On the Relatively Abelian Corpora. 



63 



when -,=^l + 2/>, -=^-2, (^^^1, ^ 

 when -=1 + 2;^, (ti^^l, 



when -=2, rti>*l, > 



> (33) 



(i) I,, II,, III,, IV,, 



VIIL, 

 (ii) VI, 



IX, 

 (iii) X. 



These promised, let us now consider t]ie relation between 

 Pi^iii) and the elementary corpora. In virtue of (31), we see that 

 the elementary corpora given in (33) are certainly contained in 

 P(m). But, as for the corpora Vi and VII, which are contained in 

 (32), l)ut not in (33), we have to make a further investigation. 



To begin with, suppose that m is odd. In this case, both V« 

 and VII can never be contained in l\m). For, if Vi be contained 



in P{_)n), tJie number sn(^-J, and consequently also ü^Y _,, V must 



be contained in P{;m). Then, from the formula 



^i'w + r)— ^-(ii — v 





it follows that F^jii) must contain the number ^'i^.i:, ), and 



consequently also sn( _;(., j. But, in terms of this last number, 



all the numbers snf^^V i = 2, 8 , can be rationally ex^n'essed. 



It follows, therefore, that Pi^m) contains all the corpora V», 



i = l,2, , and VII. Then P{:m) w^ould coincide with S{m)\ 



wdiich is impossible. Similar I'casoning applies also to Vo. ^ ;•„ 



and VII. 



Hence, making use of the lemma given in the last section, 

 we see that the corpus P{m) must be composed of the following 

 components: 



(i) I., II„ III,, IV., 



VIII,, 

 (ii) VI, 

 IX, 



i — 1,2, , excepting the value, '' 



for which -^=1+2/?, 

 when m^O (mod. -j), 

 when 1)1 = (mod. 3), 

 when m = (mod. 3-), 

 (iii) the derived corpus of V, (/ = 1 ,2, ■ • • ; V,= VII, if -t = 1 + 2o).^ 



)m 



