(i.) A quadric cone is a cone having any conic for base, and a point O on the line perpendicular to the plane of the conic through the focus for vertex.

(ii.) A sphero-conic is the curve in which any quadric cone cuts a sphere whose centre is the vertex of the cone.

In an introductory course on space-geometry, where only a passing mention of sphero-conics seems possible, I have found it useful to show directly the fundamental property that the sum or difference of the focal distances is constant, instead of following the classical method, which consists in deducing the property by the use of infinitesimals, from the fact that these focal distances make equal angles with the curve; this latter fact being in effect a direct consequence of Desargue’s theorem.

In a recent review of Prof. W. B. Smith’s “Infinitesimal Calculus,” fault is found with his statement that “In all strictness there can be no motion at an instant, and hence no speed (or velocity) at an instant. The concept of speed (or velocity) or motion will not combine with the concept of instant (or point of time) to form a compound concept.” Part of the reviewer’s criticism is as follows: “If we allow that motion at an instant is impossible, how are we to escape from Zeno’s paradoxical conclusion that all motion is impossible? How can I move from one place to another during a minute, say, if at every instant of that interval motion is impossible?”

Some idea of a system of such equations may be given in order to clear the way.

In Professor Chrystal’s Algebra, Chapter XIV., No. 18, p. 303, we have:

“Of the equations

with two similar equations in y, z, where x, y, z are unequal; prove that any two imply the third.” (Cayley.)