526 
MU. B. GOMPEETZ ON THE SCIENCE 
that if the theorem were true through life, the value of annuities on any number of 
joint lives might be with ease obtained, which would be a most beautiful and useful 
property. 
But, as I have shown in my paper of 1825, and here more fully illustrated, those 
values can only be considered as approximative constants for a limited period, though 
that period is very long, and whilst w increases from 0 to about 60, if a were =10. I 
have shown in that paper, in the Carlisle Table of Mortality of the late learned Mr. 
Milne, that the theorem affords values for at the different ages, say from 10 to 60, 
differing very triflingly from Milne’s Tables ; and having in that paper expressly stated 
and shown that they were not absolutely constant, but depend on the ages of the thi’ee 
selected lives, it need not cause surprise that the theorem does not serve for that useful 
purpose; because yi, determined by the equation would come out so 
much larger than the limits of the applicability of these first-found constants, that, if 
the method availed, it would be a contradiction to the assertion that the elements were 
variable, as any one may convince himself ; but there are cases easily pointed out where 
the ages may be so taken as to afford a result by the theorem approximatively true. 
But this observation does not deprive the theorem 'Lg^^^=d . g\ of very great and 
serviceable value ; but to make it extensively available it will require a Table for every 
age a of the constants d, g, varying from one age to the other ; though the theorem 
= C . — ‘^(6* . A’ — /« . 5') + gjf 
appears still more valuable, which has the same constants for every age, from birth to 
extreme old age, and which, from the age of about 10 to 80, will take the simpler form 
c^L^=C€*— -/i. q), 
in consequence of the portions, kef and between these limits being insignificantly 
small. 
When the differences of the selected ages are only instead of 30 years assumed further 
on 20 years, that is, when difference w=20, and if m be respectively 10, 20, 30, 40, 50, 
60, that is, if the youngest in the selection be respectively 10, 20, ... 60, we have, for 
finding d, g, q in the formula lj^=d.g\ , the following data: — 
For the selection 10, 20, 30, we have 
^(^=3-88012, -=^=-0132565, --(-^)=2-71185, ^=--051508. 
(?=8-9343 , ^=-030526 , -z.(-^)= 1-07407, 7=--11860. 
For the selection 20, 40, 60, we have 
^fZ=3-88137, -=^=-012984 , 
(?=8-9374 , ^=-027897 , 
For the selection 30, 50, 70, we have 
3-8273 , -=^= -0192535, 
e_(^=8-8127 , o_^=-044334 , 
^(-^)=2-72607, ^=--053211. 
-^(-7)=I-08829, ^=--12254. 
C-. 
^(_^)=2-30241, ,(7=--020064. 
_=(-'^)z=2-66463, "^=--0462. 
