1. If a (or its real part) is positive and less than 1, and if t (or its real part) is positive, then, according to Mellin, the case +(x) =exp(-txa) of the Euler-MIacLaurin difference E+ (n)f(p (x) dx, where in1, 2, and 0_ x <oo, can be evaluated explicitly; cf. (9) and (9bis) below. The resulting turns out to be closely connected with results which in earlier papers I obtained on Cauchy's symmetric stable distribution functions of index a (where either 0 < a < 1, as in Mellin's formula, or, to certain ends, 0 < a < 2, as in Paul L6vy's justification of Cauchy's formal result). This family of distribution functions is defined by the property that the Fourier transform of the distribution is even and is identical with e ta if 0?t<oo. In what follows, the analytical questions concerning Cauchy's distributions will be dealt with in a systematic way. Roughly speaking, there will be three issues involved: (i) the explicit form of the Laplace transform of the distribution, (ii) the connection of the Laplace transform with the Stieltjes transform, finally (iii) the application of the resulting explicit formulae to a determination of the weight function which belongs to the stratification, in terms of the symmetric Gaussian distribution (, = -2), of the symmetric stable distribution of any index /3 < 2. At the end (Section 13), there will be considered the asymptotic expansion (and its implications for Cauchy's transcendents) which takes the place of Mellin's convergent expansion if 0 < a < 1 is replaced by 1 < a < o0 (the limiting case a =1 is trivial). The values a = 2n, where n 2, 3, 4, are exceptional (but n = 1, i. e., a =2, is trivial). For this exceptional case, there will be reproduced with Professor Polya's permission an asymptotic formula which he communicated to me several years ago. For 0 < a < 2, a side issue will be the shape of the frequency curves. This issue has, however, little to do with Cauchy's particular choice of symmetric distributions, since what results holds for all symmetric distributions