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(With I figure in the text)There is at present very little knowledge of actual densities of population among the higher fungi. It is not known, for example, how or how far the mycelial soil fungi (the greater part of which are Basidiomycetes) utilize the available soil strata under various plant associations, nor is it known in what way the ring-forming habit reacts on the behaviour of an association whose dominant species are continually spreading outward like ripples on a pond in the rain. In view of the probable importance of the ecology of soil fungi as a major factor in the behaviour of plant associations of interest to the farmer, any information on such elementary points will be of value.The difficulty of obtaining this information is however considerable. One cannot determine populations of fungi merely by counting them, since in most species not all the mycelia fructify every year; and without exact or at least fiducially circumscribed estimates of population one cannot determine any of the points of interest. To obtain these estimates a rather elaborate statistical procedure is therefore needed, which in turn demands a rather extensive series of observations of the territory in successive years. Again one cannot interpret the biological significance or insignificance of local apparent variations in population density without a close study of the local topography and ecology, nor without some information on the manner of dissemination by which the fungi may pass from one spot to another. It is therefore only at this stage of the present investigation that one can begin to consider these questions.(2) ESTIMATION OF POPULATIONS This is not the place to expound in detail the statistical methods employed. The principle involved is however simple. If we are given the total number of fructifying mycelia of a given species in a given territory, on a number of occasions at the same time of year, then the longer the series the larger will be the maximum number hitherto recorded, and the greater the probability that it will be the maximum possible. Suppose we have a set of counters numbered successively i to A^, and take one from among them at random t times, with replacement; let n be the highest counter thus found. Evidently the probability that n will have a given value will be a determinate function of TV and <; and conversely, if we are given t and fi, we shall be able to infer a probability distribution for N This can be done easily if the variate, in this case the number on the counter «, is a rectangular variate; in the case of the fungi however this is not so. Instead we have a function

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