66 -^rt. 5. -T. Takenonchi : 



AVheii III is odd, the components are 



( i ) II, II;, IVi, 1 = 1, -2, , excepting the value, 



for which -^ = I + In, 

 VIII;, when ;;i = (mod. -f), 

 (n) IX, when m = (mod. 3-), 

 (iii) a derived corpus of III; (i = 1 , 2, ; III, = YI, if .- = 1 + 2/>). 



AVlien m is even, Init not divisible ])y 4, we get 

 ( i ) (ii) as above, 



(iii) III,, i = 1 , 2, , excepting the values, 



for which r:i=\L or 1+2/v, 

 YI, when m = Q (mod. 8). 



AVlien III divisible by 4, we get 

 ( i ) (ii) (iii) as above, 

 (iv) X. 



Observe that, in these lists, we miss two kinds of elementary 

 corpora, viz. V and VII. 



Bnt, is it not possible that the corpus V or VII is contained 

 in the whole corpus K\_iii] ? We are now going to shew that it is 

 impossible. AVe shall, however, confine ourselves to the proof that 

 Vi is not contained in K{m). I'he same reasoning also applies to 

 other V's and VII. 



Decompose K[;m'\ into elementary corpora as al)Ove. Then, 

 reject IVi from the components thus obtained, and recompose all 

 the rest. We oI)tai]i a relatively Abelian corpus, which we shall 

 call K'. Now, after the rejection of IVi from the components of 

 K\jn\, there may be still some elementary corpora, whose relative 



degrees are powers of 2; viz. IV,-, 'i=2,3, , or some X's, if m be 



even. Composing these elementary corpora all together, we obtain 

 a divisor of K' . We sliall call it K" . Then, every divisor of K' 

 having a power of 2 as its relative degree must necessarily l)e a 

 divisor of A'"; consequently its relative discriminant cannot con- 

 tain the factor -,. It follows therefore that K' is relatively prime 

 to Vi. Hence, if we compose Vi with K\ Ave obtain a relatively 



