250 UNIVERSITY OF COLORADO STUDIES 



For, by the first revolution w z is changed into w a and by the second 



2 

 revolution w 2 is changed back to w I . Hence, putting 2 — - 7 ==z' 2 , 



V 27 



the function w determined by the equation 



2 , / 

 w 3 — w-\ — -=+z 2 =o 



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will return with its original value after z' has described one revolution 

 in the z'-plane. Hence, setting w = /(z / ), around 3' = o we have the 

 expansion 



w== /(o)+ z r(o)+ Tr / // (o)+. 



2 



At 



2 

 1/27 



22 



z' = . £= , W I =W a =^jl 



By differentiation we find {]'(z')=w') ^w 2 .u/ — 11/ + 22 / = ; w' — 



_09/l2 » 



which for ^=0 and w/ = \l- assumes the form - . By the usual 



\ 3 o 



evaluation we find / , (o)= ±f-v|- , hence 



" b, '"Jl~Ni( ,- *T;) + - • ■ ■ 



Around B 2 we have 



^-Ji+Vi( I+ y^) + - ' 



' 3 =-(w I +W 2 ) = 2^^+. • • • 



