1881.] computing Natural and lobular Logarithms, fyc. 387 



last place. The suffix of in the second correction, diminished by 1, 

 shows how many places of the quotient, after applying the first 

 correction, are left unaffected by the second correction, that is, how 

 many places can be trusted on applying the first correction only. 

 For natural logarithms it will be seen that this never gives less than 

 5m + 1 places, that is, 2m + 1 places in addition to those determined 

 without correction. Thus in Table I, where m is never less than 3, 

 we can always obtain sixteen places. For tabular logarithms, as in 

 Table II, we must first observe a critical value in the numbers them- 

 selves. In that table the number whose logarithm is finally 

 sought, must be less than l'OOl. Hence, while in the upper part of 

 the preceding table of critical values, m will always be 3 or more, in 

 the lower part, m — 1 will always be 3 or more, so that m will always 

 be 4 or more. As far then as the quotient '0 3 434, the first correction 

 gives only 5 . 3 + 1 = 16 places, and this is the largest quotient that 

 can commence with *0 3 . If the significant figures are greater than 

 434, then m will be 4, and up to the quotient *0 4 778 we can trust 

 5 . 4=20 places, and beyond it we can even trust 19 places. Observe 

 that '0 4 9 at the bottom of this table is followed by "0 3 1 at the top 

 (II, second column), for which, also, the second corrections leave 

 5 . 3 + 4=19 places unaffected. 



But in determining the full number of places of the first correction 

 from the uncorrected quotient by equations (12) and (16), we are, 

 of course, obliged to take so many significant places, that on cubing 

 the result and multiplying by the proper coefficient, no error affecting 

 the full number of places should be committed. The number of 

 places required for this purpose is so large that if we calculated the 

 result directly, the present method of correction would be illusory. 

 Hence it is necessary to use common seven-place logarithmic tables 

 which can be trusted to six places. Consequently, we can use onlv 

 six significant places in the quotient for finding the correction, and we 

 thus introduce an error not exceeding half a unit in the last place in 

 excess or defect. On estimating the limiting effect of this error, 

 I find practically that on using six significant places of the uncor- 

 rected quotient to determine the first correction, we may trust all 

 six places of the correction found. The total number of places that 

 can be trusted, when this error is allowed for, depends on the quotient. 

 Let r be the significant places of the quotient converted into a decimal 

 fraction with one unit place. Then the real quotient is '0 m l X r, but 

 on taking only six significant places, we use as a quotient - m l Xr + 

 •0 m+6 l X 5, and the error thus made in the correction may be taken as 

 the term involving r 2 in the cube of this number divided by 12 M 2 , 

 that is, as •0 3m+8 lxl5r 2 -f-12M 2 . Then, putting 15r 2 ^12M 2 = 10 and 

 100, and finding the corresponding values of r, we obtain the critical 

 values of the quotient where the suffix of in the error of the 



