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Published online by Cambridge University Press: 18 August 2016
The following paper was originally ritten simply as a reply to Mr. Macfadyen's, which was redd before the Institute in May last. It has since been carefully revised and to a great extent recast, in consequence of suggestions receivd, to the effect that the subject was of efficient importance to be delt with in an independent paper and submitted afresh to the consideration of the Institute. In consequence of this change, a considerable amount of personal criticism, which might hav been suitable enuf for a personal reply to another paper, has been either struck out altogether, or much modified. It will therefore be understood that, in what follows, I hav somtimes had Mr. Macfadyen's arguments in view when I hav not mentiond his name
* Let a represent the new premiums of the first year, b the renewals, and c the expenses; and let a, β,γ, represent the corresponding figurs of the second year; also let x be the proportion of the new premiums that is spent, and y the similar proportion of the renewals. Then we hav the two equations ax+by=c, ax+βy=γ, whence we get .