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Extra info for Celestial mechanics. Vol. 5, Part 2. Topology of the three-body problem.

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7) IXo - IX1 + IX2 = 3 - K we compare the connectivity tc* of the covering surface S* with the connectivity ;c of S and obtain 3 - tc* = s(3 - ;c) - b, ( 6) where b is the number of equivalent simple branch points of the covering surface. But since S* and S are each images of S we have tc* = ;c = tc, and (6) reduces to (s - 1)(3 - tc) - b = 0. ( 7) Consequently, if " exceeds 3 then the transformation must be one-toone and if " = 3 the covering surface S* can have no branch point. SURFACE TRANSFORMATIONS If A is an s-to-one transformation of an oriented surface S into itself, then r = ± s is defined as the index of the transformation where the plus or minus is taken according as the transformation preserves or reverses the orientation.

Because of the continuity in the variation of rot AA 1 so long as the curve does not acquire any multiple point the stated formula continues to be true or false in this second process of variation. In the first plane the arc P 0P 1 crosses the strip a ;;;;; r ;;;;; b while P 1Q 1 lies outside of it. Hence P 0P 1 can be deformed on the strip into a rectilinear segment P 0P 1 • Moreover the arc P 1Q0Q1 crosses the strip b ;;;;; r ;;;;; c, and can be continuously deformed into the broken line P 1Q0Q1 without changing the position of P 1 , Q0 or Q1 • Hence we obtain by legitimate modification a broken line P 0P 1Q0Q1 , where these points are arranged in the order of increasing r-coordinates, while P 1 has a larger 8-coordinate than P 0 , and Q1 has a smaller 8-coordinate than Q0 • In this normal position the validity of the expression f3 - IX for rot AA, is evident.

Proof: Suppose that the transformation T has no invariant point. Let (r, ) be the polar coordinates of the ring: 1 ~ r ~ 2 and T be given by r' = R(r, ), ' = E>(r, ), and determine E>(r, ) such that > 8(2, ) and< E>(l, )on each boundary. We map the ring on SURFACE TRANSFORMATIONS the strip: 1 ;;£ y ;;£ 2, - oo < x < + oo, by y = r, x = -

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