130 Prof. Sylvester on the Numbers of Bernoulli and Euler, 
1. @. 
og ag ; Lee i 
I) = coefficient of git in Go hj\G=n+l) Gon+l) ee 
= the sum of the (n—r)ary homogeneous products of 
1 ee ace ae 
Thus, then, we are able to affirm, from what is known concerning 
Ao" ™ 
(see Prof. De Morgan’s Calculus), that the r-ary homoge- 
neous product-sum of 1, 2, 3... (which is of the degree 2r 
in n) always contains the algebraic factor n(n+1)...(n+7). 
Addendum.—Since sending the above to press, I have given 
some further and successful thought to the Staudt-Clausen 
theorem. Staudt’s demonstration labours under the twofold 
defect of indirectness and of presupposing a knowledge of the law 
to be established. In it the Bernoullian numbers are not made 
the subject of a direct contemplation, but are regarded through 
the medium of an alien function, one out of an infinite number, 
in which they are as it were latently embodied; and the proof, 
like all other mductive ones, whilst it convinces the judgment, 
leaves the philosophic faculty unsatisfied, inasmuch as it fails to 
disclose the reason (the title, soto say, to existence) of the truth 
which it establishes. I present below an immediate and a direct 
proof of this beautiful and important proposition, founded upon 
the same principle as gives the law of the necessary factor in the 
numerators (viz. the arbitrary decomposition of the generating 
function of Bernoulli’s numbers into partial fractions), and rest- 
ing upon a simple but important conception, that of relative as 
distinguished from absolute integers. 
I generalize this notion, and define a quantity to be an integer 
relative to 7 (or, for brevity’s sake, to be an 7** integer) when it 
may be represented by a fraction of which the denominator 
does not contain 7. 
The lemma* upon which my demonstration rests is the fol- 
* This lemma is the converse of a self-evident fact, and it virtually em- 
bodies a principle respecting an arithmetical fraction strikingly analogous 
to a familiar one respecting an algebraical one; viz. in the same way as a 
rational algebraical function of 2 can be expressed in one, and only one, way 
as an integral function augmented by a sum of negative powers of linear 
functions of 2, so a rational arithmetical quantity can be expressed in one, 
and only one, way as an integer augmented by the sum of negative powers 
of simple prime numbers multiplied respectively by numbers less than such 
: : : ‘ 2 . c 
primes. In drawing this parallel, the arithmetical quantity pe where 
2 es 
(aw+b)’? 
4 
c<p, is regarded as the analogue of the algebraical one 
