The Elements of Coordinate Geometry, in Three Parts; I. Cartesian Geometry. II. Quaternions. III. Modern Geometry, and an Appendix

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 Excerpt: ... Since this equation does not contain x, it will be true for all corresponding values of y and Hence, dropping the subscripts to z and y and substituting in equation (1), gives for the required equation. Letting 2pA2 = d?, and dividing through by V6iW, it may te Put under the form (7) 236. in which a2, W, d, are used for the new denominators of z2, y2, x, respectively. The Hyperbolic Paraboloid may be generated by the movement of a parabola whose parameter so varies that its arc shall follow the opposite branches of an hyperbola, the vertex remaining at the pame point. Let the origin be at the vertex of the parabola, the plane zy be parallel to the plane of the hyperbola, and the axis of x pass through the centre of the hyperbola. If the equation to the hyperbola be a%x--bx2yi = ab2, we would find, in the same manner as shown in the pre via. 169. ceding Article, that the equation of the surface would be a2 V d-' but if the equation of the hyperbola be a?z?--b?yi =--«i26i2 (which is the conjugate of the preceding one), then we would find a? + ft2 d for the required equation. 237. Problem.--Required the equation of the surface generated by a right line moving parallel to a plane and along any other two right lines. Let one of the lines, as AB, lie in the plane xy, and the other in the plane xz. Take the origin on the line 0 C. Let the equation of the line OC be z--nx, z which is the required equation, and is an equation of the second degree between three variables. The character of the surface is not readily seen from this equation, but by a transformation of coordinates it may be reduced to the form Mz2-Ny2-Qx = 0; and hence it is an hyperbolic paraboloid, (Art. 236). It is a warped surface, (Art. 222). The same surface would be generated by the line OC...