•524 
ME. B. GOMPEETZ ON THE SCIENCE 
done. In the first place, I observe that it is usual to consider the logarithm of a posi- 
tive quantity and a negative quantity the same ; thus, suppose g were a positive number 
greater than 1, its logarithm would be positive, and the logarithm of this logarithm 
might be written the logarithm of the logarithm of g ; but if g were a positive number 
less than rrnity, its logarithm would be negative ; and then Wg, if X stood for the loga- 
rithm of, and Xkg for the logarithm of the logarithm of, would have no meaning in the 
positive scale of logarithms. This distirrction I did not notice in the notation in my 
paper of 1825, though, properly, the notation ought to have beerr, as g was found to 
be a positive number less than 1, — \g)’, but I did not neglect in my calculation 
to atteird to the consequence; because, till the value of g is known as to its being 
less or greater than unity, there is a convenience, but only to be adopted with care, in 
writing \lg. 
Irr this paper, as has been already explained, I use the prostrate small I, thus c — =, 
with the loop upwards, the first to denote the Napierian logarithm of, the second the 
comrrron logarithm of; and with the loop downwards, thus = — =, to represent the anti- 
Napieriarr logarithm of, and the anti-common logarithm ; and would mean the 
comrrron logarithm of the Napierian logarithm of, if they are placed horizontally in the 
line on the left of a character ; thus -=> c- x would signify the common logarithm of the 
Napierian logarithm ofx; but if placed below the character, which may be in some cases 
more convenient, thus w, would signify the common logarithm of the Napierian loga- 
_s- 
rithm of x, and x would signify the common logarithm of minus the Napierian logarithm 
of X. 
Art. 11. It appears now time to show how to use the original formula lj^=d.g'^ , in 
order to reduce it into a form for practice, and which may be written 
-Ij, = c 
-d-\-g{l-\-g.x-\-\g^x''-\-~-g^x^+ &c.). 
where whether put to the left of a character or below it, stands for the Napierian 
logarithm of, and q, - q\ ^ q\, &c., multiplied by stand for the coefficients of 
A’, x^, of, See. in the development of c^L^; and I add, with respect to any function 
ax-\-baf-{-cx^., &c,, we would represent the anti-Napierian logarithm of it by 
&c. The value of these coefficients will be difierent according as 
the values d, q are difierent, which values, from what has been stated, will difier for 
every selection used in the vital rule of three of the three lives, as, though they have for 
a long period in every selection the appearance of constancy, they are not absolutely 
constant ; and I observe, that I use the same notation for the development of the value 
of c-Lj, in the more perfect formula which I have given, namely, 
c-L^=zCQ kf—<=-(e''.qxx—h) - j-p* J 
but I should state the formula to which I here refer is 
>L^= CS + kf — .x—h) 
