3. We might express it thus:--a few instances are not sufficient to display a law at all; a considerable number will suffice to display it; but it takes a very great number to establish that a _change_ is taking place in the law. 4. The mathematician may illustrate the nature of this substitution by the analogies of the 'circle of curvature' in geometry, and the 'instantaneous ellipse' in astronomy. In the cases in which these conceptions are made use of we have a phenomenon which is continuously varying and also changing its rate of variation. We take it at some given moment, suppose its rate at that moment to be fixed, and then complete its career on that supposition. 5. So called from its first mathematical treatment appearing in the _Commentarii_ of the Petersburg Academy; a variety of notices upon it will be found in Mr Todhunter's History of the Theory of Probability. CHAPTER II. _FURTHER DISCUSSION UPON THE NATURE OF THE SERIES MENTIONED IN THE LAST CHAPTER._ 1. In the course of the last chapter the nature of a particular kind of series, that namely, which must be considered to constitute the basis of the science of Probability, has received a sufficiently general explanation for the preliminary purpose of introduction. One might indeed say more than this; for the characteristics which were there pointed out are really sufficient in themselves to give a fair general idea of the nature of Probability, and of the sort of problems with which it deals. But in the concluding paragraphs an indication was given that the series of this kind, as they actually occur in nature or as the results of more or less artificial production, are seldom or never found to occur in such a simple form as might possibly be expected from what had previously been said; but that they are almost always seen to be associated together in groups after a somewhat complicated fashion. A fuller discussion of this topic must now be undertaken.