The necessary and sufficient condition that two generators of the should be at right angles is that their point of intersection should lie on the director sphere x2 + y2 + z2- = a2 + b2 – c2. Let a2 + b2 – c2 = r2 and a2 > b2. Then if r2 > a2, b2 > c2, and in this case the curve of intersection of the H1 and sphere consists of two closed ovals lying on opposite sides of the plane XOY. Thus every generator of the H1 meets the sphere in two real points. This result is easy to obtain analytically. For the generator

meets the sphere in points given by

This note deals with methods of determining the lengths and direction-cosines of the axes. If the lengths are 2α. and 2β, then α–2 and β–2 are the two non-zero roots of the discriminating cubic for the cylinder, and thus are easily found. The following methods apply geometrical considerations and determine both the lengths and the direction-cosines.

The following method of cutting off an nth part of a given straight line requires besides a ruler only a pair of dividers or other means of laying off on a given straight line a segment equal to the distance between two given points. In other words, we assume that, besides using a ruler in the usual way, we can mark off from a given straight line AX a segment AB, which shall be equal to the distance between two given points C and D.

The proof given by Chrystal uses infinite series, but by a simple modification it may be put in an elementary form.

The following solution of an old problem may be new to some readers.

“The ages of A and B are as 3 to 1, and in 15 years they will be as 2 to 1. Find their ages.”

The object of this note will, perhaps, be best explained by first repeating the usual steps in the solution of two simultaneous equations in x, y, where one is linear and the other is of the second degree. Let the equations be