A FEW EXAMPLES IN THE THEORY OF FUNCTIONS 247 



and substituting these values for u and v in (4), we get 



y = H=— \\m 2 -\ t. . [8m 2 — 6m 1 . (6) 



J 27 \ m ° \ mf 



When 2 moves from b to c to d to B 1} y is negative and increases, so that 

 m and consequently u decrease. With the negative sign of y goes the 

 positive sign of v, so that also v decreases. Hence, while 2 moves from 

 b over c and d to B x , w x moves from b t over c 1 and d z to the point 



4 



- . After 2 has crossed B x , y becomes positive and increases; hence 

 m increases, u increases and v decreases, w 2 moves from ~\~ over / z 

 and g x to h x . If we take for m a, proper fraction - , then 



_ 1 li, /8 6 \ 



* 27 \« 2 * \n 2 n J 



(7) 



When v is negative and increases, n must decrease; m and hence w 

 increase. In (7) the positive sign must be taken and hence for v 

 the negative. Consequently as 2 describes the path bed B z , the second 



4 



determination, w 2 , describes the path b 2 c 2 d 2 \- . Similarly, it can be 

 shown that w 2 describes the path \-} 2 g 2 h 2 when 2 describes BJgh, 



and that the third determination w 3 describes the path b 3 c 3 dA — 2"^- j 



2 

 j 3 g 3 h 3 , when 2 describes the line # = —7= from below upwardly. The 



paths corresponding to dm} and dp} can easily be sketched by com- 

 puting the points m 1} m 2 , m 3 , and p t , p 2 , p 3 by formulas (2) and (1), 

 respectively. For further details the reader is referred to Durege's 

 treatise, loc. cit. 



(d) RIEMANN SURFACE OF THE FUNCTION. 



To construct the Riemann Surface for the function w, superpose 

 three parallel 2-planes at equal distances and mark in each the branch- 

 points and the origin perpendicularly above each other. We stipulate 



