( 4r,(i ) 



Frnm tho identical equation 



Ave derive by tlie successive substitutions ;(• — aj , «o, 03 



(«1— ''s) («1— ''2) "" ~ («2— ''a) ("2— ''2) " («3— ''s) («3— ''2) " 



(",-'.,)V^"'^-^ (".-mI/"°'-"' (<.,,-«V^:'=^'- 



«1 — ^2 ®2 — ^2 - '^S — ''2 



After nniltiplyina' numerator and denominator of tliese tliree frac- 

 tions successively by («2— ";•,), ("n — «1) niid («1— «o), addition "-ives 



4 ("i-"-^(^':'~'>i)]/^'^^^ 

 "3— '2 



Similarlv by interchanging' ro and c;; we obtain 

 = ("2~«;i)(«i"^)l/'"-'"-"' + ("3-",) ("2-^) lX"lI-^ + 





and havir;g moreover the identity 



O 1= («2— frP r<'i— ^'1) + («3— «i) («2—^1) + (ffi — «2) ("3— ^'1)' 



the quantities («1— &i), («2—^1) and (o-j— ?>,) eau bi> eliminated and 

 the invariant relation between the branch-points of a reducible inte- 

 gral takes the form 



