Summary. The Schr6der and K6nig iteration schemes to find the zeros of a (polynomial) function g(z) represent generalizations of Newton's method. In both schemes, iteration functions fro(Z) are constructed so that sequences z,+~=f,,(z,) converge locally to a root z* of g(z) as O(Iz,--z*f"). It is well known that attractive cycles, other than the zeros z*, may exist for Newton's method (m = 2). As m increases, the iteration functions add extraneous fixed points and cycles. Whether attractive or repulsive, they affect the Julia set basin boundaries. The K6nig functions K,,(z) appear to minimize such perturbations. In the case of two roots, e.g. g(z)= z 2-1, Cayley's classical result for the basins of attraction of Newton's method is extended for all Kin(z).The existence of chaotic {z,} sequences is also demonstrated for these iteration methods.

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1. Introduction. This paper deals with the lack of uniqueness of the representations of the complex numbers in positional notation using Gaussian integers as bases.Kátai and Szabó [3] proved that all the complex numbers can be written in radix form using the base –n + i with the natural numbers 0, 1, 2, …, n2 as digits. They remarked that they did not assert the uniqueness of these representations but gave no further indications of any multiple expansions. The geometry of these complex bases [2] indicates that some numbers have two expansions in a given base, while a few numbers even have three different expansions.

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SUMMARY:When using MacConkey broth medium incubated a t 4 4 ' as a confirmatory test for Bacterium coli Type I, positive reactions were produced by anaerobic lactose-fermenting bacteria such as Clostridium welchii. The use of brilliant green bile broth instead of MacConkey broth was completely successful in suppressing the growth of these organisms a t a", but did not influence the growth of B a t . coli !L'ype I. Two other coliform organisms, namely, Irregular Type I1 and Irregular Type V I were found to ferment lactose at 44'. A rapid test for their differentiation from Bact. coli Type I has been based upon the ability of that organism to produce indole at M', whereas the two irregular types are indole negative. By subculture of presumptive positive tubes in brilliant green bile broth and peptone water together with incubation a t MO, an estimation of the Bact. coli content of a water sample may be obtained within 48 hr. of primary inoculation. It is now over 40 years since Eijkman showed that coliform bacteria from the intestines of warm-blooded animals would grow and produce gas in glucose broth at a temperature of 46", whereas coliforms from the intestines of cold- the milk sample into tubes of media and incubation a t 4 4 ' , or the subculture of primary positive tubes from the first 24 hr. a t 37" into MacConkey medium and incubation of these at 44". It was found (Mackenzie, 1938) that for riverwater samples, where Bact. coli was abundant, primary incubation a t 44" was as efficient as the subculture procedure; but when stored, filtered and chlorinated water samples were tested, the organisms, possibly because of attenuation, were more frequently recovered by the subculture procedure than by direct inoculation at 44'. Since, in addition, very large incubator baths would have been needed to accommodate the groups of tubes to be set up for each test, the subculture procedure for the rapid identification of Bact. coli was chosen for our laboratories and first used as a routine test in September 1939. It is now Two procedures were suggested by Wilson et al., either the direct inoculation of ,