1881.] computing Natural anal Tabular Logarithms, fyc. 385 



be represented by suffixing m to the right of that digit. Thus, "0 m l 

 is a decimal fraction beginning with m zeroes and followed by 1, 

 and nat. log 1-0 7 1 = , 8 9 8 50 7 3 8 083 6 53 7 to forty-eight places. Other 

 Avriters have used m in this sense, but it is not applicable to other 

 digits, and conflicts with the usual notation of powers, thus 230 3 looks 

 like (230) 3 =12,167,000, in place of 23,000 = 230 3 . 

 Write equations (5) and (6) thus — 



where 



nat. log (l+p)=x + c=%-\-c 1 + Cc i -\- 



tab. log (1+p) = z + t=z + t 1 + tc 1 + 



Cl = T \x*=xZ x -0833 . . . , c 2 =^=x 5 x -0125 

 tab. log c 1 = 3 tab. log a? + '920 8188-2 . . 

 tab. log c 3 =5 tab. log 03 + -096 9100-2 . . 

 tab. log x=± tab. log q + -359 7271 . . . 



* 2 =3 5 x-351 376 544 673 68 

 tab. log ^ = 3 tab. log z + -645 2501-1 . . . 

 tab. log ^=5 tab. log 2 + -545 7728-1 . . . 

 tab. log z=i tab. log ^ + -118 2500 .... 



* 1= z 3 x -441 824 842 539 87 



(10) , 



(11) , 



(12) , 



(13) , 



(14) , 



(15) , 



(16) , 



(17) , 



(18) , 



(19) . 



By means of these equations the corrections can be calculated from 

 the "quotient" (that is, the approximate values of x and z) either 

 with or without existing tables of logarithms, or the quotient x or z 

 may be calculated to which a particular value of the first correction 

 is due. 



From these has been calculated the following table of the critical 

 values of the first and second corrections, upon which the whole 

 practical use of the corrections depends. The quotients were first 

 taken to proceed from *0 m l to '0^9 by steps of '0 m l. Then the values 

 of the quotients were determined, which reduced either of the two 

 first corrections to "0JL, n being variable, from which point the suffix 

 of 0, or the number of initial zeroes, changed, giving critical values of 

 the corrections. Such quotients were then inserted in numerical 

 order. The approximate numbers were obtained from the quotients 

 on the supposition that p was small enough to make nat. log (1 +jp) 

 =p, and tab. log (l+p)=Mp, to three places of significant figures. 



The suffix of in the first correction, diminished by 1, shows the 

 number of places which are unaffected by that correction, that is, the 

 number of places in the uncorrected quotient which may be trusted 

 without corrections. The undiminished suffix shows a number of 

 places which cannot be wrong by more than one unit in defect in the 



