352 Praciical Comparison of 



+ . . . V or putting -I— = X, = - Xa" {x" + Xnx"-^ + 

 / h\a 



XV (/I— 1) 0?""^ + . . .) whence the present fluent becomes 



X Xr*~' 



+ — r*~' — , {x + X), X being here positive, because a 



k ck 



is less than unity, and putting q and c successively for :r, we have 



^-4(, + X) = 4A-X- A)=X-^ and 

 k ck k \ c c J ck 



h.r'-'^ - —r'-'' (c + X) = - ^ r'-», the difference being 

 k ck ck 



X — — (1 — r""^) r= A , the present value of the annuity : and 

 ck 



at 4 per cent, we have X = 25.497,\X=: 650, and if-— = .0115, 



c 



the formula becomes A = 25.497 — (I - r'-'^) ; the re- 



k ^ 



suits of which agree sufficiently well with the Northampton table. 

 I. Annuities at 4 per cent, upon the arithmetical hypothesis. 



22. In the quadratic hypothesis, the fluxion is — /r''"' — d a: = 



k 



- /?•--« ^ + /r'-« J^~ do; and the fluent — r^-^ - A. r'-« 

 k cck k cck 



(x^ + 2\x + 2 X^) which from a* = g to a? r= c affords us 



k 



~ _^ {q^ + 2\q + 2X«) - —r'-' + jL^r'-' (cc+ 2\c 

 cck k cck 



+ 2VX) =A A _ 2i - i:^- ^ + ,-. r!i + 3^1 Y 



k \ cc cc cc L c cc A J 



