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By Edward Kasner

Differential - Geometric features of Dynamics
Edward Kasner

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Now these functions, which vary continuously with t, can only cease to satisfy inequalities (47) after having reached values satisfying condition (45). And this, seeing that Va < I, is incompatible with condition (44). We must thus conclude that, whatever the ~s satisfying conditions (46), the functions X s will satisfy inequalities (47) for all values of t greater than T. In this manner, we can regard our theorem as proved. We see that the theorem of Lagrange is only a particular case. , Vm (vanishing, like all the functions considered here, for x, = X z = ...

Mk = ... , mk which satisfy m. Downloaded By: [University of Ottawa] At: 00:59 28 April 2010 Then, for m > I, equality (33) will give where R(m) is what R~m) becomes when we replace the (32) by the upper limits adopted above. , (34) ordered in increasing powers of the quantities q" will possess terms of which the moduli will be greater than those of the corresponding terms of series (26), for all positive values of t (they will even be greater than these moduli multiplied bye"). Now series (34) can be considered as ordered in increasing powers of the quantity] ql +q2+'" +qb and if, conforming to what has been noted in the preceding section, we take the following for an upper limit of quantities (32): M API +1-'2+ ," +11,,' our series] will not differ essentially from that which we arrived at in Section 4.

Under the indicated condition we may take k = n. , n), where the f, are holomorphic functions of the quantities lXi' becoming zero for Downloaded By: [University of Ottawa] At: 00:59 28 April 2010 (x, = (X2 = ... 'J. for t = 0). Consequently the preceding equations are solvable with respect to the quantities (Xi' and when the quantities a, are sufficiently small in absolute value we can obtain (35) where the lfJ, are holomorphic functions of the quantities ai' becoming zero for a, = a2 = ... = an = o.

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