the Values of increasing Life Annuities. 165 
to the common scale of notation, (as given in most treatises on 
i the subject,) no others for this purpose, I believe, exist any where 
i in print, but in the present work; in regard to which I have only 
j to observe, they are as simple and concise as the nature of the 
investigation would possibly admit: and as simplicity and ac- 
curacy were objects indispensably in view, so they have not been 
attained without some efforts of patience and perseverance. My 
studies are prosecuted under auspices the most unfavourable: I 
have to lament that my present situation but so ill accords with 
a disposition for scientific pursuits. 
The several formulz I shall enumerate will apply in the four 
following cases; viz. when the annuity increases in the order of 
the numbers (1.3.6. 10. 15) (1.3.5.7.9) by the squares of 
this latter series, and also by the squares of the series (1. 2.3. 
4,5.) 
Then in the first case the formula exhibiting the value of the 
_annuity will be 
a—1 —2 _ 2 =O (Be eee toe ) 
at —z)+ 6(a +1) + x2—1.(a? + 9a + 10)r+0—1.(a2 + 32+ 2)2—\ > —2. (2a) + e—1.(1—2) + 20% 
24.(a—1)3 . 
In the second: 
a.(e+1)+(2¢4+54+0—1.(2a+ rr (> (A +0) 
a.(w—1)2 
And for the two succeeding cases it will be respectively, 
a —? ay 2 2-2 /64%—1  —a 
@)+7r—1.a+ (4.(62+8) +2—1.(40? + 200+ 17) + (x—1)(2a+ 1). a(S =) 4+16.(0+1) + 2—1.0 
a.(z— 1,3 
. And, 
= ~a Gr—(r—1.20) ig 
T, (6(a+1) + (a? + 6a+7)24+02—1-(a% +204 1)2+a—1.2— (Q@--)|)* (vr) 
a\a—l] )? 
In each of the foregoing formule x denotes the ratio or amount 
of one pound for a year; and (2) the complement or double the 
expectation of human life, according as it is deduced from any 
assigned table of observations. 
The annuity may commence with the addition of some fixed 
annual payment, still increasing in the same order; as for instance 
(11,13. 16. 20) (11. 13. 15. 17), and so on, for the other series; 
and thus may be generated various forms of increasing annuities 
at pleasure. In this case the only difference will be, that we must 
augment the value previously obtained, by the value of such ad- 
ditional annuity on the given life for the total value required. 
If the annuity, being a deferred oné, does not commence until 
a given period equal 7, years, the quantity (a) must be deter- 
L3 mined 
