Mathematical Questions and Solutions, from the "Educational Times." Volume 20 (Paperback)

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. 1874 edition. Excerpt: ...of odd order have one real point at infinity besides the circular points, if of even order two which are at right angles or harmonic of the circular points. The intersections with the line infinity at the circular points are due to multiplicity of these points, not to contact with the line infinity. Now, first, the line infinity it a tangent to this envelope at each of the cireular points and no otherwhere. For the line infinity can only become an asymptote by the variable one point or one of the variable two po'nts at infinity coming to coincide with one of the circular points. In the second case the variable two points being harmonic of the circular points, if one of them coincide with a circular point, the other must coincide with it. In both cases, then, there are two curves of the pencil which have the line infinity for asymptote; and it is clear that the intersection of the line infinity with the next consecutive asymptote (i.e., its point of contact with the envelope) is the circular point at which it is an asymptote. Next, from any point at infinity not a circular point, one tangent distinct from the line infinity can be drawn to the envelope. For there is one curve of the pencil that passes through this point. If, then, the line infinity is an ordinary tangent at each of the circular points, we see that from any point at infinity three tangents may be drawn to the envelope; viz., the line infinity counting twice, and one other. The envelope is therefore of the third class, having the line infinity for double tangent whose points of contact are the circular points; that is to say, a hypocycloid of three branches. In fact, the tangential equation of the curve may be at once written down. Let i=0, j=0 be the equations to the circular...