different Tables of Mortality, 351 



the extreme cases; otherwise the agreement would have been much 

 more accurate : but it is obvious that the formula comes far nearer 

 to the direct computation from the Carlisle tables than the value 

 derived from the Northampton tables. 



19. We are now to examine the consequences of the two hypo- 

 theses in cases which require the consideration of interest or dis- 

 count, to be combined with that of the contingency of survivorship 

 at each step. In order to represent such cases, we must multiply, 

 as is well.known, the fluxion of the contingency of payment by the 

 power r*, or rather r*~', for the value as referred to the age g, r 

 being the present value of a unit payable at the end of a year : 



for instance , if we reckon at 4 per cent, compound interest. 



104 



But it must be remembered that in this mathematical sense of 



compound interest, the interest of ^100 for a quarter of a year 



is no more d^l, at 4 per cent., than it is ^16 for 4 years ; and if we 



wish to reckon at the rate of £ for a day, we must neces- 



365.25 ^ 



sarily make the interest something more than £4: for a year. In 



almost all cases occurring in practice, the difference of the two 



mode§ of considering the interest is half a year's purchase of an 



annuity, payable annually : but sometimes, for an annuity of a very 



short duration, a further correction may be required : the correction 



is, however, in all cases, very easily computed, and generally 



by taking the fluent half a period later, both at the beginning and 



at the end of the term. 



20. The present value of an annuity on a single life may, there- 

 fore, be represented by— / r*~'-f-dj7, since dj? is negative; that is, 



J k 



in the arithmetical hypothesis — / L — (\ — jLjdjr= — / 



a;dj?; and the fluent must be taken as usual from x = 



ck 



c to 07 =: ^. 



21. The general theorem for fluents of this form is fa* j?" Ax 

 ._ ^ /^ _ nx--' 7i(m--lK-^ , 7i(y^-l) (n-2) x"'' 



"^ hla \ hla hVa hVa 



2B2 



