CONNECTED WITH HUMAN MOETALITY. 
535 
the continuance, that is to say, that chance is equal to 
— ('B. See.), 
where ‘K=2 =*B+‘A‘B ; =^K=3 =’B+2 'A^B+^A 'B ; 
^K=4 ^B+3 •A.^B+2 ^A.^B+^A'B ; ^K=5 ■^B+4 'A/B+3 ^A.^B+2 ^A.^B+^A. 'B ; 
where the law of continuation is evident. And the chance that the combination A failed 
pre'siously to the failure of the combination B, is the fluent of 
QKx-\-^Ax'^-\-^Ax'^, &c.)('Bi+2 ^B.2;i’+3 &c.), 
which evidently is the excess of the chance of the combination of B failing inde- 
pendently of any connecting A, above the chance of the combination B failing whilst 
the combination A exists, and thus we may proceed to successive additional cases of con- 
ditional, contingencies with respect to more combinations. Now it is observable that in 
consequence of the great convergency of the coefficients 'A, ^A, ^A, &c., and of 'B, ^B, ®B, 
&c., and of the small values of and ^B, except very shortly after birth, which result 
from what has been previously stated, if all of them, except ^A, ’B, be considered as 
nothing, unless x be very great, the first value will be very nearly expressed by the first 
term —^'Q.x—t, because 'A.'B is small of the second degree ; and the second value in a 
similar way, for a long period in which it shall occur that both combinations fail, is 
very nearly ■|AiBi.ir’^ — giving an equal chance which shall have failed first ; and this is 
a generalization of Mr. Morgan’s hypothesis of two lives only, which enabled him to 
solve questions respecting the lives of three persons A, B, C, contingent on the life or 
death of A, provided B and C be both dead, involving contingencies of survivorship 
between them. But that this law cannot accurately exist for any possible continuous 
law of mortality, I have proved in my paper of 1825, unless of the form 
where e', e',e are constant quantities at pleasure, and a the age, and which, in an extreme 
case of e differing infinitely little from unity, is reducible into the form 
lja=g'—(fa, if g'=e'—e", and g'’=e"s, 
e' and e" being infinitely large, differing from each other by a finite quantity, and s infi- 
nitely small, and in consequence g" a finite quantity. But with respect to the two com- 
binations generally — (and thus is found the chance of one combination failing during 
the continuance of the other combination, or of its failing after the continuance of the 
other combination, and this is evidently considerably more general than the cases of 
Messrs. Morgan, Bailt, and Milne, in which there are only three lives concerned,) — I 
observe that they will give useful practical solutions however many lives are concerned, 
and however complicated the conditions of survivorships may be between them, whereas 
it may be a question if the solutions of those gentlemen, whose memory I respect, are at 
all practical; especially when there are only tables of single and two joint lives at hand. 
I therefore consider that the solutions I had given to those three-lived questions, and 
to questions of more lives, in my Tract of 1820, to which I refer the reader, which 
were to be derived from calculated Tables of annuities, are worthy of the attention of 
MDCCCLXII. 4 D 
