Some problems of diophantine approximation: Part II. The trigonometrical series associated with the elliptic ϑ-functions

where q ~ e ~i*, are convergent when the imaginary part of 9 is positive, and represent the elliptic O-functionsWhen v is a real number x, the series become oscillating trigonometrical series which, if we neglect the factor 2 and the first terms of the second and third series, may be written in the formsx The notation is that of TANNERY and Molm's Th~orie des fonefians elli~tiques.We shah refer to this hook as T. and M.