A Rudimentary Treatise On Analytical Geometry And Conic Sections

This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1871 edition. Excerpt: ...nearer and nearer to the curve (2) the further it is prolonged from the origin, or centre of the curve; and that although the interval between the straight line and the curve (in the direction of?/) becomes eventually less than any finite interval that can be assigned--however small this may be, yet it can never wholly disappear; that is, the straight line can never actually meet the curve. Hence, if through the origin o two straight lines Kl, Mn, be drawn, of which the equations are b, h y =-x, and y =---x, a a that is, such that tan Kob =-, and tan Mob =---... (3), a a v' the latter angle being the supplement of the former, these two lines will continually approach nearer and nearer to tho curve the further they are prolonged, and yet can never meet the curve within any finite distance from o, in whichever direction from o that distance be measured; that is, whether the supposed point of meeting be (x, y), or (--x, y), or (as, --y), or (--x, --y). These two lines are called the asymptotes of the hyperbola; it is plain that they embrace both branches of the curve, and embrace them so closely that they form the boundaries of separation between all the transverse diameters of the curve, and all the conjugates to these: --every straight line drawn from o within the angle Kon, or within the angle Mol, meets the curve; whilst every straight line drawn from o, without both these angles, that is within the angle Mok, or Lon, proceeds onwards in both directions free of the curve. All these inferences are fully justified by the equations (1) and (2) at p. 116: we have sufficiently seen that the straight line or lines (1) can meet the curve (2) only at a point infinitely remote, or for which x = oo; but if (1) were replaced by the line y = mix, in...