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The "Encyclopaedia Americana" defines an epi demic as a "sudden excessive prevalence of a disease in a population". This is a typical definition, and although suddenness is not always considered an essential feature of an epidemic, most would agree that it refers to disease, and that it denotes an in creased prevalence. But there are two serious difficul ties in this concept which limit its usefulness and unless they are recognized they may prove a serious hindrance to the detection of epidemic properties in several classes of disease. The first is found when the occurrences are few in number, sparsely scattered in time, and widely sep arated in relationship to the frequency of any supposed fluctuations. This problem arises with uncommon illnesses such as congenital malformations when an apparent crop of cases poses the question how many cases can make an epidemic. The true difficulty seems to be the breakdown of the concept of prevalence upon which in turn the concept of an epidemic depends. The situation is analogous to the breakdown of the gas laws when very few molecules are involved and the inappropriateness there of the ordinary con cept of pressure; or the breakdown of the theory of wave propagation in the study of low intensity radia tion. Pressure, wave motion, prevalence, and epide micity, are terms appropriate to the study of high concentrations and when we are compelled to deal with low intensities special methods of examination must be devised. The second difficulty arises from the assumption implicit in the above definition, that an epidemic is recognized on a time base. However, it is clear that space must sometimes be taken into account as well as time. A feature of many epidemics is that the geographical region of maximum prevalence moves with change in time, or stated conversely, that the time of maximum prevalence changes with change of place. The result is that a disease which is manifestly epidemic on the time scale in each of a number of small regions, may over a sum of those areas show only equivocal evidence of epidemicity in the terms defined above. Conversely, a disease of which the incidence is geographically uneven during a short period may appear more evenly distributed when a longer period is considered. These difficulties are not serious with a frequently occurring disease; either the time scale or the geo graphical area may be divided into sufficiently small classes to demonstrate variations of prevalence in terms of the other. But with uncommon diseases such as leukaemia or congenital malformations the pro cedure of subdivision leads back into the problems of small numbers. It may be impossible to examine either the space or the time variations in prevalence, without at the same time impairing the applicability of the concept of prevalence. This may not be for lack of large numbers of events for analysis, but because they occur sparsely over a large area and over a long period of time, and accumulation of further data may do little towards solving the problem. For these reasons, no doubt, most of the statements to date of low intensity space-time clustering in particular disea ses have been based upon impressions and anecdotes rather than upon rigorous analysis. A previous attempt (Knox, 1959) has been made to analyse low intensity data on time base alone and the present paper is an attempt to extend this to space and time jointly. The two purposes of the present study are to suggest a method of analysing this class of data in a more discriminating manner than has so far been employed and to employ the technique in the investigation of a particular problem.