Prefatory .—It is somewhat surprising that so little mathematical work should have been done on the subject of epidemics, and, indeed, on the distribution of diseases in general. Not only is the theme of immediate importance to humanity, but it is one which is fundamentally connected with numbers, while vast masses of statistics have long been awaiting proper examination. But, more than this, many and indeed the principal problems of epidemiology on which preventive measures largely depend, such as the rate of infection, the frequency of outbreaks, and the loss of immunity, can scarcely ever be resolved by any other methods than those of analysis. For example, infections diseases may perbaps be classified in three groups: (1) diseases such as leprosy, tuberculosis, and (?) cancer, which fluctuate comparatively little from month to month, though they may slowly increase or decrease in the course of years; (2) diseases such as measles, scarlatina, malaria, and dysentery, which, though constantly present in many countries, flare up in epidemics at frequent intervals; and (3) diseases such as plague or cholera, which disappear entirely after periods of acute epidemicity. To what are these differences due? Why, indeed, should epidemics occur at all, and why sbould not all infections diseases belong to the first group and always remain at an almost flat rate? Behind these phenomena there must be causes which are of profound importance to mankind and which probably can be ascertained only by those principles of careful computation which have yielded such brilliant results in astronomy, physics, and mechanics. Are the epidemics in the second class of diseases due (1) to a sudden and simultaneous increase of infectivity in the causative agents living in affected persons; or (2) to changes of environment which favour their dissemination from person to person; or (3) merely to the increase of suscep­tible material in a locality due to the gradual loss of acquired immunity in the population there; or to similar or other causes? And why should diseases of tbs third class disappear, as they undoubtedly do, and diseases of the first class remain so persistently?—all questions which immediately and obviously present themselves for examination.

•week by week, and became very unhealthy in appearance, whilst there was a progressive improvement in the appearance of the plants supplied with increasing amounts of auximones. Those receiving the larger amounts retained their normal healthy appearance throughout the experiment and increased in size. 6. The beneficial effect of the auximones was not due to a neutralisation of the toxic substances present in the ordinary distilled water, since comparable results were obtained with conductivity water. 7. An interchange of culture solutions, with and without auximones, showed that the plants are very sensitive to the presence or absence of these substances. 8. It is suggested that some of these growth-promoting substances may act directly as organic nutrients, and others may be of the nature of accessory food substances. I am indebted to the authorities of the Botanical Department, Imperial College of Science and Technology, South Kensington, for facilities afforded for carrying out a considerable portion of the work in their greenhouse laboratory. I wish also to express my grateful thanks to my assistant, Miss Mockeridge, B.Sc. Without her valuable help it would have been impossible to obtain a record of the numbers and weights of the plants each week.

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