38 Art. 5. —T. Taken ouch i : 



The relative différente of y with respect to h{{>) is 



(l + 2.o)-V-rJ— _i — V^Jt , ^ ., ) = a-(l-rr)(l+a-)--a;y. 



^ '-^\l+2; l + /'a- /\ 1 + rr l + ^ra- / v a / j 



Hence the relative diffcrente of C with respect to h{p) contains y 

 to the fourth power. The relative discriminant of C is therefore 

 equal to 2'3^ 



In the relatively quadratic corpus 



Q = KVT^>) = HVY^,}, 

 all tlie integers can he represented in the form 



«+/5a/T+27/ 



2 ' 



wdrere « and ß are integers in k(f'), such that 



«- — (I + 2p)ß- = (mod. 4). 

 From this congruence, it follows that 



op—ß-=(a + ß)(a—ß) = (mod. 2). ' 



Hence or—ß- nmst the divisible by 4. Therefore, from the original 

 congruence, we see that ß, and consequently also «, must Ije divisi- 

 ble l.y 2. 



Thus the numbers 1 and \/l-l-2/^ from a system of leases 

 {Mui'tmcdôasis) of Q, so that the relative discriminant of Q is 

 2^(1 4-2/0- 



§. 12. 



In this section we consider the division of periods of sn b}' a 

 power of 2. 

 Put 



then the duplication-formula for j9(«^) is 



