302 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



for which 



^ = -1 + 2*, 



V= 1-2*, 



V= -1-2/, 



we have evidently similar re- 

 gions for each. Then, by taking 

 the e'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 



£ = UV, 7] — v \/u (v — u) + 1 , 



£ = uv , r) = — v y/u (y — u) + 1 , 



$ = -(u 2 v + a/mV-F 4w 2 — 4), 



v 



$ = ~(u 2 v — V«* 4 » a + 4w 2 — 4), 



7] = V, 



( = uv 



from 



u 



U 



^z=^(u-V" 2 -4t; 2 +4), 



^ = -(« + V" 2 -4r 2 + 4), 



rj — uv, £ = a 

 f] = uv, t, = u 



00 



(b) 



(c) 

 (<*) 



(e) 

 (0 



u 



£= -( u + Vl6 — 16/ + u 2 -4v 2 -8(L + 2i)v ), 



v = u (v + 1 + 2 1) , 



$ 



M 



)#=!( 



M _ Vl6 - 16t + w 2 — 4u 2 - 8(1 + 2%)v ), 



(g) 



00 



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



Case II. — The polynomial <£(£, rj) contains multiple factors. 



Here, any points which are common to two different irreducible 

 factors of <£(£, rj), or are critical points of a single irreducible factor, 

 will be critical points of the curve 



*(£v) = o, 



