Vol. i. u u 'S'SO GENERAL PROPERTIES OK ALGEBRAICAL CURVES. ['212. thirteen given points also pass through three other given points. Hence also it follows that W ^ given points are not always sufficient to determine a curve of the nth degree, and that one other additional point at least may be required for the purpose. It is to be observed that these results are true, whether the equations of the nth degree represent proper curves of the nth degree, or systems of factors of lower degrees. Thus if the constants of an equation of the second degree are to be determined by making the curve pass through five given points, of which four are in one straight line, the resulting equation will be composed of two simple factors of the first degree, but all the arbitrary constants in it cannot be determined, because the fifth point is not sufficient to determiue the position of the second straight line. 212.] Thus although two curves of the nth degree intersect in «2 points, yet any riL points taken arbitrarily may not be the points of intersection of two curves of the nth degree: . n(n + 3) , .. , , . . , . . but ——i 1 ot them being given, the remainder, viz. ———~ wi"" oe fixed- Similarly two curves of the with and wth degrees respectively intersect in mn points; but tun points taken arbitrarily on the curve of the with degree will not be the points of intersection of it with the curve of the nth degree. Generally, however, every curve of the wth degree which , . (w» — 1) (m—2) . . passes through urn— points on a curve ot the ii ■■■ i ii (wj — 1) (wi—2) , . mm degree, will also pass through „ other and fixed points on that curve; because this number of points is, see Art. 211, less by one than the number which is required for the absolute determination of the curve; and therefore these points are the points of intersection of all curves of the nth degree which satisfy the required conditions. An extension of several of the preceding properties of algebraic curves will be found in a memoir by Mr. A. Cayley, Fellow of Trinity College, Cambridge, in Vol. Ill of the Cambridge Mathematical Journal, p. 211. |