Legendre proves a theorem which is easily seen to follow directly from Lambert's mode of investigation, viz. that if in the continued fraction 19 then, whether B, B,... be all positive or all negative, or some positive and some negative, the value of the continued fraction is irrational. He also remarks that must be irrational; for if it we should have, from (i), since tan π= =0, m = and as after some value of r the fractions 5n—7— In—&c.' m m (2r+1)n' 2r+3' &c. must be less than unity, 3 must be irrational if m and n are integers, whence 2 is irrational. The expression of tan v in the form of a continued fraction Lambert obtained by v3 treating sin v or v— +... and cos v or 1- +. in a manner analogous 1.2.3 to that in which the greatest common measure of two numbers is found in arithmetic; and Legendre deduced it from a more general theorem he had proved with regard to the conversion of the ratio of two series into a continued fraction. It may be obtained very simply by forming the differential equation corresponding to y= A cos (√2x+B), viz. y+y'+2xy"=0, whence yi)+(2i+1)y(i+1)+2xy(i+2)=0 by application of Leibnitz's theorem. From this we have whence, after determining B by putting a=0 and writing √(2x)=v, That Lambert's proof is perfectly rigorous and places the fact of the irrationality of beyond all doubt, is evident to every one who examines it carefully; and considering the small attention that had been paid to continued fractions previously to the time at which it was written, it cannot but be regarded as a very admirable work. Lambert showed, in the same memoir, that e" is irrational, so that the Napierian logarithm of every rational number is irrational. 1 We can obtain a little more information about the irrationality of e, for we have Now any continued fraction in which all the numerators are unity and all the denominators are positive integers must circulate if it be the development of an ex 1 pression of the form A+BC; so that we see that e, when a is integral, cannot be of this form. Taking the expression for the tangent in the form 1 we see that when x is an integer, cot is irrational, but cannot be of the form A+B C. 1 COS cannot either of them be rational unless cot is of the form B/C, which not being the case, sin and cos x have; so that cos is irrational. x x are irrational, and cannot be of the form BC. cannot be rational unless cos=B/C, which B2C a form which we have shown it cannot sine and cosine; that is to say, (e+e *), (e-e *), and (e+e) are irrational. It may be remarked that it is easy to show that sin is incommensurable from the series; for if sin 1=2, then (q even, as of course we may take it) and the series on the right-hand side must be intermediate in value to 1 (q+1) x2 if sin is commensurable. An exactly similar method proves the irrationality of 1 2 2 (ex-e &c., but gives no result when applied to cos or (e+e). It is probably true that both the sine and cosine of every rational arc are irrational, though no proof of this has, I believe, been given; and there is, as Legendre has remarked, very little doubt that is not only not the square root of a rational quantity, but also not even the root of any algebraical equation with rational coefficients, although the demonstration of this seems difficult. Similar remarks may be made with respect to e. *This expression can be deduced from (i) by transforming the terms of the latter thus: An instance of the application of Lambert's principle is afforded by a theorem of Eisenstein (Crelle's Journal, t. xxix. p. 96), viz. whence the series is always irrational when z is an integer greater than unity. 1 1 1 1 +... can be converted into the continued fraction The series a1 a2 + always less than ar+1, the sum of the series is irrational. Also from the equality we see that if after any finite integral b,+1 is always less than br+1, 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. GLAISHER, B.A., F.R.A.S. is of a very 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 and throwing it into the form of a continued fraction by the usual method, we have and from the manner in which the continued fraction is deduced from the series, it is clear that the nth convergent of the former corresponds to n terms of the latter. There is, however, a far more convergent fraction from which e can be computed, viz. ex-e-x a formula given by Lambert (Berlin Memoirs, 1761), who obtained it by performing 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 Géométrie;' 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√(2x), as the 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:-(1) In order to practically test the advantages of a continued fraction and a series as a formula for calculation with 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. 397. The several convergents to the value of e also seemed to be of value. and writing the convergents P1, P2, (3) so that p1 =1, p2=3......., q1=1, 4,=1..., the convergents were calculated as far as Pa (which corresponds to the quotient 150). The following Table contains the values of the convergents as far as 939 P20 920 7 71 20 7 064 900 016 612 187 878 152 462 721 P39-5 933 736 817 524 490 649 943 748 885 310 086 922 977 536 976 487 014 058 103 672 162 883, 939=2 182 899 784 489 322 239 844 266 493 459 455 750 162 013 065 305 797 591 300 833 210 159. In Ingn+1 In+19n+2 272...; so that 135 figures of the result obtained by dividing P39 by 939 are corOn performing the division to 137 places and applying the correction for the value of e was obtained to 137 decimal places, viz. which agrees e=2.71828 18284 59045 23536 02874 71352 66249 77572 47093 69995 95749 66967 62772 40766 with Mr. Shanks's calculation obtained from the series to the last figure; there is therefore no doubt of the accuracy of the result to this extent. The value given by Callet, in the introduction to his Tables Portatives,' starting with the ninth group of five, is and these figures should be 46928 08355 51550 58417 2. ..; ...47093 69995 95749 66967 6.... The thirty-ninth convergent to the continued fraction (3) gives a result as accurate as that found by summing the first ninety terms of the series (1); but there would be no great disparity between the absolute number of figures formed in the two calculations. The computation of the convergents was, however, far preferable in point of arrangement and convenience to the calculation of the successive terms of a series; for not only were the divisions in the latter replaced by multiplications, which are far more compact, but the work in the former case ran straight forward and required no copying of results. There is also another very great advantage in the continued fraction: the great difficulty of performing a piece of work to a considerable number of decimal places is the inconvenience caused by the length of the numbers; and in the above calculation we get roughly 2n figures of the result without ever having to use a number more than n figures long in the work: thus P30 and 939 contain each 67 figures, and by dividing them we obtain 138 figures of the result; this advantage is due to the fact that all the numerators in (3), except the first, are equal to unity. It may be remarked that the final division was the most laborious part of the work; the calculation of and 939 required barely 13,000 figures, the division about 18,000. P 39 We can compare the number of decimal places afforded by (3) and (1) when n is large as follows:-The number of places Pn yields* is equal to the greatest in log [} {1.6... (4n—6)} {1.6... (4n—2)}] teger contained in log and the number of places obtained from n terms of (1) is equal to the greatest integer in so that the nth convergent to the continued fraction gives more than twice as many decimal places as n terms of the series. On certain Families of Surfaces. By C. W. MERRIFIELD, F.R.S. The author had already shown that conical and cylindrical surfaces not only satisfy the general equation of developable surfaces in differentials of the second order, rt=s2, but also that on passing to the differential equation of the third order, there are two equal roots in the case of conical surfaces and three equal roots in the case of cylindrical surfaces. *See Proc. Roy. Soc. vol. xix. p. 514. |