Mr. Morgan on Survivorships. 225 
in the first or second years, and of the life of B in the third 
year. Therefore, the probability for the third year will be ex- 
pressed by — 4- l n ~ }l - a " 4-I 4. * 
J 1 2^6 I ab ' zab “ — ab X 
b — 0. a , w — a. a'-J-a , ?z — o.d' + a" r-, 
— . j . By proceeding in the same 
manner for the fourth year, the probability will be found 
and suppos- 
ing x to denote the difference between the ages of B, and of the 
oldest person in the table, and y and % respectively the num- 
ber of persons living at the two last ages in the same table, 
the whole probability of the elder life’s dying after the younger 
will be r= -±j- into b—z.a' 4 m—z.a' +a 77 + n—z. a ll ^ r a" l ^ r 
o—z. a!"-\-a "" .... 4-jy — % Now, since it is well 
known that the probability of both lives failing in x years, 
without any regard to the order of their extinction, is = 
b— z-xv' + a"+a'"+a"" 4 -a* / . , 
1 — ~ h (or supposing to be the number of 
persons living at the end of x years from the age of A) = 
— b — , it is evident that, if the foregoing series be subtracted 
from this fraction, the probability will be obtained of the 
younger person’s dying after the elder in x years. In the 
first paper which I communicated to the Royal Society on 
this subject,* I not only described the most concise method of 
computing a table of the probabilities of survivorship between 
any tw r o given lives, but computed a comprehensive one for 
persons of all ages, whose common difference was not less than 
ten years. As the contingency in this lemma is of considerable 
* See Phil, Trans. Vol. LXXV 1 II. p. 335. 
Gg 
mdccxciv. 
