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UNIVERSITY OF COLORADO STUDIES 



(l+a+a*=o) , 



the roots of the given equation w 3 -w+z=o, according to Tartaglla's 



formula, are 



( w t =p+q , 



■j W 2 =ap+a*q , (i) 



For 2 



=o,£=-*^, q=+i^~, 



and the values of the" three roots are 



w 1 =o , 



w 3 =q(a 2 —a) = + i , 

 w 3 =q(a—a 2 ) = —i . 



For z = ± 



V21 



p = q, so that at each of these points the two roots 



2 2 



iv* and w, become equal. When z is real and > — — or < — 7= , 



33 n V27 V27 



/> and 5 are real, so that w 1 is real, while w 2 and w> 3 are conjugate 



2 

 complex. When z is real and |z| <~t= , then we have the casus 



r '1/27 



irreducibilis and we can get the real solutions of the cubic by introducing 



2 

 an auxiliary angle <f> by the substitution z= ——7= cos <£ . In this case, 



as \z <— 7= 



V 27 



cos <£ is real and 



|i/ t/> .. . $\ 



>=-v -(cos *sin-l , 



\3\ 3 3/ 



li/ <6 . . 4>\ 

 ;=*-* -lcos-+^sin-l . 



\3\ 3 3/ 



The three solutions of the cubic become 



r ii * 



^=2^-.COS 3 



w 2 =-y|-(-cos- + V / 3sin-j , 

 _- 3 =^(-cos|-l/isin|). 



00 



