202 LIFE ASSURANCES. 
266. The columns M and R being constructed for assurances in a 
manner similar to N and S for annuities, if in the formula of Art. 264 
we substitute M and R, for N and S we obtain 
(a-p) (M m —M m+ „) + p (R m —R m+ „ — n. M,„ +n )__ 
D„ 
the single premium 
for an assurance on a life aged ni for the term of n years, commencing 
at £a and increasing £p each year during the term of the assurance. 
Similarly the formula of Art. 265 will become 
= the single premium 
(a + V) (M m — M m+ „) — p (R OT - R 1M+ , t —n. M m+n ) 
D m 
for an assurance for n years on a life aged m, commencing at £a and 
diminishing £p each year during the term of the assurance. 
If we substitute in the last two expressions N m _ t —N n+n _! for D m in 
the denominator, we shall have the expression for the annual premium. 
(Art. 40.) 
If n be not less than the oldest age in the table, M,„ + „ and R OT+B will 
disappear, and the expression for an increasing assurance will become 
(a-p)M m +p.R,„ 
— = the single premium for an assurance on a life 
■L'm 
aged m, commencing at £a and increasing £p each year until the 
. , „ , j ( a +P)M m —p.R m . . , . 
period of decease ; and — l-“ 6 sm gle premium lor 
an assurance on a life aged in, commencing at £a and decreasing £p 
each year until the time of death. 
The annual premium in the last two cases will be expressed by sub 
stituting N OT _i for D m . 
Required the single and annual premium to effect an assurance on a 
life aged 30 for the term of 7 years, commencing at £100, and increas 
ing ¿£'50 each year. (Northampton 3 per cent.) 
(a—-p) (M m —M OT+B ) + p (R m —R m+ „—n.M m+n ) _