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/659

 two preceding numbers become respectively

185. ''Corol. 2''. In the same manner it appears, that,

186. ''Corol. 3''. Also that $${}_{t\urcorner}\!\mathfrak{A} = v \left ( 1 - {}_t\!av^t \right ) - (1-v) {}_{t\urcorner}\!\text{A}$$ or $${}_{t\urcorner}\!\mathfrak{A} = v \left ( 1 - \frac{{}^t\!a}{a} v^t \right ) - (1-v)\,.\, \left ( \text{A} - \frac{{}^t\!a}{a} v^t \cdot {}^t\!\text{A} \right )$$.

187. ''Corol. 4''. When the assurance is on the joint continuance of all the lives, the formulæ of No. 184 become respectively

And those of numbers 182 and 183, $${}_{t\urcorner}\!\mathfrak{ABC},\ \&\text{c.} = v \left ( 1 - \frac{{}^t\!(abc,\ \&\text{c.})}{abc,\ \&\text{c.}} v^t \right ) - (1-v) \times$$ $$\left [ \text{ABC},\ \&\text{c.} - \frac{t(abc,\ \&\text{c.})}{abc,\ \&\text{c.}} v^t \cdot {}^t\!(\text{ABC},\ \&\text{c.}) \right ]$$, and $$\odot {}_{t\urcorner}\!\mathfrak{ABC},\ \&\text{c.} =$$ $$\frac{1 - \frac{{}^t\!(abc,\ \&\text{c.})}{abc,\ \&\text{c.}} v^t}{1 + \text{ABC},\ \&\text{c.} - \frac{{}^t\!(abc,\ \&\text{c.})}{abc,\ \&\text{c.}} v^t \left [ 1 + {}^t\!(\text{ABC},\ \&\text{c.}) \right ]} + v - 1$$.

188. ''Corol. 5''. According as the assurance is in the last survivor of two, or of three lives, the formulæ of No. 184 become respectively

And those of numbers 182 and 183 become

Where $${}_t\!(\overline{ab}) = {}_t\!a + {}_t\!b - {}_t\!(ab)$$, (141). and $${}_t\!(\overline{abc}) = {}_t\!a + {}_t\!b + {}_t\!c - \left [ {}_t\!(ab) + {}_t\!(ac) + (bc) \right ] + {}_t\!(abc)$$, (142).

For the values of $$\overline\text{AB}$$, $$\overline\text{ABC}$$, $${}_{t\urcorner}\!\overline\text{AB}$$, and $${}_{t\urcorner}\!\overline\text{ABC}$$, see numbers 157—159, and 161.

189. ''Corol. 6''. When the assurance is on the joint continuance of the two last survivors out of the three lives A, B, C; the formulæ of No. 184 become respectively

Those of numbers 182 and 183,

$${}_{t\urcorner}\!\overset{2}\overline\mathfrak{ABC} = v \left [ 1 - {}_t\!\overset{2}\overline{(abc)} v^t \right ] - (1-v) {}_{t\urcorner}\!\overset{2}\overline\text{ABC,}$$ and $$\odot {}_{t\urcorner}\!\overset{2}\overline\mathfrak{ABC} = \frac{1 - {}_t\overset{2}\overline{(abc)} v^t}{1 - {}_t(\overset{2}\overline{abc}) v^t + {}_{t\urcorner}\!\overset{2}\overline\text{ABC}} + v - 1$$.

Where $${}_t\! \overset{2}{\left ( \overline{abc} \right )} = {}_t\!(ab) + {}_t\!(ac) + {}_t\!(bc) - 2 {}_t\!(abc)$$, (143).

For the values of $$\overset{2}\overline\text{ABC}$$ and $${}_{t\urcorner}\!\overset{2}\overline{\text{ABC}}$$ see numbers 157, 160, and 161.

190. $$v \left [ 1 - {}_t\!\overset{m}\overline{(abc,\ \&\text{c.})} v^t \right ] - (1-v) {}_{t\urcorner}\!\overset{m}\overline{\text{ABC},\ \&\text{c.}}$$ the value of an assurance on any life or lives for the term of $$t$$ years, which was given in No. 182, may also be expressed thus:

And this, in words at length, is the rule given in No. 93.

191. When $$t$$ is not less than the greatest possible joint duration of any $$m$$ of the proposed lives, the last expression becomes $$\left ( 1 + \overset{m}\overline{\text{ABC},\ \&\text{c.}} \right ) v - \overset{m}\overline{\text{ABC},\ \&\text{c.}}$$ which is also equivalent to the first in No. 184; and, in words at length, is the rule given in No. 97, for determining the value of an assurance on any life or lives for their whole duration.

192. By substituting $$\frac{1}{1+r}$$ for $$v$$ (146) in the last expression, it becomes $$\frac{1 + \overset{m}\overline{\text{ABC},\ \&\text{c.}}}{1+r} - \overset{m}\overline{\text{ABC},\ \&\text{c.}} = \frac{1 - r. \overset{m}\overline{\text{ABC},\ \&\text{c.}}}{1+r}$$, or $$\frac{\frac{1}{r} - \overset{m}\overline{\text{ABC},\ \&\text{c}}}{1+\frac{1}{r}}$$. And 1em