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Literatur
American Journal of Mathematics, Vol. 1 pp. 5–29, 129–147, 245–260 and Acta Mathematica, T. 8 pp. 1–36.
Mécanique Céleste, T. 1, p. 82.
Mécanique Céleste, T. 1, p. 97 and Bull. Astr., T. 1, p. 65.
Mécanique Céleste, T. 1, p. 101.
About two thirds of the expense of these computations have been met by grants from the Government Grant and Donation Funds of the Royal Society.
It is perhaps worth noting that 2Q may be written in the form\(v\left( {r - I} \right)^2 \left( {I + \frac{2}{r}} \right) + \left( {\rho - I} \right)^2 \left( {I + \frac{2}{r}} \right) + 3\left( {v + I} \right).\)
A somewhat similar investigation is contained in a paper byM. Bohlin, Acta Math. T. 10, p. 109 (1887). The author takes the Sun as a fixed centre, which is equivalent to taking the Sun's mass as very large compared with that of Jove; he thus fails to obtain the functionQ in the symmetrical form used above.
Amer. Journ. of Math. Vol. 1, pp. 5–29.
Popular Lectures, vol. I, 2nd ed. pp. 31–42; Phil. Mag. vol. 34, 1892, pp. 443–448.
On the part of the motion of the moon's perigee etc. Acta Mathem. Vol. 8, pp. 1–36.
The equation of condition for thee's is easily shown to be\(e_{ - j} \left( {c + 2j} \right)^2 = \sum _i e_{i - j} \Phi _{ - \iota } ;\) and sinceФ i =Ф -i this is exactly the same as that for theb's save thate −j corresponds withb j.
It may be observed that whenV is constant (as is the case when we only consider mean motion)V 2Ψ=Θ, and Mr Hill's equation for δp becomes identical with the present one for δq. It is well to remark that what I denote byc is2c of Mr Hill's notation.
Acta Mathem. vol. 8.
The orbit in question isC=40·0,x 0=1·0334; see Appendix.
When I explained the results at which I have arrived toM. Poincaré, he suggested that there may be coalescence between a doubly periodic orbit and a singly periodic one, when the two circuits of the former become identical with one another and with the latter.
SirWilliam Thomson,On the Instability of Periodic Motion, Philosophical Magazine, vol. 32, 1891, p. 555.M. Poincaré also considers that orbits may have a temporary, but not a secular stability. Acta Mathem. T. 13, 1890, p. 101.
At least the computation was not completed, for it was found to be so troublesome, that it appeared that the work could be better bestowed elsewhere.
It would have been better to have drawn the similar curve forC=38·0, but this one suffices for the present purpose.
Méc. Cél., p. 109.
Phil. Mag., Nov. 1892.
I have now (July 1897) traced some of them.
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Darwin, G.H. Periodic Orbits. Acta Math. 21, 99 (1897). https://doi.org/10.1007/BF02417978
DOI: https://doi.org/10.1007/BF02417978