252 Mr. Gompertz’s analysis applicable to the 
chance if A and B are both to be dead, whether A dies after 
B, or B dies after A, if 1 — -~- = k . (1 ^Y^beingcon- 
stant. 
“ Note. If the constitution of both functions of life is of 
“ one and the same continuous character, which is not neces- 
“ sarily the case, unless they be taken from one and the 
“ same continuous table ; then it becomes an interesting pro- 
“ blem to search, what is the constitution of the function of 
“ life, to admit of the aforesaid equality of contingency of A 
“ dying after B or B dying after A ; on the condition of their 
“ both dying in that time ; that is to say, to find the common 
“ characteristic L which shall be independent of a, b, and a ?, 
“ such that 1 =k . ( 1 — the requisite equa- 
“ tion above given. And as this is a problem which, from its 
“ mode of solution, may be equally interesting to the Analyst, 
“ as forming one of a rather novel species of problems, I 
“ shall give its solution for both purposes ; in order to which 
“ I first observe, that unless k be unity, it must be expressi- 
ng 
“ ble in the form -g-, so that the equation may be written in the 
a 
“ form D 'r" - D = D, . — D, ; otherwise there 
A a L a 0 Li o 
a 0 
“ would not be a perfect similarity of the character L, on both 
D a 
“ sides of the equation. For the sake of brevity, put — = H , 
a a 
D b 
“ and therefore — = H^; and our equation will stand 
b 
“ H . (L — La) = H . (L, , — LA ; but by Taylor's 
a v <z+£ v b 'T x b 
