On tlie Eelatively Abelian Corpora. 35 



It follows from ^. S (iü), that the roots of the equation 



^„^1 = iu.-ü.xu-p<h:)(u-:ro.:) = (19) 



are the values of r(?^), where 



(; = 0, 1, 2, 

 /(4 + 3o)''(4-3/>>" n-0^ o .... 3'._l 



^ ' h =0,1,2, ,3'-l; 



•jr, in other words, 



o^(; + y.,j) ,- = 1,4, ,3(3"-l) + l, 



(1 + 2/;)-+- /^ =0,3, ,3(3"-l). 



It is evident that, of all the roots of equation (19), those 

 which arise from one and the same factor on the right-hand sido 

 must correspond to the same value of c in the above expressions 

 for II. But, the consideration of the parallelogram of periods of 

 r(?/), as shewn in §. 4, gives 



Avhence we infer that the roots of the equations 



correspond respectively to tlie values 



c = 2,0,1. 



Since these three equations define one and the same relative 

 corpus with respect to /{."), we may confine ourselves to the one 



This equation can be reduced to the following series of cubic 

 equations : 



//: + 37/,.,^?-l = 0, (20) 



/.■ = 3,4, ,2y?, + 2, y,= 1. 



If we consider (20) as an equation for -, its discriminant is 



