292 ESSAY ON PROBABILITIES. 



of items will exceed that of the second; and, whatever 

 may be laid by in each year, it produces a larger sur- 

 plus, and larger payments on account of claims, in the 

 next year. If, then, the surplus could increase without 

 limit, so would the dividends ; but if the surplus have 

 a limit, the dividends also have a limit : and it is plain 

 that the limit arrives, when the yearly outgoings from 

 claims and management are equal to the receipts from 

 premiums and interest of capital. A mathematical in- 

 vestigation of the conditions necessary in order that the 

 fund may increase, but not without limit, gives the 

 following result : 



Suppose an insurance office, constructed upon the 

 preceding principles, to have arrived at its stationary 

 state, with respect to influx and efflux of members, 

 and make the following suppositions : 



A The assets of the office, for precision, say Janu- 

 ary 1, 1838. 



P The real present value of all premiums from 

 members then in existence. 



C The real present value of all claims (not includ- 

 ing additions) to which the office is then liable. 



m The expenses of management till January 1, 1839- 



p The amount which will accrue from premiums and 

 interest of premiums by January 1, 1839- 



c The amount of claims (not including additions 

 from the surplus fund), which will be paid before 

 January 1, 1839. 



r The interest of one pound for one year. 



t The fraction which is taken of the tabular surplus 

 fund in the computation of the dividend. 



We suppose (as must be the case in an old office), 

 C greater than P, and (as must be the case in a solvent 

 office) A and P together greater than C. 



1. In order that there may be a surplus fund in- 

 creasing, but not without limit, find the fraction which 

 a year's interest on C is of c. Then t, or the fraction 

 of the surplus fund (or of A f P C), which enters 

 into the formation of the dividend, must exceed that 



