548 
ME. B. GOMPEETZ ON THE SCIENCE 
the well-known theorem that the sum is equal to 
P + l 
JL 
2.3 
X 
p-i 
1 p .p—l .p — 2 
6 ‘ 2 . 3 . 4. 5 
1 p .p — \ .p — 2.p- 
2 . 3 . 4 . 5 . 6. 7 
3 . 0—4 - 

&C., 
as very few of the terms will be sufficient, perhaps only the first term, or the two 
first terms, or the three first terms, in which several cases the theorem will stand 
1^+2^+ 3^+ &c x^, respectively equal to 
^p+i 
x^X 
And to explain why high powers may be required, I may first mention that the yearly 
payments which the question may require to be made may not all require to be of the same 
value, but of values depending on the time in which they are to be made, or in other 
words, some function of x : if, for example, the payments were uniformly to increase as 
X increased from one to two, to three, &c., and if the corresponding payments were to be 
one pound, two pounds, three pounds, &c., then each term of the series &c. 
which would express the function of the value required for every value of x from 1 to x^ 
to be summed, would have to be multiplied by and so be changed by that multiplier 
into A47 +Ba'^-1-C^^+ &c. ; and the conditions relative to the yearly payments may be 
such as to introduce multiplications of much higher powers of x. And in complicated 
questions, even when the complication is not great, as for instance in the case which 
may not unlikely occur, to find the present value of the reversion of an annuity to be 
granted to the joint lives A, B after the longest life of five other given lives, C, D, E, E, G ; 
were this to be attempted by the method given by authors by Tables of joint lives, even 
though we might have Tables of the value of any number of joint lives for every age 
(which we have not, and in consequence of not having such Tables very insufficient inter- 
polations are to be had recourse to), the labour would be very great. 
We should have, simple as the question appears by its enunciation, to search for the 
value of thirty-two annuities, of which one would be on two joint lives, six on three 
joint lives, ten on four joint lives, ten on five joint lives, five on six joint lives, and one 
on seven joint lives ; but if the question were to be solved by the method explained, we 
should only have to find, by inspection of the Collecting Table, if it extended to those 
high powders, severally the sums of the several progressions 
r_j_2^4.35&c.; r+2®+3«&c.; V+2^+Z^&c. &c. 
severally multiplied by coefficients, say A„ A^, Ag, &c., found "without much trouble from 
the conditions of the question, such values, A„ Ag, Ag, &c., being of series converging so 
s"wiftly that a few terms which the multiplication produced would be sufficient. 
And in the operation we should find the notation I have proposed useful, as for instance 
to write (5)234, 789(g), for -00000234, 789000000, as numbers of such description "will 
come into operation. 
As the application of the theorem requires an easy mode of finding an analytical 
expression for the anti-logarithm of an analytical expression, and of giving the analytical 
