The Elements of Plane Analytic Geometry

The Elements of Plane Analytic Geometry

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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. 1903 edition. Excerpt: ...and (xzfys) any point which does not lie on either of the curves (5) or (6). The statement we have made will be proved if we can show that (xl9yl) and (#a, jya) both lie on (4), but (xz, yj does not. In order to find out whether (xyj lies on (4) or not, we must replace the variables (x, y) in (4) by the constant values (xy and then see whether the equation is fulfilled. In doing this we must remember that u is merely an abbreviation for a certain expression in the variables x and y. This expression will therefore take on a certain constant value, which we will call ul9 when these variables are replaced by the constants xiy yl. In the same way the expression v takes on a constant value vx when x, y are replaced by xt, yv Thus equation (4) takes on, after this substitution, the form x - = o (7) and it remains to be proved that this is a true equation. By hypothesis (xl9y1) lies on the curve (5). Accordingly its coordinates satisfy (5) and we have and from this the truth of (7) follows. We have thus proved that (xiy yx) lies on (4), and by precisely the same method we show that (#a, jy9) does so. In order finally to show that (#3, yt) does not lie on (4) let us indicate by uz the value of the expression u when the variables x, y are replaced by the constants x3, y2, and by v% the value of v after the same substitution. The result of substituting xz, yt in (4) is then.s =.-(8) This equation, however, is not true, since, the point (x99yz) lying on neither (5) nor (6), neither u3 nor v% is zero. The result of this section may be stated by saying that if we have two curves whose equations are so written that the righthand members are zero, the two curves may be represented by a single equation obtained by multiplying these equations together. Tshow more

Product details

  • Paperback | 48 pages
  • 189 x 246 x 3mm | 104g
  • Miami Fl, United States
  • English
  • black & white illustrations
  • 1236546423
  • 9781236546425