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The Significance of the Divergence of the Series Solution
Published online by Cambridge University Press: 20 January 2009
In the two parts of this investigation previously published it has been shown that the solution in terms of elliptic functions represents the motion of the particular dynamical system under consideration throughout the whole range of values of s and g for which a real solution exists, except for those values for which s = 2g and k = 1, but that, on the other hand, the series solution is convergent and represents the motion only so long as
for values of s and g for which the sign of this inequality is reversed the trigonometric series representing the solution are divergent. It is of importance to investigate what discontinuities, if any, of the system correspond to values of s and g which lie on the boundary of the region of convergence; the present part is concerned primarily with showing that under such circumstances no discontinuity of the system exists, thus confirming the suggestions made in Part I., § 12.
* Baker, and Ross, , “The Vibrations of a Particle about a Position of Equilibrium,” Proc. Edin. Math. Soc., XXXIX. (1920–1921), pp. 34–57 (referred to in the sequel as Part I.)CrossRefGoogle Scholar; Baker, , “The Vibrations of a Particle about a Position of Equilibrium—Part II.; The Relation between the Elliptic Function and Series Solutions,” Proc. Edin. Math. Soc., XL (1921–1922), pp. 34–49 CrossRefGoogle Scholar (referred to in the sequel as Part II.).
† Part II., p. 48; it is evident that the sign of inequality in inequality (35) on that page should be reversed.
* Part II., §11, p. 48.
† Of. Part I., §8, p. 48.
‡ Part II., § 11, p. 48.
* Part II., § 10 (ii), p. 47.
* The plotting of the boundary curve disclosed a numerical slip in the calculations on which the orbit represented in figure 2 of Part I., p. 45, was based; the correct orbit for this case is shown in figure 3 of the present paper. This emphasises the value of finding the envelope as a cheok on the long and intricate calculations.
† Archives Néerlandaises des Se. ex. et nat. (2), 15 (1910), pp. 246–283 Google Scholar. The system discussed by Beth corresponds to the case in which s=0.
‡ Part I., §2, p. 35, eqns. (1).
* See Poincaré, , Leçons de la Mécanique céleste, I, p. 90.Google Scholar
† Part I., §2, p. 36, eqn. (6).
‡ Part I., §2, p. 36, eqn. (5).
* η1 and η2 are arbitrary constants of integration which may be added to P 1 and p 2. They are determined so that q 1, q 2, P 1, and p 2 satisfy the integral of energy
Their importance was not recognised in the calculations given in Part I.
* As previously mentioned the work given in that place contained a numerical slip.
* The complete calculations for this and the previous Parts are given in the Thesis submitted by the Author for the degree of D.Sc. in the University of Edinburgh, entitled “The Convergence of the Trigonometric Series of Dynamics,” 2 vols., which may be found in the General Library of the University of Edinburgh.
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