(463 ) 



Every redai'lble integral, with five a«signoil branch-points 

 «1, a.., «3, cj, Co, can be brought into this funn by a linear substi- 

 tution. For by means of the equations 



rtj Cg ("l «2 



«1— «3 ^1— ag 



4 0i 



pi"- + -7r)~' 



'^i — ''a 



«1 — ^3 



4/2, 



öj — «2 Cg — ag ('j — Cj o 



«1— «3 t"2~Ö3 



' ' P (« + -X^-'^3 



we can determine the ratios «i : fg • ^3 ''"d the argument «. 



A somewhat laborious calculation gives the values of the (juanti- 

 ties b and the formulae of substitution. 



Putting 



p ("3— «3) =P («3— «1) = P ("1— "2) = pv, 

 the following results are obtained 



h=— ^P i'+P*-' • . — r- ' h = — ^pv + p'o. 4- ■> ' ' 



P^'i+P «3 P i« 



+ /''«! ' 



hi-= — 2p V 4- p't 



p'H^ -\- p'u.2 



^iP"! ''2P"3 ^3p"3 



I 1 1 



P'H P»i P'h 

 h h h 

 1 1 1 



Furthermore the formulae of substitution are 



(■'■— P»i) (•'^—^1)'^ 



pW—t^ = (pu, + pu;i + pv) 

 P ^—f-2 = (P"-.', + /'"i + pi') 



'i-'-^—f/iV—ffs 



(.>—pu,)ix-b2f 

 4rf.3— gfg*— grg 



(.f— /^Kg) (.r— ^>3) a 

 4 .ï=i— ^rg ^•— j/3 



