The Genesis of the Double Gamma Functions

The following paper is the natural sequence of results obtained in two previous papers.The " Theory of the Gamma Function "* contained a discussion of the function defined by the formula and it is evident that the expression on the right-hand side of this equality may bo regarded as the positive half of the product expression for sin nz; wo may, in fact, term it the " halb-sinus " with Betti.fAgain, in the "Theory of the O Function,"% it was shown thatwhere r (z) is a quadratic function of z.If now we can associate with the letter m, each time that it occurs in this product, a complex constant r which is not real and negative, we shall obtain a pi'oduct which may be regarded as the positivo quarter of the product expression for Weiorstrass's function <r(s), and which will be therefore a natural extension of the V function.