8 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [vol.xiv. 



the two hundred and fifth decimal by a method given in J. R. Young's Elementary essay on 

 the computation of logarithms (pp. 13-14). Glaisher ' verified the first resvdt, using the con- 

 tinued fraction 



1 ' 6 10 4n + 2+ ... 



but the second result was shown by Boorman ^ to be incorrect after the one hundred and eighty- 

 seventh decimal. 



Boorman's formula is readily deduced. Since we have identically 



1 1^1. n+l-m 

 m mn~n mn 

 we obtain 



m mn n mn 

 by the substitution 



m = n—l 



in the numerator of the right-hand member. The series for the Naperian base may thus be 

 transformed into the series 



\ ij l\n mnj I mAn^ m^iij 

 1 mn m^n\n^ m^n^/ 



wherein m = 2, n = 3; m, = 4, n, = 5; m^^^, n^^l; 



Tich&nek ' and Minks verified Boorman's value of e as far as the two hundred and twenty- 

 third decimal, making use of Euler's continued fraction 



in connection with the relations 



2.1^2.3^2.5 



1 + F 



l-F 



e-1 1 1^1 1 ^ 1 



2 1 1.7 7.71 71.1001 ' 1001.18089 



Gauss* gives values of e"T ranging from 15 to 57 decimals for 13 values of n at irregular 

 intervals between the limits 1/2 and — 16. He used the formula 



giT _ j\^glog O+IO log 6-log c-log 6 . .. -I" 



The quantity N is an approximate value of e"" multipUed by aXlO*, and the quantities 

 c,d,... are the factors of N so selected that their natural logarithms may be taken from 

 Wolfram's ^ tables. 



The present contribution consists of the following tables : 



Table I: Values of the reciprocal of n\ to 108 places of decimals at intervals of unity from 

 1 to 74. 



Table II: Values of g^ to 42 significant figures at intervals of unity from to 100. 



Table III: Values of e* to 33 significant figures at intervals of 0.1 from 0.0 to 50.0. 



> J. W. L. Glalsher. On the calculation of £ from a continued fraction. Brit. Assoc. Eep. (1871), pp. 16-18. 



' J. Marcus Boorman. Computation of the Naperian base. Math. Mag. vol. I (1882-1884), pp. 204-205-, see also L'lntermedlalre des mathe- 

 maticiens, vol. 7 (1900), p. 53; G . Peano, Formelaire de mathematiques. Tome II, No. 3, p. 125. 



' F. J. Stndnifika. Ueber^die Berechnung die transcendenten Zahl e. Jahr. iiber die Fort, der Math. Bd. 23 (1891), p. 440: VortrSge fiber mono- 

 periodische Functionen. Jahr. ilbcr die Fort, der Math. Bd. 25 (1893-1894), p. 736. 



' Lemniscatische Fimctionen. Werke 3, pp. 413-432. 



'• Logarithmorum Nsturalium. (48 decimals). See Vega's Thesaurus, pp. 641-684. 



