CONNECTED WITH HUMAN MOETAEITY. 
521 
And to make this simple theorem easily available for the calculations, it will be con- 
venient to have a Table of powers of numbers increasing regularly in arithmetical pro- 
gression, with the common difference of 1, from 1 to 100, of the annexed form, which I 
call the Collecting Table, and which will be found of immense service. Such, for 
instance, as the following example ; but a more extensive Table will be given further on. 
ce. 
Sums of x. 
Sums of x^. 
3^. 
Sums of x^. 
Sums of x^. 
To continue 
to x=10, 
or more. 
1 
1 
1 
1 
1 
1 
1 
1 
&c. 
2 
3 
4 
5 
8 
9 
16 
17 
&c. 
3 
6 
9 
14 
27 
36 
81 
98 
&c. 
4 
10 
16 
30 
64 
100 
256 
354 
&c. 
5 
15 
25 
55 
125 
225 
625 
979 
&c. 
6 
21 
36 
91 
216 
441 
1296 
2273 
&c. 
7 
28 
49 
140 
343 
784 
2401 
4676 
&c. 
8 
36 
64 
204 
512 
1296 
4096 
8772 
&c. 
&c. to 
100 
&c. 
&c. 
&c. 
&c. 
&c. 
&c. 
&c. 
&c. 
It is evident that if this Table be continued to great values of x, say to 100, or high 
powers of x, which may be necessary in some cases, we shall get very high numbers for 
the value of x^ ; but these numbers, when multiplied by their coefficients, which will 
be very small, may not prevent the series from converging. 
Art. 8. But the case renders a new arithmetical notation convenient, which will be 
explained further on, as only a few of the most significant figures will be required, and 
the remainder may be considered as noughts. This analysis does not only require the 
anti-Napierian logarithm to be taken of analytical expressions, but also the reversion of 
analytical expressions into Napierian logarithms to be found; for if there be a series 
a-{~hx-\-cx^-\-dx'^, &c. when the coefficients <z, h, c, See. are a converging series, the 
Napierian logarithm of it is the Napierian logarithm of a 
+ &c. 
•ol -x-^- X A — X 
2\a 'a 'a 
* 1 /Z» c 
“Tq ( ~ x -\ — 
' 3 ya 'a 
x‘^-\-- X^, &C. ) &C., 
which may be represented by a-\-^hx-\-^cx^Ar'dx^-^ See., and would give the values of 
’c, ^d, See., or rather proceed as follows. 
Supposing the Napierian logarithm of 
1 -f BiX-^B.^X^ -f 
were equal to 
Aja7+ AjO^* + Ajir^ &c. , 
if these equations be put into fluxions, we shall obtain, after dividing by x, 
B, + 262.^ + 3633?® +4 643^ &C. . , n a OA O A A 
l + + + &c. -\-AK^x\ See., 
4 B 2 
