(170 ) 
theorie”; BACHMANN, „Die Lehre von der Kreistheilung”) or by 
extension to any uneven ~ of one of the equations 
n—l 
2 2m kr 
(= a | i n 
n ) eee | auk 
sin — 
n 
n—l 
2 m kit 
WN Tt Tl ied n 
( n ) hi al kn 
LU 
n 
proved only for prime denominators, resp. of equations analogous to 
these of the complex integers formed out of the cubic, resp. biqua- 
dratie roots of unity, in which the trigonometric functions have 
been replaced by the elliptic functions belonging to the curve of 
Kiepert, that is to the imvariants gy= 4, 9;=0, resp. by the 
lemniscatic functions (gg = 0, g3 = 4), which n» in the second parti- 
cular case is prime to 1 —j or 1+ -¢ (e.g. KRONECKER „Berliner 
Monatsberichte, 1876’’). 
Out of both definitions the different qualities of the generalised 
symbol can easily be deduced; and with its help it can also be 
shown that for the same holds good the corresponding generalisa- 
tion of the Lemma of Gauss. Now every mathematician will regard 
as natural, if only on account of the analogy with the procedure 
followed almost without exception, the definition of the generalised sym- 
bol by an extension of the Eutsr criterium (resp. of the analogon of it in 
the region under consideration), and, as it does not suffice as is 
immediately proved, will try to attain it by a corresponding gene- 
ralisation of the Lemma of Gauss (resp. of its analogon). For the 
symbol obtained in this way the relations just named are to be 
pointed out. 
In his treatise „Zur Theorie der quadratischen Reste” published 
in the 1st vol. of the „Acta mathematica” (1882) Scuerine has 
applied for the quadratic restcharacters a deduction founded in 
reality on the ideas developed above; his deductions, however, are 
rather prolix, not being limited to this subject only, and therefore 
they have not found a good reception in the books and the students’ 
