284 ESSAY ON PROBABILITIES. 



what proportions a given sum should be divided among 

 the insured. 



Let us return to the original constitution of an in- 

 surance office (page 238.). derived from the statement 

 of its main object ; namely, that it is a savings' bank 

 with a power of equalizing those results in which the 

 different durations of life would cause differences, Sup- 

 pose that such an office sets out with premiums imagined 

 to be no more than sufficient, but which are afterwards 

 found to be more than sufficient, leaving an admitted 

 amount of surplus in hand. The first thought would 

 be of restitution; namely, rendering back to each indi- 

 vidual the amount which he had bond fide contributed 

 towards the surplus. To do this properly, it must first 

 be settled whether the insurance office is one or many. 

 Does each age insure itself, or do the separate ages in- 

 sure both themselves and each other ? If the premiums 

 were properly proportioned, there would be no occasion 

 to ask this question : but if the incomers of one age pay 

 unduly as compared with those of another, then it is but 

 fair that they should receive in proportion. In the dis- 

 tribution of premiums, which I have described in p. 270., 

 it is equitable that a remedy should be provided, by 

 virtue of which those who enter the office young should 

 receive more than the rest. And it is, for this reason, 

 desirable that the proportions of the division should be 

 regulated by a true table of mortality. 



Let P be the real premium, and P -j- p the office pre- 

 mium ; and let the death of an individual take place 

 after he has been n years insured, and just before the 

 (w+l)th premium is paid. If the office had been a 

 compound interest savings' bank, the deceased would, 

 at his death, have been entitled to the following amount. 



P + p improved at compound interest for n years 

 P+p n-l 



P -f p 1 year. 



But under the conditions of insurance, the part P, with 



