On the Eelatively Abelian Corpora. 33 



Hence, putting 



we get ' 



2il + 3(\+'2o),,_,z-^^9z,-3(\+2o)^n-i=0. (17) 



Since, by (10), the number ^-'(t\) = ±V—g-^ can always be rationally 

 expressed l)y ^•''(r,,), we see that the corpus Ä"(lr' '(?'/,)) is identical 

 Avith 7v"(^/„ V— (/s), i.e. K(z/„ Vl— />). Thus we conclude that 



Now, if we put as before 



the relation (14) becomes 



where /^ =r(«/' — 1, r/j = — 3r(?/)-. 



Therefore, by successive iteration, Ave get 



f/;i-i 



Avhereyj,_i and ^,,_i are integral functions of r(?^), of degrees 3'"^ and 

 3''"^— 1 respectiveh'. Tlie recursion-formulae for/,"_i and //,,_i are 



All the values of i//, are the roots of the equation 



Since each factor in this equation is of the 3''"'th. degree, the rela- 

 tive degree of the corpus K{iji) cannot be greater than o''"l 



But, on the other hand, since l + 2/> is a primary integer, it 

 can be shewn exactly as in the preceding section, that the corpus 

 Ä'Xsn^v.), i.e. Ä"(K^v,)), is of relative degree o''~\ Therefore, since 

 the equation of degree 3''"^ for determining ^-(r/,) can be looked 

 upon as an equation for ^'('v,)"' of degree 3''''. the corpus K{)^-(^r,)^), 



