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UNIVERSITY OF COLORADO STUDIES 



that to the values of z in the first plane shall correspond in the w-plane 

 only the values w, of the first determination, to the second plane those 

 of w 2 , and to the third plane those of w 3 . These planes or sheets must 

 now be so cut and connected that by turning around B T once, one must 

 be led to a point in the second sheet vertically below the starting-point 

 in the first, because around B lf w I} and w 2 are permuted. Similarly, 

 as around B a , w 2 and w 3 are permuted, by turning once around B 2 , 



Fig. s 



one must be led from the second to the third sheet. The three sheets 

 may be appropriately designated as w r , w 2 -, and w 3 -sheets, Fig. 5. 



Passing in the z-plane a circle with the origin as a center around all 

 branch-points and other singular points which are finite and then making 



an inversion with respect to this circle z—- , a function /(z') is obtained 



which within the inverted circles has no branch-points or other singu- 

 larities except possibly the origin. Describing now a closed circle, 

 say the circle of inversion, it is possible to decide whether the roots of 



