518 
ME. B. GOMPEETZ ON THE SCIENCE 
or its equal, 
will for a long series of years, with d, g, g constant, be a very near approximation to the 
values which the data afford, provided d, g, g be determined by the vital rule of three, 
by taking x successively equal m, m+ 2 %, where m andm+ 2 ?zare somewhere near 
the limits of age through which the formula is meant to be applied, is a most valuable 
information ; on account of the application of the above law to the most complicated 
intricacies which may be proposed, by means of what I consider rather a novel branch 
of mathematical investigation, which I term vital algorithm and analytical arithmetic of 
logarithms and anti-logarithms, in very complicated entanglements and disentanglements : 
for by the law of mortality we can get the value of L,, the number living at the age 
or of La+„ or them logarithms in a series of powers of x with very converging coeffi- 
cients, as wiU be shown further on, and thence we can obtain a series of the values of 
X X &c., however many lives there may be ; or we may have the logarithms 
&c. , to which if we add the logarithm of say x into the logarithm of r, 
r being the present value of unity due in one year certain, at the proposed rate of interest, 
and deduct from this sum the sum of xL^+XL^-f-XLc &c., we shall have the logarithm 
of unity to be received in x years on the condition that all these proposed lives be in 
existence in x years time. And then finding the analytical expression for the anti-loga- 
rithm of this, expressed by a series AQ-{-K^x-\-A^x^-\-A 2 Pif‘-\- &c., in which Aj, Aa, A 3 , &c. 
express a series of very converging terms, we have the present value of unity to be 
received in x years, and then by a Table to be presented in this tract of powers of 
numbers, and the sum of these numbers from x—l to any proposed required limit, we 
can by multiplying the coefficients Aj, Aa, A 3 , &c., of the series Aj, Aa, A 3 , &c., which 
will be mostly very convergent coefficients, if not always, find the value of all the annual 
payments between any one time and any other; many examples of the ease of this 
method comparatively with the seemingly insurmountable difficulties which appear on 
these subjects, I hope to have time to lay before the reader, and to discuss many other 
branches which may be of interest. But not to keep the reader in suspense, I will 
—fi^ 
continue the subject referable to the formula 'L^=d.g\ , where from any proposed value 
of X to any far greater value, d, g, g may be considered as constants, though not actually 
so ; and now I will show how nearly the formula 
—\~\eWg^.x—h) 
between the ages of 10 and 80 agrees with the Tables ; and will then show how to find 
the constants of the equation from the commencement of life to extreme old age from 
the more complete formula 
X]L_j,=CS Jc . \g^.x — /ij-j-jU/j'*, 
where from birth to extreme old age all the quantities except x are constant, but of 
interestuig values, so that commences its significance at birth, and within less than 
one year decreases to total insignificance, but will never absolutely vanish ; the term 
