Equating the coefficients of the same powers of ✪ in (5)

and (8), we obtain

From this we see that the values of B-1 increase with

B 2-1

very great rapidity, but those of ultimately approach to

2n

* A variation of (9), due I believe to Raabe (Diff. und Int. Rechnung, 1. 412), depends on the following ingenious transformation:

where 2, 3,

are the prime numbers taken in order. This formula would be of great use if we wished to obtain approximate values of B, corresponding to large values of n, as it is well adapted for logarithmic computation.

4. If m be a positive integer and p be positive

5. Euler was the first to call attention to a set of numbers closely analogous to those of Bernoulli. They appear in the coefficients of the powers of a when sec x is expanded. Thus

formulæ analogous to (9) and (10), from which (12) may be

deduced.

* Due to Plana (Mem. de l'Acad. de Turin, 1820).

+ Schlömilch (Grunert, 1. 361).

2

6. Owing to the importance of Bernoulli's and Euler's numbers a great many different formulæ have been investigated to facilitate their calculation. Most of these require them to be calculated successively from B, and E, onwards, and of these the most common for Bernoulli's numbers is (2). Others of a like kind may easily be obtained from the various expansions which involve them. Thus from (5), multiplying both sides by sin 0,

The simplest formulæ of this nature both for Bernoulli's and Euler's numbers are obtained at once from the original assumptions

7. But direct expressions for the values of the numbers

may be found.

t

et - 1

=

Thus

8. These formulæ are capable of almost endless trans

formation. Thus, since A-10*1=

page 28), we can write (16) thus

n

&c.)

- A0-1 (Ex. 8,

&c. 02n+1

32

9. A more general transformation by aid of the formula

y3

{log (1+ yE)} 0ƒ (0) = yƒ (1) – 1⁄2 . 2ƒ (2) + &c.

2

In (19) write E' for x and operate with each side on ƒ(0′).

n-1

{log (1+ AE')} 0" ƒ (0′) = (− 1)"1 {log (1 + AE')} 0ƒ”−1(0′)

=

· E' ƒ”−1(0′) = [(x + 1) ▲ (x + 1) ▲ ... ƒ (x + 1)]x=0·

This transformation has been given because it leads to a remarkable expression due to Bauer (Crelle, LVIII. 292) for Bernoulli's numbers.

Denote by A' the operating factor (x+1) ▲, and write

for f(x) and 2n+1 for n, and we obtain from (18)

10. A series of numbers of great importance are those which form the coefficients of the powers of x when x") is expanded in powers of x. These usually go by the name of factorial coefficients.

It is evident by Maclaurin's Theorem that the coefficient of x in the expansion of x() is

DKO(n)

Although it is not

K

*

Comparing (22) page 25, and (25) page 26, we see that

coefficient of A" in the expansion of {log (1+▲)}«. That this is the case is

B. F. D.

8