and a new Theorem concerning Prime Numbers. 129 
the expansion of this last generating function will be of the form 
ie tl my 1) yg to , where [ is an integer, and therefore B,, will 
2Qnl 
be of the form pe (ue="— 1)" 
Suppose now, when sca i. is reduced to its lowest 
wees (we? — 1) 
terms, that p (a prime contained in 2n) does not appear in the 
numerator, this can only happen by virtue of p being contained 
in w??—1(w2"— 1); let now yw be taken successively 2,3,4,...(u—1), 
then »?”—1 im all these cases is divisible by p; and therefore, 
by an obvious inverse of Fermat’s theorem, (p—1) must be con- 
tained in 2n, i. e. y must be a factor of the denominator of b, 
under the Staudt-Clausen law, which proves my theorem. 
As a corollary to the foregomg, using Herschel’s transforma- 
tion, we see that if w be taken any intecer whatever, 
Qn (1 + A)*? +2(1-+ A)? 4... + (u—1) on 
a oy eat 
be 7S ag ar Nae 7 Beas ir, Ne 
L k. = —l 
2 a K—3 ‘iad 
Rete Pitt BE AN ee BES - 
=" on 2 
ay = ny 
id AY At 2+ pe ee iis Ee 3 Ae 
and if we write 0°"*’ instead of 0°”, the result vanishes. For 
the case of w=2, this theorem accords with one well known. 
As this subject is so intimately related to that of the Herschelian 
differences of zero, I may take this occasion of stating a proposi- 
tion concerning the latter, which (simple as it is) appears to have 
A’0” +7 
escaped observation, viz. that is in fact the expression 
\r 
for the sum of the homogeneous products of the natural num- 
bers from 1 to 7, taken n and z together. For 
1 
(v—n)(v7—n+1)...(¢%—l1)z 
las 
Bae We Nn writer beter ieee 4th 
Ilr lLa—n  (e—n4+1) ° (v—n+2) or 
Hence obviously 
1 | ey (Oe VAG 3 rl a) Ye 
im? —r(r7—1)"4+r. 5 (r—— 2)" + se, }, 
Phil. Mag. 8. 4, Vol. 21, No, 188 Feb, 1861. K 
