16 REPORT 1871. 



An instance of the application of Lambert's principle is afForded by a theorem 

 of Eisenstein (CreUe's Journal, t. xxix. p. 96), viz. 



14.1 ,14. 1 4. _ 1 I l-z 1 1-z- 



■whence the series is always irrational when z is an integer greater than unity. 

 The series — — — H +••• can be converted into the continued fraction 



«! 02 ff3 «4 



1 "' "'' "^ • so that if after any finite value of r, Cr^+o'r is 



■) 



0,4- Oj— «!+ flg— a2+ a^ — a^ + Sic. 



always less than Or+i, the sum of the series is irrational. Also from the equality 



1_J_ 1 _ _ 1 61 h 



b^ b,b,^ b,b.A ' ' b,+ b,-l+ b,-l+8cc. 

 we see that if after any finite integral br+1 is always less than br+\, the sum of the 

 series is irrational. 



On the Calculation of e {the base of the Napierian Logarithms) from a 

 Continued Fraction. By J. W. L. Glaisheb, B,A., F.E.A.S. 

 The series by which e is defined, viz. 



is of a verj' convergent class, so that it would be reasonable to expect that no 

 better formula could be found for its calculation. Taking the series in the form 



1-1-1- 1 + 1 - 

 -e-^ 172 + 17273 ••■' 



and throwing it into the form of a continued fraction bj' the usual method, we 

 have« 



^_1^ J__]^ J^ ^ . 



e 1+1+2+3 + ...' 

 and from the manner in which the continued fraction is deduced from the series, it 

 is clear that the ?«th convergent of the former con-esponds to n terms of the latter. 

 There is, however, a far more convergent fraction from which e can be computed, 

 viz. 



lul = A J_ 1 1 (2) 



2 1+ 6+ 10+...4W + 2+...' ^ -* 



a formula given by Lambert (Berlin Memoirs, 1761), who obtained it by per- 



forming on ^- an operation similar to that affording the greatest common 



measure of its numerator and denominator. Another investigation is given by 

 Legendre in the Notes to his ' Geometric ;' and this is reproduced in the Notes to 

 the French translation of Euler's ' Introductio ad Analysin.' It can also be very 

 easily obtained from the differential equation 



corresponding to y=e Vf^x)^ j^g ^j^g fraction for tan v was found in the previous 

 paper. 



The continued fraction (2) is much more convergent than the series, and I 

 was tempted to calculate the value of e from it for two reasons : — (T) In order to 

 practically test the advantages of a continued fraction and a series as a formula for 

 calculation "vsnth respect to the arrangement and performance of the operations ; 

 and (2) to decide between two different values of e which have been given — the 

 one by Callet in all the editions of his ' Logarithmes Portatives,' and the other by 

 Mr. Shanks in his ' Rectification of the Circle,' and Proc. Roy. Soc. vol. vi. p. 39/. 

 The several convergents to the value of e also seemed to be of value. 



