. Journal . r=40, ^ = 0,with interest at 3 per-cent, an approximate value of \io^t^ix:i+twas thus deduced of -1156, as compared with the value of -1166,calculated from the 0^^^ Tables. The above process is laborious, and not very satisfactory. Theformulas can, however, be much simplified, and the work reduced,by eliminating the terms involving a in formulas (8) to (11).Since a= — ra = rlogioS (where s is the Makeham constant) wehave, in formulas (8) and (9), and ,.;?.+.= 10°+^ ^slOC; so that, in both cases (as is well known), the annuity-value can becomputed at the special rate i, where {l-\-i
. Journal . r=40, ^ = 0,with interest at 3 per-cent, an approximate value of \io^t^ix:i+twas thus deduced of -1156, as compared with the value of -1166,calculated from the 0^^^ Tables. The above process is laborious, and not very satisfactory. Theformulas can, however, be much simplified, and the work reduced,by eliminating the terms involving a in formulas (8) to (11).Since a= — ra = rlogioS (where s is the Makeham constant) wehave, in formulas (8) and (9), and ,.;?.+.= 10°+^ ^slOC; so that, in both cases (as is well known), the annuity-value can becomputed at the special rate i, where {l-\-i)~^ = sv; and,similarly, in applying the method illustrated by formula (12), 1906.] Asstirances, v:iih Allowance for Selection. 79 g=log/l + j) will be replaced by § = loge(l+ i) =«^i~ + ^-By thus eliminating the terms involving a, the series given informulas (JO) and (11) will be much simplified, and, whenarranged in ascending powers of C, that is of —^c^, will runaccordino; to the following scheme :. + 1 rt ^ V v-. — •> + ^ ; -f li —-1 i i: V^ 1 5 c. 5 + :i X ■* 1 _ 1 + 1 ^ ^ y, ,. -|~ — ^ ^ ^ M + ^T^ 5 1 ^ >*v s g ~- — ^ 1 -?- ! + X ii O 1 1 ^_^ . 1 + 5i 1 5 ^ ^ ^ Sd ^ U£ -TI + ^^ -* p s ^*v ^•^-v ■^v ■^ s ■ ^^ rY ^ T- - ; X ^» P T ?: • >^ >^ ■**V •^^ S O I 1 L 1 1 1 i ] ~ i,- : 5= — : + + + 4- + + 80 Ajiproximate Method of Valnation of Whole-Life [Jan. This shows clcarl)^ the law followed by the successive terms ; andthe general term, say the coefficient of C involving the mthpower of M, will be equal, where m > n, to n 1 - , /»-(« +Air)«= ,- —r/~(« + ^)^ \m \m — vJ ^ ^ \m, — v ^ and, where w? = /*, to —p [(/c + -»|r) —/c]. Summing these expressions for all values of m from w to co,we have -|»l M2- .,„_j («+iv: _ M- ^„^ which can be shown to be equal to « ^ ^ - ■ This is the coefficient of C», and, multiplying this coefficient by[ — ^y, we have, as the coefficient of c-, this expression agree
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