Let 0 < r < 1 and let f(z) be regular for |z| ≤ 1.

Then from Cauchy's integral

we have the inequality

If Z is a fixed point on the surface of the earth (assumed spherical) and P is the North Pole, then the locus of a point X, which moves in such a way that the angle α between the great circle arcs PX, ZX is constant, is called a curve of constant bearing.

In the figure, ABC is a triangle with B > C; AF is made equal to AC, AE bisects the angle A, and BDEG is a rectangle. It is easily seen that the angles marked α are equal, that B − α = C + α, and hence that α = ½ (B − C). Then:

The pencil of quartic curves

where x, y, z are homogeneous coordinates in a plane, was encountered by Ciani [Palermo Rendiconli, Vol. 13, 1899] in his search for plane quartic curves that were invariant under harmonic inversions. If x, y, z undergo any permutation the ternary quartic form on the left of (1) is not altered; nor is it altered if any, or all, of x, y, z be multiplied by −1. There thus arises an octahedral group G of ternary collineations for which every curve of the pencil is invariant.

Since (1) may also be written

the four lines

are, as Ciani pointed out, bitangents, at their intersections with the conic C whose equation is x2 + y2 + z2 = 0, to every quartic of the pencil. The 16 base points of the pencil are thus all accounted for—they consist of these eight contacts counted twice—and this set of points must of course be invariant under G. Indeed the 4! collineations of G are precisely those which give rise to the different permutations of the four lines (2), a collineation in a plane being determined when any four non-concurrent lines and the four lines which are to correspond to them are given. The quadrilateral formed by the lines (2) will be called q.

In many cases, when a colony of micro-organisms such as moulds, yeasts or bacteria grows on the plane surface of a solid medium (e.g. agar), starting from a single cell, the colony tends to grow as an ever expanding circle. The reason for this is that every cell, if free from competition, can multiply at roughly a constant rate in all directions in a plane, limited by the fact that territory occupied by one cell cannot be occupied by another. For the purposes of the present discussion, we can assume, as a first approximation, that the whole process is two-dimensional.

In this article is described the construction of a thread model of a hyperboloid of one sheet (H) and its asymptotic cone (C). It ia simple to make, requiring only cardboard and thread, and can be made collapsible and of pocket size if desired. The model consists of two hinged pieces of cardboard (intersecting planes π and ) on which are drawn circles SH, respectively in which the planes meet H, and the concentric circles SC, respectively in which the planes meet C. A number of generators of the same system on H are now represented by threads joining SH and , and the corresponding parallel generators of C are represented by threads joining SC and . In order to ensure that these generators are well spaced, those of C are taken at equal eccentric angles apart in a principal elliptic section. The main theorem used in the design is that if l a generator of C, then the tangent plane to C at points of l meets H in two generators both of which are parallel to l