easy to obtain an expanded expression for this, it is very easy to calculate its successive values in a manner analogous to that used in Ch. II. Art. 13.

Let C

numerical value of the coefficient of x in the expansion of x("). Then since x("+1) = (x — n) x(1), we obtain

and we can thus calculate the values of C+1 from those of and we know that the values of C1 are 1, 0, 0,...

1

11. Let us denote by C," the numerical value of the coefficient of in the expansion of x(") in negative powers of x, so that

к

expansion of log (1 + z)" by Maclaurin's theorem. Thus this expansion may be written

A formula analogous to (22) can also be obtained by means of Art. 13, Ch. II. This gives for numerical values

and thus from the values of C-1 those of C can be obtained. The values of C, are of course 1, 0, 0, .....

12. Analogous series are those of the coefficients when x" and x" are expanded in factorials.

-n

1

Again in (25), page 26, put ux= and (D)=D"1, and

*It will be seen that, as in the analogous case we could expand {log (1+ z)}* in terms of C, we can expand (e2-1)* in terms of -(x+1)

In fact

13. There is another class of properties of Bernoulli's numbers that has received some attention; these relate to their connection with the Theory of Numbers. Staudt's theorem will serve to illustrate the nature of these properties. It is that

Ban-1=integer + ( − 1)" (1 + 2m+1)

where m is a divisor of n such that 2m+1 is a prime number. Thus, taking n=8, we have (since the divisors of 8 are 1, 2, 4, 8)

It will be found on reference to page 91 to be 77. Staudt's paper will be found in Crelle (xx1. 374), but a simpler demonstration of the above property has been given by Schläfli (Quarterly Journal, vi. 75). On this subject see papers by Kummer (Crelle, XL. XLI. LVI.). Staudt's theorem has also been given by Clausen.

14. To Raabe is due the invention of what he names the BernoulliFunction, i. e. a function F(x) given by

when x is an integer, and which is given generally by AF(x)=x". He has also given the name Euler-Function to the analogous one that gives the sum of

1n - 2n + 3n- &c. + (2x-1)n

when x is integral. See Brioschi (Tortolini, Series II. 1. 260), in which there is a review of Raabe's paper (Crelle, XLII. 348) with copious references, and Kinkelin (Crelle, LVII. 122). See also a note by Cayley (Quarterly Journal, II. 198).

15. The most important papers on the subject of this Chapter are a series by Blissard (Quarterly Journal, Vols. Iv.-Ix.) under various titles. The demonstrations shew very strikingly the great power obtainable by the use of symbolical methods, which are here developed and applied to a much greater extent than in other papers on the subject. They include a most complete investigation into all the classes of numbers of which we have spoken in this Chapter; the results are too copious for any attempt to give them here, but Ex. 15 and 16 have been borrowed from them. The notation in the original differs from that here adopted. Ban there denotes what is usually denoted by B2n-1 See also two papers on Anom and its congeners by Horner (Quarterly Journal, Iv.).

16. Attempts have been made to connect more closely Bernoulli's and Euler's Numbers, which we know already to have markedly similar properties. Scherk (Crelle, Iv. 299) points out that, since tan

(+)

secx+tan x, the

expansion of this function in powers of x will have its coefficients depending alternately on each set of numbers (see (7) and (11), of this Chapter. This idea has been taken up by others. Schlömilch (Crelle, XXXII. 360) has written a paper upon it. It enables us to represent both series by one expression, but there is no great advantage in doing so, as the expression referred to is very complicated. Another method is by finding the coefficient of x" in the ex

pansion of

1

ae* - 1

from which both series of numbers can be deduced by

taking a = ±1 (Genocchi, Tortolini, Series I. Vol. III. 395).

17. Schlömilch has connected Bernoulli's numbers and factorial coefficients with the coefficients in the expansions of such quantities as Dnf (log x),

Dn

Dm (1710), &c. (Grunert, VIII. IX. XVI. XVIII.) Most of his analysis could be

" f(x) rendered simpler by the use of symbolical methods. This is usually the case in papers on this part of the subject, and the plan mentioned in the last Chapter has therefore been adhered to, of giving characteristic examples out of the various papers with references, instead of referring to them in the text. We must mention, in conclusion, that the numbers of Bernoulli as far as B11 have been calculated by Rothe, and will be found in Crelle (xx. 11).

3. Obtain the formula of page 107, for determining successively Bernoulli's numbers, by differentiating the identity

6. Apply Herschel's Theorem to find an expression for a Bernoulli's number.

7. Demonstrate the following relation between the even Bernoulli's numbers:

9. Prove that the coefficient of 02 in the expansion of

10. Express log sin x and log tan x in a series proceeding by powers of a by means of Bernoulli's numbers.

[Catalan, Comptes Rendus, LIV.]

11. Shew that the coefficient of

B

[log (1-e) dt - z log z is B numerically.

in

n

0

n-1