^40 
ME. B. GOMPEETZ ON THE SCIENCE 
enjoyed by any number of joint lives were calculated, they would in many cases be 
more convenient than the value of annuities on joint lives ; as, for instance, if we had 
to find the reversion of an annuity on three lives now aged a, 5, c, we should only have 
to search the Table of the reversion on three joint lives, and we should have the value 
at once : but if the calculation were to be made by methods above, and we represent the 
anti-Napierian logarithm of the present expression, as we did that of the former expres- 
sion, by &c., the present value of the annuity of which the first payment 
is to be in the time w, and the last in the time z, will be represented by the expression 
^"+P'S ;| 
^.» + 2_|_p3g 
/yi^ + 3 
vv 5 
&c.. 
where it may be observed that if the condition of the deaths, say of a, (3, y, did not enter 
the question, this formula ought to be the same as the first, which would require P to 
be 1, which may appear a paradox; but the paradox is solved if we consider the con- 
dition would be the same if they entered in the question or not, if they could not die ; 
that is to say, ifp„, &c., and consequently P be =1. Then, by means of the 
Collecting Table, the value of the above series will be given. 
Art. 21. The valuation by the old methods of annuities depending on such compli- 
cations, by the aid of Tables of only two joint lives, as the value of the annuities on 
three, four, five, &c. joint lives can only be obtained from them by very troublesome 
interpolations of very inaccurate results, evidently calls for an improved method. For 
instance, if the annuity depended on the joint existence of three lives of the present 
ages of a, J, c after the existence of all the lives d, e, /, the labour and insufiiciency hi 
point of accuracy of that mode of valuation will immediately appear ; because, the 
chance of each of them individually being living in x years being for the sake of ease 
represented by a, b, c, d, e, f respectively, by multiplying out at length the chance of 
the conditions required being fulfilled, that chance will be then represented by 
abcy,\—dy,\—ex.\—f—abc—abce—abcd—abcf-\-abced-\-abcfe-\-abcfd—abcdef. 
It appears that by the old methods, with only Tables of two joint lives, we should have 
to interpolate the value of one of three annuities of three joint lives, three of four dif- 
ferent annuities of four joint lives, three of five different annuities of five joint lives, 
and one annuity of six joint lives, and the results of every one of these interpolations 
would be inaccurate. 
Art. 22. On special risk, considered with respect to single lives, and on many lives in 
connexion ; and the valuation of certain contingencies ; and of property dependmg on 
those contingencies and under influential connexion ; I am not aware whether or not any 
one has gone before me on this important subject; but I have not, in my official practice 
in the science of assurance, lost sight of its existence. And to draw the reader’s atten- 
tion at once to the subject, I will suppose we had the two problems to consider — the first, 
to assure a sum on the extinction of the coexistence of two coexistent lives A, B, com- 
monly called two joint lives, that is, to be payable at the death of the first of the two 
