^536 
ME. B. GOMPEETZ ON THE SCIENCE 
scientific inquirers. For instance, the first of them and the following, in which, instead 
of there being only three lives concerned, there are any number : the first question of 
those gentlemen is to find the value of the assurance of the hfe' of A provided in the 
lifetime of B and C ; and my solution, expressed in the notation of that paper, is as 
follows : — -p representing one year, if the tables are calculated for annual payments; 
n the period from which the assurance is to commence ; 711 the period to which the 
assurance is to continue ; r the present value of unity due in one year certain at the 
rate of interest involved in the question ; then the value calculated from the Table of 
annuities for every unity assured, the present ages of A, B, C being «, h, c respectively, is 
L, 
p 
a -p, b-\p, c-\p ^ I 
T ^ 
b, c 
a—p, h — \p, c—\p 
^h-\p, c-\p I 
T A 
a, b — \p, c—^p 
■^b, c 
an extremely small and insignificant quantity being neglected. 
Art. 17. This, with Tables calculated for two joint lives, and the annuities for 
those interpolated from them, instead of being impracticable, as the other solutions 
are, ufill be found perfectly simple and easy. And in my second Tract, namely that 
of 1825, I gave Tables for calculating the values of annuities at any likely rate of 
mortality to be adopted, and at any rate of interest, and applicable when the lives are 
subject to different rates of mortality, and the same theorem for the value of assurance, 
onutatis mutandis, however many lives B, C, D, E, &c. are proposed to be jointly li-^fing 
at the death of A. And with regard to art. 7 of the paper of 1825, I will only in this 
place direct the reader to my solution to the question, which is one of the more intricate 
nature of those given, and solved by those gentlemen. The question being the value of 
the assurance on the first death of A and B, which shall be second or third of the deaths 
which shall happen of the three lives A, B, C, my solution to this question in my fii’st 
paper is simply 
r 
T r 
T 
r 
JL 
2 • 
'p 
'p 
n 
n 
n 
n 
0, 
a, G — m 
that is, half the sum of the assurances of B, of A, of B, C, of A, C, less the assurance of 
the joint lives A, B, C, all of which are easily computed. This solution, when I obtained 
it, seemed to me so enormously superior to those which those learned gentlemen had 
given to it, that I found it necessary to show my readers that they depended on the 
same principles, by reducing the portions of my solutions to the portions of the earlier 
solutions ; and the remaining questions of that section are of a similar superiority to 
the earlier solutions. 
Art. 18. But in making this remark I wish to express myself indebted to those gentle- 
men for their labours, whose names remain honoured in the memory of scientific persons 
who have cultivated different branches of science, and to state I allude to the comparison 
that my readers may follow an improved path if they meet with one, and to remind 
those who may follow with benefit for science others who have trodden down the aspe- 
