estimation of the value of life contingencies. 
253 
^ Cl x 2, ^ 
“ theorem L a+x = L a + x • -p -f- — • -jr -f* &c. a • being taken 
^ b x 2, ^ b 
“ constant ; and also = L & x ‘ ~ 4” ~ ' + & c - 
“ being taken constant ; therefore if these be substituted in 
“ the last equation, it is evident that as the thing is to re- 
“ main true, whatever x may be, the homologous powers of 
“ x must destroy each other ; and consequently we must 
“ have, by making the comparison of the coefficients of those 
“ powers in the equation, • —£■ = • -y ; J == H b 
j — i ^ jj^ 1 1 ^ 
“ • -pr ; • -v- f - = &c. ; and as this is the case what- 
“ ever a and 6 are, it follows that each side of the equation 
i L a 
<e must be constant; therefore putting H^* — g, H 
l b 
“ g and h being constant quantities, we have And 
L a S 
L a h 
“ taking the fluent of this, we have hyp. log of jj = a ; 
“ p being some constant quantity ; and this may be reduced 
“ to the form L a = y>. e°. a , e standing for the number whose 
“ hyperbolical logarithm is — ; and taking the fluent again, 
“ we have L a — e' — e".e a ; e t e' and e" being independent 
“ qonstant quantities, though e" is = — p . -|-the independence 
remains because p . y is arbitrary. And as the equation L a = 
“ pe°a, makes L = p. — .c a a 2 = — a L ; L = —a L ; 
c <* r g 5 a a g a’ 
MDCCCXX. L 1 
