280 
If the separation is carried out completely, we find 
k=M—1 } 
e(a,b,a)—=(—1) B (k—1)p18(M—k, ko) + 
k=M—1 
(lt BS (lk (kare (lk, a) & (Mk, a). 
If @ is made to tend to zero, we have again in Eurer’s notation 
e(M—k,k,0)=g¥—*— pl , (kh, 0)—p*, 
but (Ll, M—1, a) and «(1,a) would, for a=0O, become divergent 
series. For these functions, however, bolds true 
Lim te (1, M—1,a) — €(1,a)e(M—1, a} = — q¥ 
zl 
and making use of this, we find for « >>1, consequently 6< M—1 
k=M—2 
gap ED Was eh (gi — 9) = 
k=b+1 
k=M—2 
Fl SS (DE (k= Dar pk — (M—2)5_1 g'!-1. 
k=a 
For the case a=1, 6= M—1 the difference M even or odd 
appears. 
For M odd both sides vanish identically, for M —2N, we find 
on the contrary 
9 
gNA =p —p? + pt—...+(—l)*.$p’, 
a result for p(2N-—1,1) that shows some resemblance with the 
one that was obtained for p (J/—1,1) in the ease MZ odd. 
In the equation deduced above we may now successively write 
a= 2,3,...., M—1. In this way a number of equations is found, 
from which it would seem possible to solve q’,g’,....q'—!. Eurer 
considers a similar system of equations for the special cases 1/=3, 4...,15. 
He finds that several relations of dependence exist between these 
which are partly connected with the general relation . 
qe + qua =p + pM, 
If J/ is odd there remain just enough independent equations in 
MA 
the cases considered by Eurer to solve q°, q°,...q 2 and ewe 
for MN, however, q?%—! is in general the only unknown 
quantity that can be solved, while there moreover remain, apart 
from the relation 2g = pV -+ p?\, which procures the value of qÀ, 
a certain number of relations between the quantities g’, q’,....q%—t. 
For the case M —6 there remains, as soon as g’* and q’ have 
been found. just one relation, which then allows to determine gq’. 
For M=8 we find gq’ and q‘ and further one relation between 
g’ and g’*. In the same way q° and q* may be solved for “= 10 
and two relations subsist between q°, g* and q*. 
