302 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



for which 



^ = - 1 + 2i, 



^= 1 - 2z, 



r;= - 1 - 2i, 



we have evidently similar re- 

 gions for each. Then, by taking 

 the c's all small enough, we 

 cover the whole unshaded re- 

 mainder of the ^-plane by five 

 circles within each of which there 

 is a development as required. 



The sets of parametric form- 

 ulae, derived by using the inter- 

 mediate transformations, are 





V 



rj = V Vw (y — m) -f 1 , 

 Tj = — V Vu (v — u) + 1 , 



V = 



$ = -(u'^v + Vu*v' + iu^ — 4), 



V, 



^ = -(u'^U — V"*V2 +42^2—4), 



y] = v. 



^ = uv 

 i = uv 



from 



?/. 



i = V2(^i-Vii'-4:V-'-^ 4), 



i = '^{ii+Vu^-^v^ + 4), 



rj = uv, ^ = u 



' 2^ 



2^ 



r} = ic(v + 1 + 2/), ^ = u 



u - Vl6 — 1 6 ^• + m2 _ 4 y2 _ s (^1 ^ 2t) y ), 

 7) = u{v + 1 i- 2i), ^=u 



(a) 

 (b) 



(c) 



(«) 

 (0 



(g) 



00 



with three more sets similar to (g) and (h). 



Cask II. — The polynomial (f>(e, ^) contains multiple factors. 



Here, any points which are common to two different irreducible 

 factors of (ft (f, rj), or are critical points of a single irreducible factor, 

 will be critical points of the curve 



<^(i;^) = o, 



