( 4r.8 ) 



arc given, a binary biquadratic «.,' must b(3 sought which has a given 

 cubic y^/ with the roots «i, aj, «3 among its first polars. So there 

 is an identical equation of tlie form 



ii-om which wc deduce 



S «0 + ^«'l _ ■^ «1 + «3 _ ' «2 + «3 _ - «3 + «4 



In the first place these equations determine z, moreover for the 

 coefficients of a.,* two relations remain, expressed by any two of 

 the four following equations 



6 ylo Ao - 12 /li Ai + 6 Aq A2 = A, i, 

 2 vis Ao — 6^1 A2 + 4 vlo A3 = ^i«, 

 4.I3A1— Gvlj A2 + 2 vlo Ai = ^2«. 

 A. Ao — 12 A. A3 + 6 vli A4 = vlg 2. 



There are still two other conditions for a.,*, namely that the fourth and 

 the fifth of the given quantities are roots of the covariant I A* + 27 «v*. 

 By the four conditions together the quantic «.v'' is entirely deter- 

 mined, and thereby the sixth branch-point c, of the integral. 



We obtain the cxamj^lo treated by Goursat by putting 



^1 = , «1 = 0, «2 = • 

 Then we have 



Ao =0, Ai = «0 «3' ^3 = -5- ^A) ('i^ ^3 = 0, A,4 = — 2 a-i^, 

 i z= 2 ff„ «4, , j := — G «3~ «D , 

 the coefficients « satisfying the equation 



A o «„ «3 — A.2 «0 «4 — .'lo U'/" = 0. 



Lastly, 



I A* + 2 j a J' = 4 «0 (3 a:i ,r + «+)(— «0 «3 a" + «„ u.^ .r^ — 4 «3-^), 

 therefore t'l, ("2, cg are the roots of the cubic 



Co «2 -j- 3 C^ J 3 _|- (73 — _ f^^ (^3 ijS ^ ^(,jj „^ .J.3 _ 4 „g2^ 



