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Mr. A. J. Ellis. On the Potential Radix [Feb. 3, 



By the positive potential radix of natural logarithms, I mean a 

 table containing 10 r , 2 r , (l'l) r , (l'0 w l) r and their natural logarithms 

 generally from r=l to r=10, but for 2 r it suffices to go to r=3, and for 

 (l""l) r to r=S, and from m=l to m— any required amount. By the 

 negative potential radix of natural logarithms, I mean a table containing 

 the numbers (1 — '0 CT l) r within the same limits, and the negatives of 

 their natural logarithms. If the improved bimodular method of my 

 former paper be used, the number of places which can be determined 

 from *0 m l as a quotient without correction is 3m + 3. By any other 

 method we cannot secure more than 2m places. I was led to the con- 

 struction of a potential radix by the bimodular method. In the case 

 of the numerical radix, the ratio of any two consecutive numbers 

 l'0 m r for a constant m and variable r, continually diminishes, but 

 sufficient was gained for the action of the method, if the ratio remained 

 constant, that is, if the consecutive numbers were the consecutive 

 powers (l'O m l) r , having the constant ratio T0 M 1. Again, as such a 

 power is very nearly equal to a number l'O m r in the numerical radix, 

 that is, as 1 + r x *0 m l + Jr (r— 1) X (•0,»1) 3 + • • ■ is very nearly 

 = l-\~r X 'OmX, it became easy by the action of the method to obtain the 

 numerical from the potential radix. The same is true for the negative 

 radixes. Although from a potential radix the logarithm of a number 

 could be obtained with the same accuracy as from a numerical radix, 

 yet the process is much longer with the former, and hence it appears 

 that the real use of the potential radix is to calculate the numerical 

 radix. This is still more the case for the negative potential radix, 

 which does not succeed in diminishing the work at all, and is here 

 simply introduced for calculating the very useful negative numerical 

 radix. 



The calculation of — 



nat. log (l + -0 m l) = -0 m l-iX-Q 2M1+1 l+ix -0 3lw+2 l- . • . 

 — nat. log (1 — -0J.) = -0JL + J x '0 2m+l l +Jx -0 3m+2 l + . . . 



is very easy, even when 0, that is for l + 'l, although in that case 

 tedious, and is easier the larger m is. It is better to calculate these 

 logarithms as checks for all values of m required, but it is actually 

 not necessary to calculate more than that for the largest value of m to 

 the requisite number of places. Thus, to fifty- two places (a sub- 

 script number denoting the number of times that the digit to which it 

 is appended has to be repeated) — 



f nat. log l-0 4 l = -0 6 9 5 50 4 3 5 83 3 53 4 16 4 8 9 52 2 5 2 9 5 3 4 9 2 54 

 \ -nat. log (1--0 4 1) = '0 4 10 5 50 4 3 5 583 8 50 4 1427 583 928 682 5407 



r nat. log 1-0 I4 1=-0 15 9 10 50 14 3 7 



\ -nat. log (1--0 14 1) = -0 14 10 15 50 14 3 7 . 



