64 Art. 5. —T. Takenoucbi : 



If m be even, but not divisible by 4, tbe relative degree of 7^^^ 

 is equal to that of F{ '^^ \. Hence 



p(,„) = p(^) . 



all the components of which are given in (34). 



Next, let m be divisible by 4. In this case, one or more X's 

 ma}^ be contained in P{in). Composing these X's all together, Ave 

 obtain a relatively Abelian corpus, which we shall call X^,. Simi- 

 larh', composing all the X's contained in Â'(m), we obtain an Abelian 

 corpus X.v. It is evident that X^, is a divisor of X,. But, X^ can 

 never coincide with X,. For, if X^,=X,. then P{m) must contain 



sn(-^ä), where 2' is the highest power of; 2 contained in m. That 



this is impossible can be shewn by exactly the same reasoning as 

 we have done for Vi and VII in tlie above. 



It must be remarked, however, that the same proof does no 

 longer hold for V, and VII in the present case, since the assumption 



that -o, -o, are all odd is essential in that proof. In fact, both 



Vf and VII are completely contained in P(m). For, since X^,<:X,, 

 the relative degi'ee of X_,, is at most half that of X,. But, since 

 S{m) is relatively quadratic with respect to P{iii), it follows that 

 the relative degree of X^, must be exactly half that of X„ and all 

 the other elementary corpora in S{'ni) must be completely contained 

 in P{in). Thus Ave see that P{)n) is composed of 



(i) I^, Hi, III^, IVi, Vi, ^=:l,2, , excepting the values, 



for which ^^ = 1+2,0 or 2, 



VIII;, when vi = (mod. t:?), 



(li) VI, VII, when m = (mod. 3), 



IX, when m^O (mod. 3-), 



(iii) X. 



Now, from (o4) and (ou), the following conclusions can be 

 deduced. 



All the elementary corpora arc included in the division-corpora 

 P{m), if m be made to assume all the positive integral values j or at 



W35) 



