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/660

 $$\frac{\frac{1}{r} - \overset{m}\overline{\text{ABC},\ \&\text{c.}}}{1+\frac{1}{r}} = \overset{m}\overline{\mathfrak{ABC},\ \&\text{c.}}$$ is the proposition enunciated in No. 81; $$\frac{1}{r}$$ being the value of the perpetuity (112).

193. Examples of the determination of the single premiums for assurances, and of the derivation of the annual premiums from them, have been given in numbers 82—88, also in 95 and 96; but by the algebraical formulæ given here, the annual premiums may be determined directly, without first finding the total present values of the assurances.

194. Example 1. Required the annual premium for an assurance on the life A now 50 years of age, interest 5 per cent.

195. Ex. 2. What should the annual premium be for an assurance on the last survivor of three lives A, B, C, now aged 50, 55, and 60 years respectively, rate of interest 5 per cent.?

196. Ex. 3. Required the annual premium for an assurance for 10 years only, on a life now 45 years of age, interest 5 per cent.

What has been advanced from numbers 99 to 109, needs no algebraical illustration.