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sum which bears the same proportion to the whole annual interest 

 that the period bears to the whole year." But there are considera- 

 tions affecting, not the interest, but the principal, which enter largely 

 into the solution of the problem. The date at which both interest 

 and principal are due is the end of the year ; it is evident, therefore, 

 that not only ought a less half-year's interest to be paid at the end 

 of the first six months than at the end of the second, but also that 

 the principal itself, if repaid at the end of the first six months, is 

 less valuable for the next period than it would have been if suff'ered 

 to complete its year. The solution of the problem has accordingly 

 been made to depend on the following assumption — " That both 

 principal and interest recommence a new year at the date of the 

 payment of the latter." I believe I am correct in saying that this 

 is the form in which the problem is usually solved, and I have no 

 objections to make to it ; but I can conceive circumstances, in con- 

 nection with life assurance payments, to which it is not strictly ap- 

 plicable ; and I have thought that it would not be unacceptable to 

 those who take an interest in the subject, if I presented the solution 

 of the problem in a new form, obtained by viewing it in another 

 light. With the practical bearing of any solution, I have no con- 

 cern ; it is the province of the actuary to ascertain, in any case pre- 

 sented to him, whether the one or the other hypothesis is applicable. 

 But I do not think it would be difficult to point out examples of the 

 operations of banks and life assurance companies where the interest 

 must be regarded as simply the payment of a sum before it has be- 

 come due, the capital out of which that sum has accrued being con- 

 tinued in its steady progress to the end of the year. However this 

 may be, whether the problem have a practical bearing or not, it- is 

 easy to see the propriety of the following hypothesis as the basis 

 of a theoretical solution of the question — " That the interest charge- 

 able for short periods of time may be deduced from considerations 

 which affect the interest alone." This hypothesis obviously presents 

 us with the following problem, which we have solved : — 



Problem. — To find the interest which must be paid at the end 

 of a fractional portion of a year, so that, being presumed to accu- 

 mulate at the same rate and in the same way in which it has itself 

 been produced, it shall, at the end of the year, amount to the exact 

 portion of the whole annual interest which would then have been 

 payable. For example, to find the interest of L.lOO for a quarter 



