278 
MU. E. AV. BARNES ON THE THEORY OF THE 
Sul^stitute Wo —a for a in the fundamental difterence equation and we find 
oS;, (o)]^ d“ — ®) — ^«(wo — « ] Wo) + + 
and therefore, since S,;(wo —alwo) = ( —S,;(rt|wo), we have 
oS«(a)^ + w,j — a) — o&„ (wo — a) — (— S;i(rtlwo) -f iB/i + i(w]). 
If therefore we put 
f{a) = (-)" 
a 
+ ~ + i ("i)> 
a)i 
otS„ (wj^ d“ Wo — a) + — iB/i + i(wo) 
we see that / (o.) is an algebraic solution of the dinerence ecpiation 
J-[a + W|) — ,/{<0 = S/;(«iwo) + iB,,4,^(wo), 
and tiierefore can only differ liy a constant from oS„(«). 
Determine tliis constant by making a = 0 and we have the relation 
oS„((r) = (—)“ [oS„(a)j + Wo — a) — oS,,{wj -f- Wo)] + - lB/i + i(wo) 
"l 
+ — iB,, +1 (wo). 
W, 
When n is odd the last two terms cancel each other, and when n is even ^B,;^.^(wo) 
vanishes. 
Hence oS?^,(«|w^, oj.,) — ( — )" [oS„(w. + wo — o) — oS,,(ojj -|- Wo)]. 
From the fundamental difference ecjuations we see at once that 
a / , X S,i+i(o|a)i) , 8,(+i(o|wo) 
oh„(w^ + wd = -—---— 
' n -{■ 1 11 + 1 
= lK/i + l(wi) -f iB,,+ ;(wo), 
and therefore we have tlie relation stated. 
§ 11. We may now show that, when n is even, 
Bo/2, 
oB„(Wi, Wo) = ( —)“ -^'(wff ^ + w/ 1 ), 
a simple expression for the even double Bernoullian numliers wliich corresponds in 
some degree to the fact that the even simple Bernoullian numliers vanish. 
On differentiating Avith regard to a the result of the previous paragraph Ave ffnd 
oS n (u I W^, Wo) ~t“ ( — oS (wj -j- Wo -— (x) = 0. 
From the fundamental difference equations Ave haAm 
oS;j (a -f- w^ + Wo) — oS„ {a + w^) — .oS,, \ci ff- wo) + 
