By R. Switzer

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Extra info for Algebraic Topology, Homotopy and Homology

Example text

Put equal to u : a 3a8 ax u* = FA (u), in which we can easily verify that F4 (u) has two real roots between — c2 and -{- c2 (the two other roots are real outside those limits or imaginary according to A 2 — V being ;> than 0); those two roots indicate the limits between which i\ swings to and fro according to a course indicated by elliptic functions. For the case 2/;33 > bx2 for instance, thus for four real roots ux <^ u2 <^ ιιΆ < u4, that course becomes: ,u„ Λ m" w i . _ . , - ^ \ , Λ / u — ιιΊ -j- —-——-—-, where sn = sn ] h t \ / and pivq — tV(j A - ·A (A'-V)!

But now an arbitrary symmetric transformation of S4 can be replaced by a reflection preceded or followed by a double rotation; which is represented by a reciprocal interchange of Sr and Si preceded or followed by a rotation of Sr and one of Si; therefore: The arbitrary symmetric transformation of S4 is represented by an interchange of Sr and Si in arbitrary positions. Let us now consider that for such an arbitrary interchange of Sr and Si a system of coordinates a of Sr is placed on a system of coordinates β of Si whilst that system of coordinates β of Si itself is placed on a system y of Sr ; then we can replace the interchange by a "reciprocal interchange" placing a on β and β on #, followed by a rotation of Sr, placing « on y, or also by a rotation of Sn placing a on y, followed by a reciprocal interchange, placing y on β and β on y.

171 16 ( 731 ) If we put iü2 — Αμ* = af, and RAV + Aq Ar = bH, and if we represent the vector ax φλ ί + α2 φ2 j + αζ φζ k, by (a) φ, we can write the six equations of motion : (a) — 2^1μ.

, then φ remains equal to rf% i. e. if the initial rotation of S4 is a rotation // to a principal space of inertia, then the motion remains a rotation // to that space.