in which the series is always convergent, so that the formula affords a method of deducing the logarithm of one number from that of another. As particular cases we have, by putting q = 1, _ _ | p - 1 /p - 1\³ /p - 1\^5 | log(e) p = 2 | ----- + (1/3)( ----- ) + (1/5)( ----- ) + &c. |, |_ p + 1 \p + 1/ \p + 1/ _| and by putting q = p + 1, _ _ | 1 1 1 | log(e)(p + 1) - log(e)(p) = 2 | ------ + (1/3)--------- + (1/5)----------- + &c. |; |_ 2p + 1 (2p + 1)³ (2p + 1)^5 _| the former of these equations gives a convergent series for log(e)p, and the latter a very convergent series by means of which the logarithm of any number may be deduced from the logarithm of the preceding number. From the formula for log(e)(p/q) we may deduce the following very convergent series for log(e)2, log(e)3 and log(e)5, viz.:-- log(e)2 = 2( 7P + 5Q + 3R), log(e)3 = 2(11P + 8Q + 5R), log(e)5 = 2(16P + 12Q + 7R), where 1 1 1 P = -- + (1/3) · ------ + (1/5) · ------ + &c. 31 (31)^3 (31)^5 1 1 1 Q = -- + (1/3) · ------ + (1/5) · ------ + &c. 49 (49)^3 (49)^5 1 1 1 R = --- + (1/3) · ------- + (1/5) · ------- + &c. 161 (161)^3 (161)^5 The following still more convenient formulae for the calculation of log(e)2, log(e)3, &c. were given by J. Couch Adams in the _Proc. Roy. Soc._, 1878, 27, p. 91. If 10 / 1 \ 25 / 4 \ a = log -- = -log ( 1 - -- ), b = log -- = -log ( 1 - --- ), 9 \ 10 / 24 \ 100 / 81 / 1 \ 50 / 2 \ c = log -- = log ( 1 + -- ), d = log -- = -log ( 1 - --- ), 80 \ 80 / 49 \ 100 / 126 / 8 \ e = log --- = log ( 1 + ---- ), 125 \ 1000 / then log 2 = 7a - 2b + 3c, log 3 = 11a - 3b + 5c, log 5 = 16a - 4b + 7c, and log 7 = ½(39a - 10b + 17c - d) or = 19a - 4b + 8c + e, and we have the equation of condition, a - 2b + c = d + 2e. By means of these formulae Adams calculated the values of log(e)2, log(e)3, log(e)5, and log(e)7 to 276 places of decimals, and deduced the value of log(e)10 and its reciprocal M, the modulus of the Briggian system of logarithms. The value of the modulus found by Adams is