62 



Art. 5. — T, Takenouchi 



Now, it is evident that 



S(m) ^ P(m) ^ P\m), 



and fS(^vi) is at most relatively quadratic with respect to iXm), and 

 JP(m) at most relatively cubic with respect to P'^{m). l]ut the rela- 

 tive degree of P\m) is in general \(f{m), the only exceptional case 

 being when î«— '2. Hence, Avhen m is odd, the relative degree of 

 F{m) must be ],(f{m). AMien m is even, the relative degree of P(?«) 

 must be either \(p{m) or \(p{m). ]jut, since l\m) cannot be relative- 

 ly c[uadratic with respect to FXm). the value y^{w) is inadmissible. 

 Hence F^n) is of relative degree \(f{m), and consequently coincides 

 with F''{m), provided that m be even. 



Between the three kinds of division-corpora ^S', F, P", theres 

 exist the relations: 



S{m) ^ P{m) ^ P(-;'>)P(<''-') , i (31) 



If Ave decompose >y(-fO, i = 1, '2, , into elementary corpora, we 



get the con:!ponents of the folloAvin'g types: 



(i) I„ II„ III,, IV„ V„ when -,=#=l + iV. -,=^2, a^^l, 



VIII,, 

 (ii) VII, 

 VI, 

 .IX, 

 (iii) X, 



when -, — A+ 2// 



when -,=2, 



«i>l, 



(:m>2 



(32) 



i'he sufiix I is attached to the components in (i) to make clear 

 their dependency upon - . 



The corpus F{-'-''), -/. being an odd prime, can be decomposed 

 into two components (cf. §. 9.). Tlio one is of relative degree 

 ^f('-), and is no other tlian F^-i); the other is of relative degree 

 y^(~i)'''~\ <^ii^^ i^^ relative discriminant is a power of - . The former 

 is a divisor of /S'C-,); and the latter is a corpus composed of VIIL or 

 IX or X. Hence, if avc decompose P«0 into elementary corpora, 

 Ave get : 



