and a new Theorem concerning Prime Numbers. 133 
And this relation obtains for any value of yu other than 2, which 
(or a power of which) could be contained in 2n. When p=2, 
the & series wiil nof all of them be the doubles of relative inte- 
gers to 2; but the v series, on account of the factor I1(2n), will 
obviously, up to v2,—, inclusive, all contain 2 and ».,=13; conse- 
quently go, will be twice (an integer gud 2)+1, and B,, will still 
be (an integer relative to ~) + : as before. Hence it follows from 
the lemma that (—1)"B,,=an absolute integer + 2 or 
1 
B,= an integer +(—)”"= ie 
which is the equation expressed by the Staudt-Clausen theorem*. 
My researches in the theory of partitions have naturally in- 
vested with a new and special interest (at least for myself) every- 
thing relating to the Bernoullian numbers. I am not aware 
whether the following expression for a Bernoullian of any order 
as a quadratic function of those of an inferior order happens to 
have been noticed or not. It may be obtained by a simple pro- 
cess of multiplication, and gives a means (not very expeditious, 
it is true) for calculating these numbers from one another with- 
out having recourse to the calculus of differences or Maclaurin’s 
theorem, viz. 
Be. igo B, Biot . 2 Pat et 
Ten ~~) irey’ Tren—s tray enw 
Bos B. 
Qn—4 At LM Se tae 2S 
et... + (2 —)it@n—4) 114 
+ (227-2 1) Bra =i 
Tl(2n—2) 112’ 
in which formula the terms admit of being coupled together from 
end to end, excepting (when n is even) one term in the middle. 
To illustrate my law respecting the numerators of the num- 
bers of Bernoulli, and its connexion with the known law for 
the denominators, suppose twice the index of any one of these 
* I ought to cbserve that in all that has preceded I have used the word 
imteger in the sense of positive or negative integer, and the demonstration - 
I have given holds good withcut assuming B, to be positive. That thisis 
the case, or, in other words, that the signs of the successive powers of ¢ 
. et—] ate 
at alternately positive and negative, may be seen at a glance by 
z 4 
putting t=2’/ — 16, and remembering that all the coefficients in the series 
for tan @ in terms of 6 are necessarily positive, because (a) tan 6 obvie 
ously only involves positive multiples of powers of (tan @) and (sec 6). 
