]^^ Art. 5. —'J'. Takenouchi : 



Hereafter we shall call prime integers of the forni (8) simply 

 primär J/ integers, 



§• S. 



Let jJ- Ije a primär}^ integer relatively prime to 3, and m its 

 norm, then 



m = 1 (mod. G) ; 



and the equation of the (w- — l)th degree 



is irreduci])le in /'(/O- ^^he roots of this ecjuation are 

 ^, = sn(r^-^], ^'^^O,!,^, ,m-2, 



y «lenoting a primitive root of f^. 



Now, to find the discriminant of this equation, we make use 

 » of the equality: 



(sn a — sn ü)(sn u + sn v) [siVjt + v) — sn(?/, — r)] 



= 2 sn V CD u tin u su(?^ + v) sn(« — y) 



_ "Isn-it sn?? sni'2^+y)sn(?/, — v) 



tinflu sn/r», 



(10) 



i I 



which can immediately he verified 1)y the addition-theorem of the 

 function sn and the relation (ß). If we put 



H = y — , V = 



ft 



/., /:= 0, 1, -2, ■•-, >ii--2, / * ;/ (mod. '-^^ï 



then, corresponding to eacii of the (m—l){m — S) different comhina- 

 tions of values of P^ and ?^\ we ol)tain an equation of the form (10). 

 Multiplying all these equations side hy side, we get 



_ 0(".-l)(«.-3) ,/(«-S)_ 



Hence, |)utting 



