ON UNIFORMLY CHANGING ANNUITIES. XXV11 



of this application of the calculus, which appear to us 

 of much value, nor are we at all sanguine in expecting 

 any." The tendency of such an assertion is to en- 

 courage those who study the subject, to stop short of 

 the differential calculus in their mathematical studies. 

 Now I assert, 1. That the calculus aforesaid may,, as 

 evidenced in the results of chapter IV., lead to most 

 valuable rules in the estimation of complicated proba- 

 bilities. 2. That if the calculus be not serviceable in 

 the deduction of the law of mortality, it is from defect 

 of observed data. As soon as larger and more correct 

 tables of the numbers living are obtained, the differen- 

 tial calculus is ready to furnish methods for correcting 

 those now in use. 3. That the differential calculus may 

 be made to give important simplifications of processes, 

 and to render the tables already constructed immediately 

 available for purposes to which no one now dreams of 

 applying them. 



If v be the present value of I/., to be received at the 

 end of a year, and tyv be the present value of a con- 

 tingent annuity of I/., then that of an annuity which is 

 to be 11. at the end of the first year, 2/., 31. , &c., at 

 the end of the second, third, &c. years, is vfy'v, where y'v 

 is the differential coefficient of $v. Now 1 -f r being 

 the amount of II. in one year, we have 



dv 1 d$v _ ^ d<f>v 



~& = (l+r)2 ' 1o = ~ 'dr ( + r > 



dtyv d<j>v 



and the annuity v--~ = (1+r) 



dv dr 



Now tables of annuities of I/, being calculated for a 

 succession of values of r differing by '01, we have 



01 x = &<pv - 1 A2<Jw + I A3</w - I A4<?w + &c. 



Substitute the value of thence obtained, and we 

 dr 



