The Messenger of Mathematics Volume 2 (Paperback)

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. 1864 edition. Excerpt: ...of the curve. First, let K be negative: then the cusp is on the side of the vertex without the triangle of reference, and the asymptote parallel to BG falls within the triangle. The curve must therefore be that drawn in fig. 46, or an orthogonal projection of it. Secondly, when K increases and takes the value zero the curve becomes rectilinear and coincides with the asymptotes, two of which close up and coincide with the diameter while the third cuts it at the point A. Thirdly, if K be positive and less than 4 there is only one asymptote which is parallel to BG, and without the triangle of reference. The curve consists of-two branches meeting to form the cusp and both approaching the same asymptote as in figure 47. Fourthly, let - = 4. This case only differs from the last in that the asymptote has moved off to an infinite distance. The curve takes the form of the semi-cubical parabola, or a projection of it, (fig. 48.) Fifthly) let #4. Again we have three rectilinear asymptotes. The asymptote parallel to BG is still without the triangle of reference but now on the opposite side, beyond A. The curve will have the form of fig. 49, orthogonally projected, and is of the same character as when K is negative, (fig. 46) except that it is differently situated with respect to the triangle of reference BC, being in this case a chord of the cusp branches in the other of the hyperbolic branches. It is easy to see that figures 46 and 49 present precisely the same curves and not merely curves whose general characters are similar. For consider the curve of fig. 49. Select any chord B'C of the hyperbolic branches parallel to the asymptote, and suppose the curve were referred to the triangle AB'C. Then all the necessary conditions are fulfilled, in order...