guarantees that the analytical method will work, it does explain the rationale of using a Farey series-type criterion on roots of unity. In the case of Waring's problem, one takes a sufficiently high power of the generating function to force the situation in which the singularities, organised into the so-called singular series, predominate. The less wasteful the estimates used on the rest, the finer the results. As Bryan Birch has put it, the method is inherently wasteful. That does not apply to the case of the partition function, which signalled the possibility that in a favourable situation the losses from estimates could be controlled. Vinogradov trigonometric sums Later, I. M. Vinogradov extended the technique, replacing the exponential sum formulation f(z) with a finite Fourier series, so that the relevant integral is a Fourier coefficient. Vinogradov applied finite sums to Waring's problem in 1926, and the general trigonometric sum method became known as "the circle method of Hardy, Littlewood and Ramanujan, in the form of Vinogradov's trigonometric sums". Essentially all this does is to discard the whole 'tail' of the generating function, allowing the business of in the limiting operation to be set directly to the value 1. Applications Refinements of the method have allowed results to be proved about the solutions of homogeneous Diophantine equations, as long as the number of variables is large relative to the degree (see Birch's theorem for example). This turns out to be a contribution to the Hasse principle, capable of yielding quantitative information. If is fixed and is small, other methods are required, and indeed the Hasse principle tends to fail. Rademacher's contour In the special case when the circle method is applied to find the coefficients of a modular form of negative weight, Hans Rademacher found a modification of the contour that makes the series arising from the circle method converge to the exact result. To describe his contour, it is convenient to replace the unit circle by the upper half plane, by making the substitution , so that the contour integral becomes an integral from to . (The number could be replaced by any number on the upper half-plane, but is the most convenient choice.) Rademacher's contour is (more or less) given by the boundaries of all the Ford circles from 0 to 1, as shown in the diagram. The replacement of the line from to by the boundaries of these circles