Page:Supplement to the fourth, fifth, and sixth editions of the Encyclopaedia Britannica - with preliminary dissertations on the history of the sciences - illustrated by engravings (IA gri 33125011196181).pdf/940

648 a year at the given rate, we must alter the value of $$v$$ accordingly.

For two joint lives, the complement of the elder, determined from the fraction $$\frac{3}{r+5}$$, being $$a$$, and that of the younger, deduced from the deaths in an equal number of years, $$b$$, we have for the binary combinations of the survivors, after x years, $$(a-x)(b-x)$$, and the fluent will be $$-pv^x (ab - (a+b)(x+p) + x^2 + 2px + 2p^2)$$, which, corrected and divided by $$ab$$, gives the value of the annuity $$p - \frac{p^2}{ab} (a+b-2p) - \frac{p^2v^a}{ab} (a-b+2p)$$; and this, with the deduction of half a payment, agrees with the tables calculated on Demoivre’s hypothesis, taking the same complements of life.

But for three lives we have no such tables, and this method of calculation becomes therefore of still greater importance. Employing here the fraction $$\frac{3}{r+7}$$ for the oldest life, we must determine the complement $$a$$ for this life, and those of the two younger, $$b>$$ and $$c$$, from an equal period. The combinations will then be $$(a-x)(b-x)(c-x) = abc - (ab+ac+bc)x + (a+b+c)x^2 - x^3$$, which we may call $$d - ex + fx^2 - x^3$$; hence the fluent is found $$-pv^x \left ( d - e(x+p) + f(x^2 + 2px + 2p^2) - (x^3 + 3px^2 + 6p^2 x + 6p^3) \right )$$ this, when $$x$$ vanishes, becomes $$-p (d - ep + fp^2 - 6p^3)$$, and calling this $$-pg$$, the corrected fluent will give the value of the annuity $$\frac{pg}{d} - \frac{pv^a}{d} (g - en + fa^a + 2fpa - a^3 - 3pa^2 - 6p^2 a)$$. Thus, if the ages are 10, 20, and 30, and the rate of interest 4 per cent. we find, in the Northampton tables, the survivors at 30 4385, of which are 3199; and at 46, the survivors are 3170; whence $$a = 57{\cdot}7$$, and $$b$$ and $$c$$ found also from periods of 16 years after the respective ages, are 68·5 and 91·7. Calculating with these numbers, we find the value of the annuity $$10{\cdot}954 - {\cdot}5 = 10{\cdot}454$$. Dr Price’s short table gives it 10·438; and Simpson’s approximation from the tables of two joint lives 10·563, which is less accurate in this instance, even supposing such tables to have been previously calculated.

It would, indeed, be easy to form, by this mode of computation, a table of the corrections required at different ages for Simpson’s approximation, since these corrections must be very nearly the same, whether Demoivre’s hypothesis, or the actual decrements of lives be employed, both for the two joint lives, and for the correct determination of the three. But the value thus found would still be less accurate, with respect to any other place, or perhaps even any other time, than the immediate result of the mode of calculation here explained.

It may, perhaps, save some trouble to subjoin a table of the values of $$p$$ and their logarithms.

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