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An instance of the application of Lambert's principle is afforded by a theorem of Eisenstein (Crelle's Journal, t. xxix. p. 96), viz.

1
1
1

1 1 1 1 1-72
1+*+
+ t.

1- % - -&c.'
whence the series is always irrational when z is an integer greater than unity.

1

1 1 The series + to.. can be converted into the continued fraction

а. а. az a 1 a,

-; so that if after any finite value of r, a,?ta, is azt az -a, + az-a, + a, - Ag+&c. always less than ar+1, the sum of the series is irrational. Also from the equality 1 1 1

1 b
.+
5 6,62
b, b, b,
+ ,

&c. we see that if after any finite integral br+1 is always less than br+1, the sum of the series is irrational.

2

3

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e

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e

e-1

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.
The series by which e is defined, viz.

1+1+1.2+
+ 1.2.3+....,

.. (1) 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

1-1-1- 12+ 1.2.3 and throwing it into the form of a continued fraction by the usual method, we have

1 1 1 2 3

;

el+ 1+ 2+ 3+... 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.

1 1 1
1

(2) 1+ 6+ 10+... 4n+2+... a formula given by Lambert (Berlin Memoirs, 1761), who obtained it by performing on an operation similar to that affording the greatest common

ertemeasure of its numerator and denominator. Another investigation 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

dy – 2x d2y y

=0, dix

dx2 corresponding to y=eV(25), 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 caleulation 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.

2

et-e

6

6

:

е

e

.

n.

Taking (2) in the form

2 1 1 e=l+

(3) 1+ 6+ 10+..' and writing the convergents P1, P2, so that P2=1, p.=3..., 91=1, 9=1...,

li la the convergents were calculated as far as

Р39

(which corresponds to the quotient

930 150). The following Table contains the values of the convergents as far as

P20

920 Pm

In. 3 19

7 4 193

71 5 2 721

1 001 6 49 171

18 089 7 1 084 483

398 959 8 28 245 729

10 391 023 9 848 456 353

312 129 649 10 28 875 761 731

10 622 799 089 11 1 098 127 402 131

403 978 495 031 46 150 226 651 233

16 977 719 590 391 13 2 124 008 553 358 849

781 379 079 653 017 14 106 246 577 894 593 683

39 085 931 702 241 241 15 5 739 439 214 861 417 731

2 111 421 691 000 680 031 16 332 993 721 039 856 822 081

122 501 544 009 741 683 039 17 20 651 350 143 685 984 386 753

7 597 207 150 294 985 028 449 18 1 363 322 103 204 314 826 347 779

501 538 173 463 478 753 560 6731 19 95 453 198 574 445 723 828 731 283 35 115 269 349 593 807 734 275 559 20 7 064 900 016 612 187 878 152 462 721 2 599 031 470 043 405 251 089 952 039 P39=5 933 736 817 524 490 649 943 748 883 310 086 922 977 536 976 487 014 058 103

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.

12

+

e

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Since

Pn Pn+1

Pn Pn+ Pn+1
In

an In+2 In+1
Pn

2

2
+(-)*+

2n9n+1 9n+19n+2
Pn
2

2 we see thate differs from

by less than

; if n=39,

=(134 ciphers) 9n Inqn+1

Inenti 272...; so that 135 figures of the result obtained by dividing Pag by 939 are correct. On performing the division to 137 places and applying the correction for 2

the value of e was obtained to 137 decimal places, viz. 9,96+1

e=2.71828 18284 59045 23536 02874 71352 66249

77572 47093 69995 95749 66967 62772 40766
30353 54759 45713 82178 52516 64274 27466

39193 20030 59921 81741 35966 29043 57..., which agrees with Mr. Shanks's calculation obtained from the series

1 1

+ t...

1.2 1.2.3 1871.

2

e=1+1+1+

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P 39

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

.. 46928 08355 51550 58417 2...; and these figures should be

...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 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. We can compare the number of decimal places afforded by (3) and (1) when n

Pn is large as follows:- The number of places yields* is equal to the greatest in

In teger contained in

Inln+1

= log (){1.6... (41-6)}{1.6... (4n-2)}]
2
22n-4
{1.3... (2n-1)}? = log

r(2n)
4n

r(n+) 1 n2n + $ 22n++

64n natten (after substituting V 211 nne-n for T(n +1)) = log {

= - +4n and the number of places obtained from terms of (1) is equal to the greatest integer in

log r(n)=n log - + log 2 - logn; so that the nth convergent to the continued fraction gives more than twice as many decimal places as n terms of the series.

P 39

log

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2n

1)34]=

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= log [ = log [

n

2-2* {I2+1}}']

|
}]
(3) ?" } = 2n log + (An–5) log 2– log n;

24n45

2n

n

e

n

e

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=s, 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.

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An examination of the surfaces described by the motion of a plane

parabola of any order with its diameters parallel to a fixed right line showed that the conan SITT dition of a pair of equal roots in the equation of the third order,

d d3
(

+λ.
+ %=0,

CALIFORNIA dr dy considered as an equation in 2, was satisfied by all surfaces traced out by a plane parabola moving parallel to a fixed line and enveloping any curve in space whatever. As singular cases, he noticed the spindle made by causing a parabola (whether fixed or of variable size) to rotate round any diameter, the ruled surface with a director plane, and developable surfaces.

He also showed that when three of the roots were equal, the surface necessarily reduced to a plane or a cylinder.

These results are, however, restricted by the method of generating the surface. In fact, for the case of three equal roots, when the partial differentials of the third order are in continued proportion, Mr. Cayley has shown that the resulting equations can be integrated and that the integration gives a more general result.

Note by Mr. CAYLEY,
The general integral of the equations

B
B

8

B can be found, viz. gives r= funct. s, and = ở gives

B 7

gives s= funct. t. But r= funct. s is integrated as the equation of a developable surface (p instead of z), viz. we have, say, p=ar+hy+g (a and g functions of h) and

da

a

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a

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dg

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Similarly 8= funct. t gives
q=hx+by+f,

db df

dh

dp Observe that the constants have been taken so that =h, =h; but in order

dy that h may in the two pairs of equations mean the same function of x, y, we must

1 have a'= ū fr

g'dh b=

dq de

g. that is

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xhd
фл

or writing a=ph, g=xh, we have

dh p=sph+yh+xh, q=hx+y wbere

ap'h+y+x'h=0. The last equation gives h as a function of x, y; and the values of p, q are then such that dz=pdx +qdy is a complete differential, so that we obtain z by the integration of this equation. A simple example is

p=shạx hy, q=-hx+y log h, hx-y=0; that is

1 y2

ce q whence

1
yo log

y
2

p=-=-x+ylogy

من أه

2

213?

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On Doubly Diametral Quartan Curves. By F. W. NEWMAN, Emeritus Professor of University College, London. This paper aimed to detail the form of the curves, and point out the simplest modes of investigating their peculiarities

. It distributed the general equation into three groups of tive, five, and four families, and was accompanied by seventy-six diagrams. If we call

Ax' +2Dc%y? +By:+2E.x2+2Fyo+C=0 the general equation, the first group of five families is when A or B or both vanish, the second group when D or F or both vanish, and these together nearly include all the forms. For in the third group, from which either no term of the original equation vanishes, or only C, three of the families are at once reducible to the second group by putting either y+fo=y"?, else r+e=x'; then the proposed curve of (xy) is visibly at most a mere variety of the preceding, being either of the same species with, or of a lower species than, that of (ry') or of (xy)In the case of B=3", D=82, F=-f'?, this reduction is impossible ; but then by operating on : instead of y, it becomes possible unless also Â=*, E=-e"; that is, the method fails only when A, B, D are of one sign, and E, F of the opposite. The analysis, thus limited, readily yields the same result, that the forms have nothing new.

A cross division of the species is into Limited and Unlimited loci. All are Centric, the origin being the Centre. Finite forms are Monads, Duads, and Tetrads. Monads are :-1. A symmetrical oval (say, a Shield), as from

r*yo+a'yo+b?z? = m. 2. An Oval with undulating sides (Viol or Dumb-bell), as aʻyo =(mo—2*)( 2o+na), m2>n?. 3. A Lemniscate or Double Loop, as aậy? =(m2-x2)x?. 4. A Scutcheon, with four sides undulating, as B?y=f? IV (m2-x2)(x2 +n”), when mo>n?, and f? <mn. 5. Orals in Contact, as ß* y* =(m2 x2)x2. 6. Pointed Hearts, crossing obliquely at the centre, as

Bạy? = mn+82ra †v {(m2—~2)(x2 +12)}, when mona > 2mn, and 82 >

m2_n

7. Hearts in Contact ; the same equation

2inn as before, only with 82 =

8. Intersecting Hearts, as 2mn

Bạy:=}(mo+n2)+v {(m2 —~2)(x2 +n")}, when m2>n2. 9. Intersecting Ovals ; the same equation as in 8, only with maZn?. All curves are here deemed Monads which can be drawn without taking the pencil off the

paper. Duads are:-1. Twin Ovals (rot singly symmetrical on opposite sides), as

a'y? =(m2-x2)(x2-no). 2. Twin Beans (or Hearts, Dicuamos). 3. Pair of Sandals (Disandalon): this has always two double tangents parallel, yet the disposition of the four points of contact is not the same in all cases. (They form a rectangle when D=0; they are in lines diverging from the centre when F=0.) 4. Pair of unsymmetrical Lemniscates, which I call Four Kites.

Tetrads can only consist of unsymmetrical Ovals, symmetrically disposed.
Of the infinite curves, one very limited class may be called Parabolic, those in

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m? -n2

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