estimation of the value of life contingencies . 245 
L x: p, ? .L r + x m ^r+n • (L«+Jm '.p, ? + p + n • L ?+ „) since 
according to hypothesis L^ +n+ i m = ^p+n — T m ^p+n anc * 
t T = L , mL , ; and by notations Art. 4. Sec- 
tion 1, L„ +Jnl:?i} -stands for L #+ , + j M x %+»+!„■ And we 
therefore have the fluent generated, whilst a? from » becomes 
= n+m, that is » L .r:/>, ? — • — ^L' r +» * ( L «+|m -.p, q 
n + m 
• L ?+m ) or its equal -(L r+n L. +n+W2 ) .h n+ i m .p }q 
— (L r +ri — L r+n+JW ) . (L/»4-« —p +n+m ) • (L ?+ „ L ?+n+m ). 
And here it should be remarked, if m represents one year or 
less, as will be frequently the case in the application of this for- 
mula to questions of practice, that generally the part (L r+rt 
J-V+n + m) • (L p +n L p + tl + m ) . (L q +n ’^q + n + m) 
sufficiently small to be wholly neglected ; but should greater 
accuracy be required, the case would be rare if it might not 
be considered constant throughout the possible duration of 
. the joint lives. 
Article 10. By writing L ?+w — L ?+JJ+m for wL' f+B , &c. we 
shall also have 0 
n + m 
^-‘x:p,q % ^r+x 
~ J x:p,q 
^ r +x {^r + n 
x \^“p+n, q+n ]T ^p+ni^q+ri 2 
' + « ^/)+w + m) "f" t(^ + « ^ + M + m) * + n ^p+n + m)\ 
: p, q ' -f # ( ^ 
r •+ n •{■n + m 
Kk 
) X {y L »:A,+A 
MDCCCXX. 
