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By Luther Pfahler Eisenhart

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Let (tlnt. tl~t) be the replacement pair. Suppose f : tl5 -7 D is a defining map for D, and n = 1,2,3, or 4. Let Fnl = i, and let 6nl be the subdisc of tl5 such that fno(6n d = tlnl . If = 1,2,···, or Pn, and fnt is defined, let 6nt be the subdisc of tl5 such that in(t-I) (6nt ) = tl nt · Then fnPn : tl5 -7 Dn will be a defining map for Dn. Note that if 1 ~ t ~ Pn, then fntl(tl5 - U~=16ns) = fl(tl5 - U~=16ns). If n = 1,2,3, or 4, and 1 ~ t ~ Pn, let X nt = 6nl U 6 - n2 U ... U 6nt . Clearly X nl C X n2 C ...

Of Wyoming Bruce A. Anderson (1966 at Iowa) Arizona State University Donald V. Meyer (1962 at Iowa) Central College Thomas C. Hutchinson (1966 at Iowa) University of Cincinnati Orabi H. Alzoobaee (1962 at Iowa) Univ. of Baghdad William L. Voxman (1968 at Iowa) University of Idaho Lloyd L. Lininger (1964 at Iowa) SUNY at Albany John P. Neuzil (1969 at Iowa) Kent State University J. Brauch Fugate (1964 at Iowa) University of Kentucky R. Richard Summerhill (1969 at Iowa) Kansas State University Richard M.

5. If Hypothesis A holds, then we have an S-case in Ba. 6. Suppose that we have an S-case in B a , and W is a neighborhood of Ba. Then there exist modifications D' and E' of D and E, respectively, such that (1) (D' U E') n Ba = 0, (2) D' c D U Wand E' C E U W, and (3) each L t has Property P with respect to D' and E'. Proof. We shall use the notation of the preceding part of this section. Let E j be the disc on BdBa bounded by S and containing Xj. We claim that E j n D = 0. Otherwise, there is a curve of intersection '"Y of D in E j .

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