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 equations 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. 67) that when a line is at an infinite distance its equation takes the form z=0, we learn that equations in Cartesian coordinates are only the particular form assumed by trilinear equations when two of the lines of reference are what are called the coordinate axes, while the third is at an infinite distance. 70. We wish in conclusion to give a brief account of what is meant by systems of tangential coordinates, in which the position of a right line is expressed by coordinates, and that of a point by an equation. In this volume we limit ourselves to what is not so much a new system of coordinates as a new way of speaking of the equations already in use. If the equation (Cartesian or trilinear) of any line be Xx+μy + vz=0, then evidently, if λ, μ, v be known, the position of the line is known; and we may call these three quantities (or rather their mutual ratios with which only we are concerned) the coordinates of the right line. If the line pass through a fixed point x'y'z', the relation must be fulfilled x'λ + y'μ + z'v=0; if therefore we are given any equation connecting the coordinates of a line, of the form aλ + bμ+ cv = 0, this denotes that the line passes through the fixed point (a, b, c), (see Art. 51), and the given equation may be called the equation of that point. Further, we may use abbreviations for the equations of points, and may denote by a, ẞ the quantities x'λ + y'u+z'v, x"λ+y′′μ+z′′v; then it is evident that la+mß=0 is the equation of a point dividing in K a given ratio the line joining the points a, B; that la=mß, mß=ny, ny = la are the equations of three points which lie on a right line; that a + kß, a – kß denote two points harmonically conjugate with regard to a, B, &c. We content ourselves here with indicating analogies which we shall hereafter develope more fully; for we shall have occasion to show that theorems concerning points are so connected with theorems concerning lines, that when either is known the other can be inferred, and often that the same equations differently interpreted will prove either theorem. Theorems so connected are called reciprocal theorems. - Ex. Interpret in tangential coordinates the equations used in Art. 60, Ex. 2. Let a, ẞ, y denote the points A, B, C; mẞ- ny, ny - la, la - mß, the points L, M, N; then mß + ny − la, ny + la − mß, la + mß - ny denote the vertices of the triangle formed by LA, MB, NC; and la + mẞ+ny denotes a point 0 in which meet the lines joining the vertices of this new triangle to the corresponding vertices of the original : mß + ny, ny + la, la + mẞ denote D, E, F. It is easy hence to see the points in the figure, which are harmonically conjugate. CHAPTER V. EQUATIONS ABOVE THE FIRST DEGREE REPRESENTING 71. 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 coordinates 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. 72. A homogeneous equation of the nth degree in x and denotes n right lines passing through the origin. y Let a, b, c, &c., be the n roots of this equation, then it is resolvable into the factors 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 It is proved, in like manner, that the equation (x − a)” – p (x − a)"−1 ( y − b) + q (x − a)"−2 (y — b)3...+ 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. 35). Ex. 3. What locus is represented by x2 - 5xy + 6y2=0? Ans. x-2y=0, x-3y=0. Ex. 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 ? 73. 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 ab, 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 coordinates 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 (Art. 14) 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 49, 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 + Cy3 = 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 B-4AC be positive, as at once appears from solving the equation; they will coincide if B" - 4A C=0; and they will be imaginary if B-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. 74. To find the angle contained by the lines represented by the equation x-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. 25) a-b 1+ ab |