134 Prof. Sylvester on the Numbers of Bernoulli and Euler, 
numbers to contain the factor (p—1)p*, where p is any prime; 
then this number will contain the first power of p in its deno- 
minator; but if the factor piis contained in double the index in 
question, but (p—1) not, then p‘ will appear bodily as a factor 
of the numerator. 
It has occurred to me that it might be desirable to adhere to 
the common definition of “ Bernoulli’s numbers,” but at the same 
time to use the term Bernoulli’s coefficients to denote the actual 
os 
coefficients in — ; so that if the former be denoted in 
2(e! 
general by B,, and the latter by @,, we shall have 
Bon= (—)"'B,, 
comin —! 0 
In the absence of some such term as I propose, many theorems 
which are really single when affirmed of the coefficients, become 
duplex or even multifarious when we are restrained to the use 
of the numbers only. 
Postscript.—The results obtained concerning Bernoulli’s num- 
bers in what precedes, admit of being deduced still more suc- 
cinctly; and this simplification is by no means of small im- 
portance, as it leads the way to the discovery of analogous and 
unsuspected properties of Huler’s numbers (namely the coeffi- 
cients of 
in the expansion of sec @), and to some very re- 
n 
markable theorems concerning prime numbers in general. 
In fact, to obtain the laws which govern the denominators and 
numerators of Bernoulli’s numbers, we need only to use the fol- 
lowing principles :—(1) That « being a prime, 2u”=0, or = —1 
to the modulus jz, according as ~—1 is, or is not, a factor of m, 
—the second part of this statement being a direct consequence 
of Fermat’s theorem, the first part a simple inference from its 
inverse. (2) That e“*—1 is of the form pt-+p72?T, where T is 
a series of powers of ¢, all of whose coefficients are integers rela- 
tive to ~, except for the case of ~=2, when e“*—1 is of the form 
2¢+2T. We have then (u2”—1)(—)"~"B, = —II(2n) x coeffi- _ 
ele Wey e(h—2)t 4 wn & 
ett — ] 
virtue of principle (2)) =I— , where I is an integer relative to 
cient of ¢?”-! in a by actual division (in 
#, containing n, and —R= -< (Vt Oe +(e — 1) 
Hence (—)"B,= an integer relative to w or to such integer + 
