1904.] LAMBERT — EXPANSIONS OF ALGEBRAIC FUNCTIONS. 171 



These first approximations are 



I, as^, — az',aiz^y — aii^ , where a satisfies the equation 



5' 

 This shows that at the origin there is one separate branch, and 

 two separate cycles of two branches each. 



To determine the behavior of the function at 2 = rfc i, place z = 

 / ± 1 in equation (i). There results 



i! _ _ 



(4) w^ (h= 2z' Z'') W* 55 (z' -f- l)- (-t- 22' Z' / — O, 



which for a first approximation may be written 



(5) W^ ± 2Z'W'^ —2' =0. 



Tne first approximations are 



7^ = ±(4/ 2'^ 



from which it is seen that at the branch-points z = dz i five branches 

 of the function hang together in a cycle. 



To determine the behavior of the function at the point z = co, 

 'w= 00, substitute in (1)2 = —, w = —, whence 



(6) 1^ (i - zy w'" — z'^ (I — z!') lu' — z''" = o. 



Equation (6) for a first approximation at (?/^ = o, 2' = o) may 

 be written 



(7) —W Z W 2 =0. 



Equation (7) has two roots each equal to — — 2'\ Increasing 



4 



mation only the lowest powers of 2' in the several coefficients, 



eachofthefiverootsof (6)by-f — 2'' and retaining for a first approxi- 



/r.^ 4*^ /5 4"'' , /4 , 4' ,4 ,3 4 ,6 ,2 ,10 / ,10 



(8) ^^w~^z' id +-V2 "Z^ — — 2 w — 2' w' — z' =0. 



Equation (8) shows that at the point (w = 00 , 2 — 00) the func- 

 tion has five separate branches, that is the point at infinity is not a 

 branch-point of the algebraic function. 



