J'appelle cubiques gauches équilatères les cubiques qui ont trois asymptotes rectangulaires deux à deux. Ces cubiques gauches possèdent des propriétés qui rappellent les propriétés classiques de l'hyperbole équilatère.

1. Let AB be the half-side of any n-gon, OB its in-radius (r), and OA its circum-radius (R). Draw OA11 to bisect ∠ AOB and AA1C ⊥ to it meeting OB in C. Then A1B2 to AB is the half-side of a 2n-gon having the same perimeter as the n-gon, OB1 its inradius (r1), and OA1 its circum-radius (R1).

The excentric angle notation in the ellipse is extremely useful, and in part we can replace it by the hyperbolic sine and cosine in connection with the hyperbola.

Take the hyperbola x2/a2/-y2/b2/= 1, then the coordinates of any point on it may be written acoshφ, bsinhφ, for cosh2φ–sinh2φ=1. The objection to its use in all cases is that the hyperbolic cosine of an angle is always positive, so that (acoshφ, bsinhφ) can only represent any point on the branch on the positive side of the axis of y, for any point on the other branch we must take its coordinates as (–acoshφ, bsinhφ).

In text books of Plane Coordinate Geometry, two methods are usually given for investigating the condition that the general equation of the second degree:

may represent a pair of real or imaginary straight lines.

§1. Let (x)u denote the function

then

Definition.—If tivo angles hare the same vertex and the same bisector, the sides of either angle are isogonal to each other with respect to the other angle.

Thus the isogonal of AP with respect to ∠ BAC is the image of AP in the bisector of ∠ BAC. It is indifferent whether the bisector of the interior ∠ BAC be taken, or the bisector of the angle adjacent to it; the isogonal of AP remains the same.

In the consideration of Question 12612 appearing in the Educational Times for January of this year, proposed by the Rev. Dr. Haughton, F.R.S., of Trinity College, Dublin, the following Diophantine Equation suggests itself: