1. The chief purpose of this paper is to demonstrate the existence of a plane quartic curve with eight undulations, an “undulation” being a point at which the tangent has four-point contact. It is shown that the curve

where (x, y, z) are homogeneous point-coordinates and f a constant, has undulations at the eight points

The curve has, in addition to these undulations, eight inflections which are; in general, distinct. But there are two geometrically different possibilities of their not being distinct, and in either instance they coincide in pairs at four further undulations. Thus two types of curve arise without any ordinary inflections at all, their 24 inflections coinciding in pairs at 12 undulations.

We suppose that f(t) is integrable in the Lebesgue sense m (π, π) and is periodic with period 2π. We denote its Fourier series by

Then the allied series is

1. Introduction. In an earlier paper1 the function

was considered, having on its circle of convergence, taken to be |z|=1 only isolated essential points of finite exponential order, situated at the points eiav (v = 1, 2, .., k). It was there proved,2 in the cases k = 2, k = 3, that the upper density of small coefficients is positive only if the points eiav are situated at some of the vertices of a regular polygon inscribed in the circle of convergence and if the singularities are virtually identical or linearly related. In this case the sequence of small coefficients possesses a density. The coefficients cn can be interpolated in the form

where the Gv(z) are integral functions of order less than 1, so that the result stated may be regarded as expressing an arithmetical property of a certain type of integral function at positive integer points.