244 UNIVERSITY OF COLORADO STUDIES 



This again seems contradictory to the equivalent result from (i), 



2 



where w 2 — w 3 , for z = ± — — , was found, and can be explained as above. 



To illustrate the values of the function w, for real values of z draw 

 the graph of the cubic w 3 —w + z=o in a Cartesian plane zw, Fig. 3, 

 which clearly shows the results obtained by the discussion of the roots. 



(b) Branch Points. 



Considering in Tartaglia's formula the expression 



and representing the roots in a complex plane w, let z describe a circle 



2 

 of radius p around the point -7-= as a center and not including the 



F27 



2 . 24. 



origin. By this assumption z — y==p.e ie and z + -y= = —= + p. e i9 , 



V27 F27 V 27 



so that 



VK-'»^H-;(-^r""H''-(rir>'") 



« 



.9 

 2 



p.e*».^-£=+pe*j 



K 27 ' K 27 



If now z describes the complete circle; i. e., when d increases by 



.9 

 27r, nothing is changed in this expression except e* which goes over 



into e\* / = — 0*2 . Hence after one complete revolution of z about 



the point -7= , the expression becomes 

 V 27 



V;(-*±^- 2 ^) i 



in other words p and q are interchanged. The same can be proved for 



2 

 the point — 7= . From (1) it appears that at these points the roots 

 V27 



w a and w 3 as given in (1) are equal. While z in the z-plane moves 



