538 
ME. B. GOIkIPEETZ ON THE SCIENCE 
and the Napierian logarithm of the present worth of unity certain in one year to he f, 
find the anti-Napierian logarithm of 
&c., 
and the anti-Napierian logarithm of 
IBj -(- Cj . -j- -{- C2 . x~ -j- B3 -|- C3 &c. , 
which will be easily done by the method explained above, and will only require a few 
terms in consequence of the coefficients of x in the above expressions being small ; and 
calling the first analytical anti-logarithm 
1-f 'A.r-f &c., 
and the second 
&c., 
and putting the fluent of minus the fluxion of the first 
— QAx-\-2.'^Axx-{-^.^Ax^x\ &c.), x(l + ^B.r+^BA’^-f&c.)='Ki‘+*K.ri:-f &c., 
then the value of the assurance for the term commencing in t years and ending in x 
years will be 
+ 1 . 3K &c., 
and will not require many terms, and the equation is thus solved without the Tables of 
the values of annuities ; and in a similar way are other questions of this species ; and 
the solution in the more compounded cases of art. 7 of my first paper, such as example 1, 
taken from Messrs. Baily, or Milne, or Mr. Moegan’s original paper, which is the 
foundation of them, containing an interfnediate conditional contingence of survivorship, 
and which in my paper above alluded to contains the double fluent reducible to a single 
fluent multiplied by a variable quantity, and a single fluent added, and the solution in 
much more intricate cases, and containing any number of lives, and even a long string of 
intermediate contingencies which would involve double, triple, quadruple, &c. fluents, 
&c., may be effected. As a simple example, suppose there were three separate com- 
binations of lives, which I will call A, B, C combinations of lives, and it be required to 
find the value of an assurance on the failure of the C combination after the failure of 
the B combination, the A combination having failed previously, provided that failure 
should happen during certain given periods. First, find by the method above the analy- 
tical value of the contingency that the B combination shall fail after a given time, the 
combination A having failed previously ; and suppose this to be 
&c. ; 
and having found the anti-Napierian logarithm of 
1 + K IC IC 
and having found the Napierian logarithm of the separated chances of every life in tlie 
combination B being living, and added the sum to the Napierian logarithm of 
and having found the anti-Napierian logarithm of this sum, and multiplied this by K, 
