275 
Mathematics. — “On some numerical series considered by Euurr.” 
By Professor Kruyver. 
(Communicated in the meeting of April 28, 1916.) 
In Nov. Comment. Petrop. XX, 1775. page 140, Eurer wrote a 
paper entitled: ‘‘Meditationes circa singulare serierum genus.” He 
treated in it certain numerical series of the form 
Be lores LE RES. ENC L 
Best sis a n 114 2P = BP as nb’ 
in which « and > are positive integers, and his aim was to prove 
Ef 
that the series p(a,8), which he denoted by | { :) might be 
e yr 
expressed integrally and rationally for all values of « and 3 by the 
values which &(s) assumes for positive integer values of s. 
Writing «a 4 = M, Eurer indicates, how by a peculiar and 
ingenious calculation a system of equations may be deduced, which 
involve the quantities  (k, Mh) for k = 2,3.4... M—1 as unknown 
quantities, so that everything is reduced to the solution of this system 
of equations. He considers in this way the cases M =3,4,...,15 
and tries, by means of a bold induetion, to arrive at a general 
solution of the problem from those special cases. He does not com- 
pletely suceeed in this. It appears from his investigations that the 
cases M even and M odd are quite different. For M odd the equa- 
tions considered by him may be solved, for M even, however, they 
are dependent on each other, and his efforts to express g («, #) in 
this case by §-values miscarry. It still remains an open question 
whether for M/ even in general, such an expression in any way can 
be found. In what follows a direct evaluation of y (u, 8) is given 
for M odd, while I shall further shortly consider the results at which 
Evuer arrived for the case JM even. 
It may be observed beforehand that g (4, M—k) and y (M—A, hk) 
for 1< k< M —1 are connected in a simple way. 
We evidently have viz. 
gy (k, Mk) + gy (Mk, k) = (U) (M—k) + OM) 
and in particular 
p (N,N) = HEN)" + 48(2N), 
M— j 
so that for given J/ only 5 quantities at most, are to be calculated. 
= 
I start from certain pretty simple trigonometrical series determined 
by the equations 
18* 
