where there are six rows and five columns, and the determinant formed according to the rules of multiplication must vanish. But this is 1, 0 (12)2, (13)2, (14)2, (15)2 = = 0, where (12), &c. denote the lengths of the common tangents to each pair of circles. If we suppose the circle 5 to touch all the others, then (15), (25), (35), (45), all vanish, and we get, as a particular case of the above, Dr. Casey's relation between the common tangents of four circles touched by a fifth, in the form Ex. 6. Relation between the angles at which four circles whose radii are r, r', r”, intersect. If the circler have its centre at the point 1 in Ex. 4, r' at 2, &c. we may put for 122 = r2 + p22 — 2rr' cos 12, &c. in the determinant of that example which becomes then 1, r2+2 -2rr' cos12, \1, r2+r'”2_2rr""'cos14,r'2+r' -2r'r'"'cos24, r''2+p'''2_2r"p""'cos34, ""cos43 = 0, subtracting from each row and column the first multiplied by corresponding square If in this we let cos 21 = cos 31 = cos 41 = cos 0, we have the quadratic in mentioned at the end of Art. 132 e. CHAPTER X. PROPERTIES COMMON TO ALL CURVES OF THE SECOND DEGREE, DEDUCED FROM THE GENERAL EQUATION. 1 133. THE most general form of the equation of the second degree is ax2 + 2hxy + by2 + 2gx + 2fy + c = 0, where a, b, c, f, g, h are all constants. It is our object in this chapter to classify the different curves which can be represented by equations of the general form just written, and to obtain some of the properties which are common to them all.* Five relations between the coefficients are sufficient to determine a curve of the second degree. For though the general equation contains six constants, the nature of the curve depends not on the absolute magnitude, but on the mutual ratios of these coefficients; since, if we multiply or divide the equation by any constant, it will still represent the same curve. We may, therefore, divide the equation by c, so as to make the absolute term = 1, and there will then remain but five constants to be determined. Thus, for example, a conic section can be described through five points. Substituting in the equation (as in Art. 93) the coordinates of each point (x'y') through which the curve must pass, we obtain five relations between the coefficients, which will enable us to determine the five quantities, α &c. 134. We shall in this chapter often have occasion to use the method of transformation of coordinates; and it will be useful * We shall prove hereafter, that the section made by any plane in a cone standing on a circular base is a curve of the second degree, and, conversely, that there is no curve of the second degree which may not be considered as a conic section. It was in this point of view that these curves were first examined by geometers. We mention the property here, because we shall often find it convenient to use the terms "conic section," or "conic," instead of the longer appellation, "curve of the second degree." to find what the general equation becomes when transformed to parallel axes through a new origin (x'y'). We form the new equation by substituting x+x' for x, and y+y' for y (Art. 8), and we get a (x+x')3+2h(x+x') (y+y′)+b(y+y')2+2g (x+x')+2ƒ (y+y')+c=0. Arranging this equation according to the powers of the variables, we find that the coefficients of x, xy, and y', will be, as before, a, 2h, b; that the new c, c' = ax" + 2hx'y' + by" +2gx′ +2ƒy' +c. Hence, if the equation of a curve of the second degree be transformed to parallel axes through a new origin, the coefficients of the highest powers of the variables will remain unchanged, while the new absolute term will be the result of substituting in the original equation the coordinates of the new origin.* 135. Every right line meets a curve of the second degree in two real, coincident, or imaginary points. This is inferred, as in Art. 82, from the fact that we get a quadratic equation to determine the points where any line y = mx + n meets the curve. Thus, substituting this value of y in the equation of the second degree, we get a quadratic to determine the x of the points of intersection. In particular (see Art. 84) the points where the curve meets the axes are determined by the quadratics ax2 + 2gx+c=0, by2+2fy + c = 0. An apparent exception, however, may arise which does not present itself in the case of the circle. The quadratic may reduce to a simple equation in consequence of the vanishing of the coefficient which multiplies the square of the variable. Thus xy+2y2+x+5y + 3 = 0 is an equation of the second degree; but if we make y = 0, we get only a simple equation to determine the point of meeting of the axis of x with the locus represented. Suppose, however, that in any quadratic Ax+2Bx+ C=0, the coefficient C This is equally true for equations of any degree, as can be proved in like manner. T vanishes, we do not say that the quadratic reduces to a simple equation; but we regard it still as a quadratic, one of whose Now this quadratic roots is 0, and the other x= = 2B + A= 0; and we see by parity of reasoning that, if A vanishes, we ought to regard this still as a quadratic equation, one of whose roots is same thing follows from the general solution of the quadratic, which may be written in either of the forms. x= B±√(BAC) с = · B = √(B2 — A C) ; the latter being the form got by solving the equation for the reciprocal of x, and the equivalence of the two forms is easily verified by multiplying across. Now the smaller A is, the more nearly does the radical become + B; and therefore the last form of the solution shows that the smaller A is, the larger is one of the roots of the equation; and that when A vanishes we are to regard one of the roots as infinite. When, therefore, we apparently get a simple equation to determine the points in which any line meets the curve, we are to regard it as the limiting case of a quadratic of the form 0.x + 2Bx + C=0, one of whose roots is infinite; and we are to regard this as indicating that one of the points where the line meets the curve is infinitely distant. Thus the equation, selected as an example, which may be written (y+1) (x + 2y + 3) = 0, represents two right lines, one of which meets the axis of x in a finite point, and the other being parallel to it meets it in an infinitely distant point. In like manner, if in the equation Ax+2Bx + C=0, both B and vanish, we say that it is a quadratic equation, both of whose roots are x = 0; so if both B and A vanish we are to say that it is a quadratic equation, both of whose roots are x = ∞. With the explanation here given, and taking account of infinitely distant as well as of imaginary points, we can assert that every right line meets a curve of the second degree in two points. 2 136. The equation of the second degree transformed to polar coordinates is (a cos❜0+2h cos✪ sin 0 + b sin30) p2 + 2 (g cos 0 + f sin 0) p+c=0; and the roots of this quadratic are the two values of the length of the radius vector corresponding to any assigned value of 0. Now we have seen in the last article that one of these values will be infinite, (that is to say, the radius vector will meet the curve in an infinitely distant point,) when the coefficient of p2 vanishes. But this condition will be satisfied for two values of e, namely those given by the quadratic a + 2h tan 0 + b tan'0 = 0. Hence, there can be drawn through the origin two real, coincident, or imaginary lines, which will meet the curve at an infinite distance; each of which lines also meets the curve in one finite point whose distance is given by the equation 2 (g cose +f sine) p + c = 0. 2 If we multiply by p3 the equation a cos*0+2h cose sin 0+b sin*0=0, and substitute for p cose, p sine their values x and y, we obtain for the equation of the two lines ax2+2hxy + by" = 0. There are two directions in which lines can be drawn through any point to meet the curve at infinity, for by transformation of coordinates we can make that point the origin, and the preceding proof applies. Now it was proved (Art. 134) that a, h, b are unchanged by such a transformation; the directions. are, therefore, always determined by the same quadratic a cos'0+2h cos e sine+b sin❜0 = 0. Hence, if through any point two real lines can be drawn to meet the curve at infinity, parallel lines through any other point will meet the curve at infinity.† *The following processes apply equally if the original equation had been in oblique coordinates. We then substitute mp for x, and np for y, where m is and n is sin sin w This indeed is evident geometrically, since parallel lines may be considered as passing through the same point at infinity. |