known quantities, a, a', B, B', in terms of the coefficients of the general equation; and then these values being substituted in the fifth give the condition required. The five equations actually From the first four we can at once form two quadratic equations for determining a, a', ß, ß', as indeed we might have otherwise inferred from the consideration that these quantities are the reciprocals of the intercepts made by the lines on the axes; and that the intercepts made by the locus on the axes are found (by making alternately x = 0, y = 0, in the general equation) from the equations Cy2+ Ey + F = 0. Ax2+ Dx + F = 0, Now if the locus meet the axes in the points L, L'; M, M'; it is plain that if it represent right lines at all, these must be either the pair LM, L'M', or else LM', L'M, whose equations are (ax + By − 1) (ax + ẞ'y − 1) = 0, or (ax + ẞ'y − 1) (a'x + By − 1) = 0. B Multiplying out, we see that might not only have the value F given before aẞ' + Ba', but also might be aß + a'ß'. The sum of those quantities DE F which, cleared of fractions, is the condition already obtained. Ex. To determine B so that x2 + Bxy + y2 − 5x − 7y+ 6 = 0 may represent right lines. The intercepts on the axes are given by the equations whose roots are x = 2, x = 3; y = 1, y = 6. Forming, then, the equation of the lines joining the points so found, we see that if the equation represent right lines, it must be of one or other of the forms (x + 2y − 2) (2x + y − 6) = 0, whence, multiplying out, B is determined. (x + 3y-3) (3x + y − 6) = 0, * 74. To find how many conditions must be satisfied in order that the general equation of the nth degree may represent right lines. We proceed as in the last Article; we compare the general equation, having first by division made the absolute term = 1, with the product of the n right lines (ax + By − 1) (a'x + B'y − 1) (a′′x + ẞ”y − 1) &c. = 0. Let the number of terms in the general equation be N; then from a comparison of coefficients we obtain N - 1 equations (the absolute term being already the same in both); 2n of these equations are employed in determining the 2n unknown quantities a, a', &c., whose values being substituted in the remaining equations afford N-1-2n conditions. Now if we write the general equation A + Bx + Cy + Dx2 + Exy + Fy2 + Gx3 + Hx2y + Kxy2 + Ly3 + &c., it is plain that the number of terms is the sum of the arithmetic N-1-(n+3); N-1-2n = n(n − 1) 1 2 = 1.2 *75. To find how many conditions must be fulfilled in order that the general equation of the nth degree should represent n right lines, each passing through a given point. We should now compare the general equation with the equation {y-y-m(x-x')} {y -y" - m'(x-x')} &c. = 0. There are now but the n unknown quantities, m, m', &c., to be determined; hence the number of conditions is * 76. To find the number of conditions which must be fulfilled in order that the general equation may represent n right lines, all passing through the same point. We now compare the general equation with - {y − y' – m(x − x')} {y — y' — m' (x − x')} &c. = 0. Beside the n unknowns m, m', &c., there are also the two x'y' to be determined; hence the number of conditions 77. BEFORE we proceed to the general investigation of the curves represented by the general equation of the second degree, it seems desirable that we should examine the equation of the circle, which ranks next to that of the right line in simplicity. To find the equation of the circle whose centre is the point (ab) and radius is r. Expressing (Art. 5) that the distance of any point from the centre is equal to the radius, we at once obtain the equation (x − a)2 + (y − b)2 = r2. If the axes be oblique, we have (Art. 6) (x − a)2 + (y − b)2 + 2 (x − a) (y – b) cosw = p2; but we shall seldom use oblique axes in questions relating to circles. COR. 1.—The equation to rectangular axes of the circle whose centre is the origin is x2 + y2 = r2. COR. 2.-Let the axis of x be a diameter, and the axis of y a perpendicular at its extremity, then the co-ordinates of the centre are obviously (r, 0), and on substituting these values for a and b, the equation of the circle becomes x2 + y2 = 2rx. The two forms just mentioned are the simplest which the equation of the circle can be made to assume by a particular choice of axes; and are those which most frequently occur in practice. 78. By comparing the equations found in the last Article with the general equation of the second degree, Ax2 + Bxy + Cy2+ Dx + Ey + F = 0, we can ascertain the conditions that this latter equation should represent a circle. If the axes be rectangular, it is evident that B must = 0 and A = C, in order that when we divide by A the equation may be capable of being put into the form (x − a)2 + (y − b)2 = r2, or x2 + y2 − 2ax − 2by + a2 + b2 − p2 = 0. - If the axes be oblique, we must compare the general equation with the equation (x − a)2 + (y − b)2 + 2 (x − a) (y − b) cosw = r2, and we find that in this case the general equation will represent B a circle, if A = C, and = 2 cosw. If the general equation of the second degree, referred to rectangular axes, fulfil the conditions B = 0, A = C, we can find the radius of the circle represented by it, and also the co-ordinates of its centre, thus fully determining the circle, both in magnitude and position; for, comparing the equations, y2 4A2 The rule, then, for bringing the equation of any circle to the form (x − a)2 + (y - b)2 = r2, may be expressed as follows: "By division make the coefficients of x and y = 1, transpose F, and then complete the squares by adding to both sides the sum of the squares of half the coefficients of x and y.” Ex. 1. Find the co-ordinates of the centre and the radius of Since F does not occur in the values just found for the coordinates of the centre, we learn that two circles will be concentric if their equations only differ in their constant term. 79. We consider in this Article the effect of two or three particular suppositions on the general equation. (1.) If F = 0 the origin is on the curve. For then the equaX tion is satisfied by the values x = 0, y = 0; that is, by the co-ordinates of the origin. The same argument proves that if an equation of any degree want the absolute term, the curve represented by passes through the origin. it (2.) If D2 + E2 = 4AF; it appears from Art. 78 that the radius of the circle vanishes, and that the equation may be reduced to the form It is plain, that this equation can be satisfied by the co-ordinates of no point save those of the point (x = a, y = b); hence it has been common to say, that the equation just written is the |