ARTHENOGENESIS in the animal kingdom has been termed an "evolu-'tionary dead end". This view stems from the writings of DARLINGTON (1937), WHITE (1948) , SUOMALAINEN (1950) , and others who suggested that only two genetic consequences result from such a system of reproduction: (1) complete homozygosity; or ( 2 ) complete heterozygosity. These authors described the mechanisms presumed to yield these results for several parthenogenetic species; unfortunately, none of these descriptions utilized rigorous mathematics to analyze the genetic properties of parthenogenetic populations.The validity of the above hypotheses was challenged by CARSON (1967a) who suggested that automictic parthenogenetic reproduction can maintain heterozygosity. He stated that in those species of insects (Drosophila mangabeirai, Solenobia lichenella, Devorgilla canescens etc.) which can restore zygoidy either by inhibition of meiosis I or by central fusion, loci which are absolutely linked to the kinetochore could be maintained in a state of permanent heterozygosity. Furthermore, with the addition of chromosomal rearrangements, the amount of heterozygosity preserved in parthenogenetic species could be increased. Thus, he suggested heterozygosity could be maintained in these organisms by at least two mechanisms: (1 ) absolute linkage to the kinetochore; and (2) chromosomal rearrangement. WHITE (1970), reversing his previous views on heterogeneity in parthenogenetic animals, appears to take a position similar to that of CARSON (1967a) ; however, his arguments also lack mathematical rigor.The mathematics needed as a basis for students of natural parthenogenetic populations was developed in part in a previous study (NACE, RICHARDS and ASHER 1970) where a model based upon a mapping function was used to estimate gene-kinetochore distances and "inbreeding" in parthenogenetically produced frogs.* It was noted that although the linkage data were derived from artificial parthenogenetic reproductions, the analytical model should apply to all systems of parthenogenesis, natural or artificial, which restore zygoidy either by inhibition of meiosis I1 or by terminal fusion. Although the paper presented a mathe-1 Part of a thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The University of Michigan Horace H. Rackham School of Graduate Studies. a These investigations were supported by a National Institutes of Health Genetics Training Grant 5 TO1 GMOOO71-12, and in part by the National Science Foundation grant GB 8187, and by The University of Michigan Computing Center.* The term inbreeding does not truly apply to the parasexual process described here. No term exists which describes a system of reproduction leading t o increased homozygosity independent of the mechanism of reproduction. Rather than formulate a new term, we will use the term "inbreeding" in quotes when it relates specifically to parthenogenetic animals.