(and therefore, likewise, a sin A + B sin B+ y sin C = 0), represents a right line situated altogether at an infinite distance from the origin. 65. We saw (Art. 63) that a line parallel to the line a = 0 has an equation of the form a + C = 0. Now the last Article shows that this is only an additional illustration of the principle of Art. 36. For, a parallel to a may be considered as intersecting it at an infinite distance, but (Art. 36) an equation of the form a + C = 0 represents a line through the intersection of the lines a = 0, C = 0, or (Art. 64) through the intersection of the line a with the line at infinity. 66. We have to add, in conclusion, that Cartesian co-ordinates are only a particular case of trilinear. There appears, at first sight, to be an essential difference between them, since trilinear equations are always homogeneous, while we are accustomed to speak of Cartesian equations as containing an absolute term, terms of the first degree, terms of the second degree, &c. A little reflection, however, will show that this difference is only apparent, and that Cartesian equations must be equally homogeneous in reality, though not in form. The equation x = 3, for example, must mean that the line x is equal to three feet or three inches, or, in short, to three times some linear unit; the equation xy = 9 must mean that the rectangle xy is equal to nine square feet or square inches, or to nine squares of some linear unit; and so on. If we wish to have our equation homogeneous in form as well as in reality, we may denote our linear unit by z, and write the equation of the right line Ax+ By + Cz = 0. Comparing this with the equation Aa + BB + Cy = 0; and remembering (Art. 64) that when a line is at an infinite distance its equation takes the form z = 0, we learn that equations in Cartesian co-ordinates are only the particular form assumed by trilinear equations when two of the lines of reference are what are called the co-ordinate axes, while the third is at an infinite distance. CHAPTER V. EQUATIONS ABOVE THE FIRST DEGREE REPRESENTING RIGHT LINES. 67. BEFORE proceeding to speak of the curves represented by equations above the first degree, we shall examine some cases where these equations represent right lines. If we take any number of equations, L = 0, M = 0, N = 0, &c., and multiply them together, the compound equation LMN, &c. = 0 will represent the aggregate of all the lines represented by its factors; for it will be satisfied by the values of the co-ordinates which make any of its factors = 0. Conversely, if an equation of any degree can be resolved into others of lower degrees, it will represent the aggregate of all the loci represented by its different factors. If, then, an equation of the nth degree can be resolved into n factors of the first degree, it will represent n right lines. 68. A homogeneous equation, of the nth degree between the variables, denotes n right lines passing through the origin. Let the equation be Let a, b, c, &c., be the n roots of this equation, then it is resolvable into the factors (-a)( - ) - c) &c. - 0, α and the original equation is therefore resolvable into the factors (x-ay) (x-by) (x − cy) &c. = 0. It accordingly represents the n right lines x ay = 0, &c., all of which pass through the origin. Thus, then, in particular, the homogeneous equation represents the two right lines x - ay = 0, x - by = 0, where a and b are the two roots of the quadratic 2 + q = 0. It is proved, in like manner, that the equation (x − a)" − p (x − a )n-1 (y − b) + q (x − a)n-2 (y - b)2. . . + t (y −b)" = 0 denotes n right lines passing through the point (a, b). Ex. 1. What locus is represented by the equation xy = 0? Ans. The two axes, since the equation is satisfied by either of the suppositions x = 0, y = 0. Ex. 2. What locus is represented by x2 - y2 = 0 ? Ans. The bisectors of the angles between the axes, x + y = 0 (see Art. 43). Ex. 3. What locus is represented by x2 - 5xy + 6y2 = 0 ? - 2y = 0, x- - 3y = 0. Ans. XEx. 4. What locus is represented by x2 - 2xy sec 0 + y2 = 0 ? Ans. x = y tan (45° ± 10). Ex. 5. What lines are represented by x2 - 2xy tan 0 — y2 = 0 ? Ex. 6. What lines are represented by x3- 6x2y + 11xy2 — 6y3 = 0 ? 69. Let us examine more minutely the three cases of the solution of the equation x2 - pxy + qy2 = 0, according as its roots are real and unequal, real and equal, or both imaginary. The first case presents no difficulty: a and b are the tangents of the angles which the lines make with the axis of y (the axes being supposed rectangular), p is therefore the sum of those tangents, and q their product. In the second case, when a = b, it was once usual among geometers to say that the equation represented but one right line (x - ay = 0). We shall find, however, many advantages in making the language of geometry correspond exactly to that of algebra, and as we do not say that the equation above has only one root, but that it has two equal roots, so we shall not say that it represents only one line, but that it represents two coincident right lines. Thirdly, let the roots be both imaginary. In this case no real co-ordinates can be found to satisfy the equation, except the coordinates of the origin x = 0, y 0; hence it was usual to say that in this case the equation did not represent right lines, but was the equation of the origin. Now this language appears to = us very objectionable, for we saw (Arts. 14, 15) that two equations are required to determine any point, hence we are unwilling to acknowledge any single equation as the equation of a point. Moreover, we have been hitherto accustomed to find that two different equations always had different geometrical significations, but here we should have innumerable equations, all purporting to be the equation of the same point; for it is obviously immaterial what the values of p and q are, provided only that they give imaginary values for the roots, that is to say, provided that p2 be less than 4q. We think it, therefore, much preferable to make our language correspond exactly to the language of algebra; and as we do not say that the equation above has no roots when p2 is less than 4q, but that it has two imaginary roots, so we shall not say that, in this case, it represents no right lines, but that it represents two imaginary right lines. In short the equation x2 - pxy + qy2 = 0 being always reducible to the form (x - ay) (x − by) = 0, we shall always say that it represents two right lines drawn through the origin; but when a and b are real, we shall say that these lines are real; when a and b are equal, that the lines coincide; and when a and b are imaginary, that the lines are imaginary. It may seem to the student a matter of indifference which mode of speaking we adopt; we shall find, however, as we proceed, that we should lose sight of many important analogies by refusing to adopt the language here recommended. Similar remarks apply to the equation Ax2 + Bxy + Cy2 = 0, which can be reduced to the form x2 - pxy + qy2 = 0, by dividing by the coefficient of x2. This equation will always represent two right lines through the origin; these lines will be real if B2-4AC be positive, as at once appears from solving the equation; they will coincide if B2 - 4AC = 0; and they will be imaginary if B2-4AC be negative. So, again, the same language is used if we meet with equal or imaginary roots in the solution of the general homogeneous equation of the nth degree. 70. To find the angle contained by the lines represented by the equation x2- pxy + qy2 = 0. - Let this equation be equivalent to (x − ay) (x − by) = 0, then the tangent of the angle between the lines is (Art. 40) the product of the roots of the given equation = q, and their dif COR.—The lines will cut at right angles, or tan φ will become 1 in the first case, or if A + C = 0 in the second. Ex. Find the angle between the lines x2 + xy - by2 = 0. x2-2xy sec0 + y2 = 0. Ans. 45°. * If the axes be oblique, we should find, in like manner, 71. To find the equation which will represent the lines bisecting the angles between the lines represented by the equation Ax2 + Bxy + Cy2 = 0. Let these lines be x ay = 0, x-by= 0; let the equation of the bisector be x Now μy = 0, and we seek to determine μ. (Art. 22) μ is the tangent of the angle made by this bisector with the axis of y, and it is plain that this angle is half the sum of the angles made with this axis by the lines themselves. Equating, therefore, tangent of twice this angle to tangent of sum, we get 2μ a + b = |