g — pl = tap + (M—3).-1 + (M— Bi pt? 
+ (MD + (gl PMA 
+ (a + (2)s—1} p°, 
(a + B= M odd, ea odd , a >1) 
and 
4 
gul — 4 pil — - 
8 
= pil pp A ot 
(M odd), 
while we may further always make use of the relation 
q + qe = p* = p fv — pe 4 pu 
if « and 6 are both greater than 1. 
On page 183 Eurer thinks to be allowed to write the results 
mentioned here on the ground of what he found for the special cases 
M=3, 5, 7,...,15. In his general formulae the signs, however, 
are not quite correct, which appears, when they are compared with 
his results for the case M/—= 11, which are mentioned at length on 
the preceding page. 
The preceding method for determining ¢ («, 3) does not apply to 
the case: « + 8— M is even. In order to arrive at some result in 
this case Etrer's method may be followed in substance. Some 
modifications, however, are necessary, for the divergent series admitted 
by Eurer, must be avoided. 
We consider the function 
nao gia | J 1 1 neo gi 
s(la,b,a) == = — -- en ede i= = 
n= me 14 n=? n(n — 2)? 
2b 
‚ZN 
and decompose each fraction 
|: 
n(n — hy)? 
into-a sum of fractions with the denominators 
nye: * wig SRL 8 eee nijd, 
(n—ayAM—-1 ,  (n—A)*AAl-2, ... , (n-—2)029, 
in which M is put for a + 4. 
Thus the function «¢ (a, 6,«) is decomposed into series of the form 
le SS 0 e nis 
= ——_ == ¢(M—k,k, a) 
n=2 nN M—kjk 
„Zn 
and in others of the form 
N= pris. A= © pil I= pia 
os = 5 =e(k,a) e(M-—-k,a). 
(w= a)M-kgk— j=, Ak 2, AME 
n= 
„Zn 
