All the statements in propositionIII.6 can be checked, as before, by solving the flow equations. The graphs in Fig.6 illustrate these facts clearly. TableI provides a comparison of the results for S 2 and H 2 at various orders. 12 TABLE I: Comparison of flows on S 2 and H 2 at 2nd order and 4th order FIG. 6: b ′ $^{2}$( λ ′ ) vs λ ′ for 4 th order flow GLYPH<Minus> 20 0 20 40 60 80 100 0 50 100 150 200 Λ ' b ' GLYPH<Space> 2 GLYPH<LParen1> Λ ' GLYPH<RParen1> (a) b ′ 0 = 2, α ′ > 0 GLYPH<Minus> 30 GLYPH<Minus> 25 GLYPH<Minus> 20 GLYPH<Minus> 15 GLYPH<Minus> 10 GLYPH<Minus> 5 0 0.5 1.0 1.5 Λ ' b ' GLYPH<Space> 2 GLYPH<LParen1> Λ ' GLYPH<RParen1> (b) b ′ 0 = 1, α ′ > 0 0.0 0.2 0.4 0.6 0.8 1.0 1 2 3 4 Λ ' b ' GLYPH<Space> 2 GLYPH<LParen1> Λ ' GLYPH<RParen1> (c) b ′ 0 = 1, α ′ < 0 S $^{2}$( K > 1) H $^{2}$( K < 1) (2 nd ) a ′ 0 > 0,ancient soln. b ′ 0 = 1, soliton α ′ > 0 shrinking b ′ 2 0 < 1, b ′ −∞ = 1,converging b ′ 2 0 > 1, b ′ −∞ = 1,expanding (4 th ) a ′ 0 > 0, ancient soln. b ′ 0 ≤ 1, ancient , b ′ −∞ = const. α ′ > 0 shrinking b ′ 0 > 1 , eternal, expanding b ′ 0 = 1 . 3048, soliton (2 nd ) a ′ 0 = 1, soliton α ′ < 0 a ′ 2 0 < 1, a ′ ∞ = 1 b ′ 0 > 0 immortal,expanding a ′ 2 0 > 1, a ′ ∞ = 1 (4 th ) a ′ 0 = 1 . 3048, soliton any b ′ $_{0}$, immortal soln. α ′ < 0 a ′ 2 0 > 1, a ′ ∞ = const. expanding a ′ 2 0 < 1, a ′ = const.