254 Mr. Gompertz’s analysis applicable to the 
“ &c. ; all the conditions are fulfilled by fulfilling the condi- 
L L a 
“ tions that H . and H . are constant. Whence we 
a a a fl 
“ find the constitution is L = e' — e" . e a ; where e' .e, e’’ are 
“ any constants at pleasure. And if I mistake not, it is the 
“ only function which will admit of this equality of chance 
“ continuously between persons whose mortality is formed 
“ from the same function ; but if e . be = 1 + e, and £ be infi- 
“ nitely small, this will stand simply L a = e f — e" (1 -}- at ) ; 
“ or writing g' for e' — e" and g" for e" e, this becomes 
“ Ly=g ' — g"a> the formula of equal decrements in equal 
“ times/’ 
Moreover the same idea of l 
a+x 
zk.l 
"b+x 
— , causes 
the fluent of 
— a -~ — to be equal to the fluent of k . 
a ‘ b 
, which if taken to commence with x equal to o, will be 
x l 
1 
2 
^a- t-x , + x i , i ^“a+x 1 ^i+x , 
• 1 — xl +~l— = — T + T~r TT— + 
a b a o 
L , 
“T ~J 7~~ 3 an< ^ this * s therefore not only the fluent in ques- 
a, b 
tion in the hypothesis of constant decrements. 
Article 3. If at the interest to be made for money, one 
pound discounted for one year were represented by r , and 
we wished to have the present value of one pound, to be re- 
