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Published online by Cambridge University Press: 18 August 2016
An article published in this Journal last January (vol. viii., p. 316), proposed itself the problem of furnishing a test whether the degree of uniformity, or the reverse, observed in a table of statistics, required to be assigned to any other cause than the relative magnitude of the numbers of combinations which each figure in the table represented. The formulæ obtained in that article gave a general solution to the problem proposed, by giving an expression for the ratios of the numbers of such combinations which would be embraced under any figure in the table; and, consequently, expressing the relative frequency with which each number should be expected to occur from the mere consideration of counting the combinations represented by it.
* So far, this investigation is substantially identical with one which is well known—the language of the above being, for the sake of clearness, adapted to the case here contemplated. The simplifications which follow, admitting the definite working out of the problem in the most useful practical form, are believed to be new.
The mathematical demonstration of the stability of average results, a principle on the practical truth of which the whole fabric of assurance business rests, has, I believe, hitherto rested on very general grounds borrowed from investigations of La Place, which were chiefly worked out in a totally different direction, and apply only to the problem now under consideration in the case where the numbers are very large. In other words, it has been proved that the results of extensive transactions of this kind approach to stability, but no definite relation has hitherto been given between the extent of transactions and the degree of stability to be expected.