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...

*"synopsis" may belong to another edition of this title.*

(No Available Copies)

If you know the book but cannot find it on AbeBooks, we can automatically search for it on your behalf as new inventory is added. If it is added to AbeBooks by one of our member booksellers, we will notify you!

Create a Want