392 Mr. A. J. Ellis. Improved Bimodular Method of [Feb. 3, 



correction changes. The results are given under "5. Full Correc- 

 tions," in Tables I and II. 



But although it is by no means difficult or very troublesome to use 

 the formulas (13) and (17) for finding the first correction, it is 

 always inconvenient to use two tables. It would be manifestly 

 impossible to give a table of corrections to six figures within reason- 

 able limits. Hence, leaving the " full correction " to be found, when 

 desired, by these formulas, I append a table of " short corrections," 

 so as to obtain twelve places of the result from Tables I and II at 

 sight. The thirteenth place has been allowed for, so that the result 

 may be thoroughly trusted, but in the " completion " an error of one 

 unit in the twelfth place may easily creep in unless " full corrections " 

 are used. These "short corrections" have been calculated from the 

 formulas (15) and (19), by assuming successive values of the first 

 correction, as "0 13 5, '0 n 15, '0^25 and so on, and calculating the 

 corresponding value of the quotients. But in the table itself these 

 corrections are entered as '0 n l, *0 n 2, &c. The limiting correction is 

 reached when the corresponding quotient is the next least to that due 

 to the number FOOL These twelve places are fully as many as are 

 required for ordinary purposes, and for them only thirteen out of the 

 eighteen places in the tables should be used. 



Section IV. — Bimodular Tables and Examples. 



Table I applies to natural logarithms giving from nine to sixteen 

 places, according to circumstances, with no corrections, twelve places 

 with short corrections, and fourteen to sixteen places with full 

 corrections. 



Rule to find the logarithm from the number. — Reduce the given 

 number to the form of a decimal fraction with an integer less than 10. 



Multiply and divide by such whole numbers less than 13 as will 

 reduce the number to one less than 1*1, as shown in Section II. 



Find the next less number in "1. Table for Interpolation," and 

 first subtract it from the reduced number, then omit the decimal 

 point, and multiply by 2, forming the " dividend;" secondly, add this 

 next less number to the reduced number, and then omit the decimal 

 point, forming the " divisor." 



Divide the dividend by the divisor by simple contracted division to 

 as many places as are required. Correct the quotient, as may be neces- 

 sary, by the table or formula of correction, No. 6 or 5. 



Add the logarithms of the divisors and the arithmetical complements 

 of the logarithms of the multipliers used in forming the reduced 

 number, to find the full corrected logarithm. 



Table II applies to Briggs's or Tabular Logarithms, giving from 

 nine to sixteen places, according to circumstances, with no correc- 



