164 
LIFE ASSURANCES. 
+ 1 *Pf. 
(in, nij, m,g, &c.), t—l 
^ ^ ^ P(m, raj, j»2, üc.), I + 1 P(,™> »*i> ”<i, &«•)> 2 *'* ^ ‘Pfai>m\, *»2> ^ 1 
4l+a 1 . 
| (m, m v in g Sc.) > j 
1 i-ll •> 
and 
2 r".p — 
O (m, m 1( m 2 ,&c.), 
(m, n»i, nig, &o.) 
“ - r { 1 +| , 
and since the value of an annuity for t—1 years is the same as the 
value of an annuity for t years, diminished by the value of the ith pay 
ment, which in this case is r‘. p- , we obtain 
’ 1 (m, ml, m2, &.C.), I ’ 
(m,tnx, m2, 4c.)^— 11 ’ 7 -Pirn, m^, m it &c.). i+^m, m v m%, *c.)^| ^(m, n»i, rnj.&c.)^ 
y* | J — P ,p — 1 Q _ E} (i H ■ 
t (in, 1»1, t' ' ' (til, IHj, W2> *°0 ^ * 
when there is only one life this formula becomes 
A^—rl l+a (m) [ — «(„,) =r{l— ^Pm,i} —’(1— /•)«(«) • 
(i (. ^ J n <i 
198. By Davies’s method— 
The single premium for a temporary assurance is found by sum- 
ming the first t terms in the numerator of Art. 189, and dividing them 
by the denominator l m .r m . The sum of the first t terms is (from the 
construction of column M) evidently equal to the difference between the 
number in column M opposite the present age of the life and the 
number in the same column opposite the age t years older : 
M m —M m+t 
199. When columns D and N only are given, the value may be 
found without previously calculating the value of the annuity, thus: