400 



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



the necessity for frequent divisions, as by 1*07, 1*0005, 1*00001, &c., 

 very simple, it is true, but rather lengthy. These are avoided by the 

 use of the negative numerical radix. 



Taking the preparation a, b, c as before, we begin with c, and 

 instead of dividing out by 1*07, we only see how often 1*07 will go in 

 the three first significant figures 705, and finding it to be 6, we 

 multiply c by 0*6, producing d, and subtracting this from c, we obtain 

 d— 1 — "06. After this the number of times that 1006 goes in 6321 

 is 6, the first significant four places of decimals. Hence we multiply 

 e by '0 2 6, subtract the result / from e, and obtain ex (1 — *0o6). It is 

 evident that by this process the number of zeroes with which the 

 decimal fraction commences can be increased by at least one at every 

 multiplication by (1 — '0 m r) a number in the negative radix. We 

 stop when sufficient zeroes are obtained to apply the improved bi- 

 modular method for a sufficient number of places. We might, how- 



1 -07 055 515 875 555 555 555 5(55 



c 



6 423 330 952 533 333 333 333 



d=cx -06 



1 -00 632 184 923 022 222 222(222 



e=c— d=GX (1— -06) 



603 793 109 538 133 333 333 



f=ex -0,6 



1-00 028 391 813 484 088 88(8 888 



g = e-f=ex(l- '0,6) 



20 005 678 362 696 817 777 



h=gx -0 3 2 



1-00 008)386 135 121 392 071 111 



Jc=g-h=gX(l- -0 3 2) 



335 392 704 979 237 550 



Z=2M(&-1 -0 4 8) 



2 -00 016 386 135 121 392 071 111 



m=k+l -0 4 8 



•000 001 676 826 111 397 547 6 



= l-+-m 



•0 17 2 083 1 



correction 



•954 242 509 439 324 874 590 1 



tab. log 9 



•301 029 995 663 981 195 213 7-1 



comp. tab. log 5 



•026 872 146 400 301 340 372 



-tab. Jog (1- -06) 



•002 613 615 602 686 687 981 2 



-tab. log (1- -0 2 6) 



•000 086 867 583 428 580 794 6 



-tab. log (1- -0 3 2) 



•000 034 742 168 884 033 200 5 



tab. log 1 -0 4 8 



11 -o 



tab. log 10 11 



11 -284 881 553 684 748 111 782 8 



sum = tab. loo* N 



o 



ever, have continued the process till the number of zeroes were half 

 of the decimal places, and then the " divisor " would be practically 2, 

 and hence the rule of multiplying the difference (which would be the 

 remaining significant figures) by the bimodulus and dividing by the 



