( 457 ) 



evidently O and c/:, eaeli taken twiee, and, as 



/A* -f 2,;«,'-= — 2i.v" , 



we mnst take A = 1 and ," = 2. So in general the quantities r are 

 determined liy the equation 



As soon as we regard as known o)ie root c^ of this e(iuation we 

 ean eonstruet a reilncil>l(' int(>gral. For then we ean put 



a, a/ = (.r — «i)(.c — Oo)G'- — «3) > 

 (*■ a;. + 2i «.') ^ (.<■ - -■,)(■' - ''.2)C'- - '3) . 



whenee it follows that 



R. — ' - 7?i ^-* = (> Al , 



(/.(■ (Ij' 



^ "t 



1/ /,'i R., n,, h'i = — ^ ]/ a, «/ . — L_ (j ^,* + 2 j «/) , 



so that 



(.,- — h^) dx 



■-/ 





is an integ-nd of the first kind relative to the quai'tic ./', whirli is 

 transformed by the substitution 



p^V — fi pW — fa 



('1 (■'• - ''Ol''' - /'])■' ^3 {-^ - '■2)(^ - ''2)' 



_ pW—t3 1 



~ «3 (.r — e.){x — b-if ~ (>i (.(• — fl'i)(.c — «oX-'' — «.;) 



into an elliptic integral with the variable pW. 



It is now clear that in this way it is possible to eonstruet the integi'al 

 out of five assigned bramdi-points. If the five braneh-points "1,02, "31 '"n ^2» 



