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. 1894 Excerpt: ...in four points at B; one is accounted for by cutting the other branch; three are to be accounted for by contact with its own branch. Moreover, z(x + z)=x hence 2 and x + z must have the same sign in the immediate neighbourhood of B. Considering the curve as a whole, y2z(x+z)=x, hence the curve can be only in certaiu ...
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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. 1894 Excerpt: ...in four points at B; one is accounted for by cutting the other branch; three are to be accounted for by contact with its own branch. Moreover, z(x + z)=x hence 2 and x + z must have the same sign in the immediate neighbourhood of B. Considering the curve as a whole, y2z(x+z)=x, hence the curve can be only in certaiu divisions of the plane, viz., those in which 2 and x+z have the same sign. F1r..' 33. Collecting these results in a diagram, shading parts of the plane in which the curve cannot lie (Fig. 33), and noting that the curve cannot cross any of the lines x = 0, y = 0, 2 = 0, x + z=0, except at B, C, it is at once evident that the parts of the curve are joined as indicated by the dotted lines, the two ends P joining through infinity, and similarly for the two ends Q. If it be required to determine more exactly how this junction is effected, the asymptotes must be found by means of the equation of the line infinity. Ex. 2. xi+xtyz-y3z = 0. This passes through B, C. For the shape at B, y = l, xi+x2zz=0; hence z = x, and there is a point of undulation at B, with BA as tangent. For the shape at C, 2 = 1, xi +.v2y-y3 = 0; hence there is a triple point at C, the tangents being y = 0, x y = 0. Writing the whole equation in the form zy(y2-x2) = x, it is seen that so long as z is positive, yy2--x2) must be positive, hence the three tangents at the triple point divide the plane into regions, from the alternate ones of which the curve is excluded. To obtain more information as to the curve, take lines through any vertex of the triangle, for example, A. Any such line y=iz meets the curve where If p. be positive, this gives two real values for x: z, positive and negative; if fi be negative, =--v, the equation is xi-vx2z2 + v3z = 0. This gives four real values ...
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