This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1914 Excerpt: ... and.,, oN2 4/S2 4a 4(W--ab) (tana-tany3)2 = _-T = &2 (i) The angle included between the lines is a/3, whose tangent is tan a tan 3 2 V (/i2--ah)----or l+tanatan/3 b(l+a/b) Hence the included angle is-2-/(h2-ab) tan-1--a + b We note that the lines are parallel if tan a = tan 8, or h2--ab; they are perpendicular if a 8 = Tt, or tan (a 8) = oo; the condition for this is that a + b = 0. If h2--ab is negative the lines are not real. Note that these results will hold for any two straight lines whose joint equation is of the form ax2 + 2hxy + by2+px--qy+r = 0, since the directions of the lines are determined by the terms of the second degree in the joint equation (see § 58). (ii) If y = x tan 6 bisects the angle between ax2 + 2hxy + bf = 0, 6 = (x+/3), or (u + a + y3). In either case tan 2 0 = tan (a + /3) _ tana + tan/3 _ 2h _ 2h 1--tan a tan/3--b(l--a/b) a--b But if y) is a point on the bisector, tan 6 = y/x, and 2 tanf l _ 2y/x _ 2xy 1-tan2 0 1-2/a;2 x2-y2' Therefore 5---5 = z xi--yl a--b Hence the joint equation of the lines bisecting the angles between the lines ax2 + 2 hxy + by2 = 0 is x2--y2 _ xy a-b " h' Note I. If h--ab is negative, we have noted above that the lines are not real. Let ab = A' + A;2, another way of expressing the fact that h'--ab is negative. We may write the equation a'x2 + 2ahxy+aby' = 0, or atxt + Zahxy + tfyt + kty = 0, or (ax + hy) + (fey) = 0. But if the sum of two squares vanish, and the quantities are real, each of the two vanishes. Hence the only point satisfying the equation is that for which ax + hy = 0, ky = 0, that is, the origin itself. Note II. The expression for the tangent of the included angle involves a square root which may have either sign. The tangent will be positive or negative as the angle selected i...
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