Mr. Morgan on Survivorships. 39 



preceding problem; then will the value for the 1st year be 



S a — s.c — d.b' , 7 t, j , a — s.dtf , 



= — j — into \- a — s .0 — 0' .a 4- 4- 



a b cr 2 ' ■ 2 ' 



a— s . r—d . b—V r ■, , S , s— t . d—e . b" , 



; for the 2d year == — r — ; into + 



2 •* a be r z 2 ' 



' 7 T7~i — T11 1 s—l.eb" , s — t.d — e.b — b -\-b" , t—t.d—e.b' 



s-t .b -b'+ b". e H h i- H ; 



_j_ 5 ~ • c ~ — t an( j so on f or the 3d, 4th, and remaining years. 

 These several fractions may be expanded into twelve different 

 series, whose sum may be found = S into ^— ^ x V — - A — ABC 



AC ' AB «. . AK , -A 1 . HBC 



_j_^._^___L-^ _[- -— — x-AT- ABT-f — 



' 2 zr zc ' 2 a 2 cr ' 2 ar 



ds 



x a _l NC + NB — NBC — x 1 + NBT. 



1 ■ 2a cr ' 



If the lives be equal, the value, by the first two rules, will be 



r-i . V— C+CC—CCC . dd 



into — x 1 + CI T 



2 r ' 2 c. r. r ■ 



2r ' 2 c cr ' 2 cr 



1 + a CT — CCT ^ "' CKK J.-^ x2 CK- CCK ; and by 



1 2 C C ' 2 C J 



the third rule it will be = S into '-' ■ v-i+ia—hi , 



•>. r 



r—\ . V— C + CC—CCC 



2T 2C 



A 7—7^^ dd 



CCK — CK + —^— x 1 + CT — — x 1 + CTT. In both 



1 2c r ' 2cc r ' 



cases, all the fractions after the first destroy each other ; so that 



s . r_i 



the general rule becomes = ^ x V — C -}- CC — CCC, 



which may be proved to be the true value, from other principles. 

 The solution of this problem might have been derived from 

 that of the first problem* in my third Paper, and from that of 

 the 6th problem -f in my last Paper on this subject, " by finding 

 " the value on the death of A, should his life be the first that 



* Phil. Trans. Vol. LXXXI. page 248. 



f Phil. Trans, for the year 1794, page 253. 



