332 
MR. E. W. BARNES ON THE THEORY OF THE 
Hence, provided a is positive with respect to the Hs, 
.S.'(«) = ,hfL 
and for all values of a, and w.. 
-1 
(1 — (’“'"'ip (1 — 
dz. 
^0 (^-^5 I Oi.dj — 
§ 49 . Secondly, ecpiate terms involving the first powei’ of e. 
We (jhtain the asymptotic equality 
/HI (/Jil 
— log II n (« + + 
= 0 /)■},, — 0 
a 
{o(s, « j fo).,) 
ft = 0 
(t + ¥}Pd — log (o) [pwj + qaq)' 
~ rdo (o) h [(^<^1 + log (pap + qaj_,) — {po)J^ log (pwi) — (qa)o)'Hog qoj^ 
“h n (p -|- q) — n lc)g n [(iSj^o {ji -|- ajj -f- ajo) (pwj -j- qoi-2) — (o H~ P^i 
— oSi®- [a + cao)pa;o] 
— 'i} (oS|^^'^(ri -f- ajj -j- aj.i) (p*^! “h V^’) log (^^^1 “h ~i~ ^\) P^i log 
— jS/^* (« fi- (0.2) (JC0.2 log qW;>] 
— [oSj^ (rt -|- ajj -|- a>o) log (pa)j^ -|- qw,,) — oS]^ (ft fi- ajj) logpaij 
— oS/ (ft + 0J.2) log qaj J 
— log u [oS]^ (ft -|- a)| -f ajo) — oSj^ {ci -\- a)|) — 2^1^ “h ^2)] 
{ci + a)^ + a)^) ob m + \P^ 4 “ ^^*1) oh {ct + a),i) 
(pajj + qw.^“‘ (pajj)"' {ixaP" 
valid for all values of 5, ft, and a;,,, provided the logarithms have their principal 
values witli res])ect to tlie axis of — ("1 + aJo). 
In order that tlie lal)Our of writing down cumhrous formulae like the one just 
obtained may he diminished as much as possible, we propose to introduce a symbolic 
notation suggested by Cayley’s notation of matrices.'^' 
If y’(z) he any function of 2:, we shall represent symbolically 
+ 
(p 
. = i{ni 4 - l)vin" 
J{z fi- a;^ + a;,,) — J {'- + ajj) — ^(z + aj,,) 
by F2 [/(^ + aj)], the suffix 2 denoting that we are dealing with two parameters. 
Thus the difference equation fw double Bernoullian numhers (§ 17 ) is written 
Slmilarl}^ 
H [,S. (. + ca) ] = - .,S„ (z) + Z». 
F2[,,S/~^(ft. -j- a>)paj log/jaj] denotes the function 
* Cayley, ‘ Collected Y orks,’ vol. 2. The corresponding theory for multiple gumma fimetions will be 
developed by employing u sym1.)olic notation uh initio. 
