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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1877 Excerpt:...between the extreme places is 223 days, these elements must represent, within the limits of the errors of observation, the entire series of observations on which the normals are based. It may be observed, also, that the successive approximations, in the case of intervals which are very large, do not converge with the same degree of rapidity as when the intervals are small, and that in such cases the numerical calculation is very much abbreviated by the determination, in the first instance, of the assumed values of P', P", Q', and Q" by means of approximate elements already known. For the first determination of an unknown orbit, the intervals will generally be so small that the first assumed values of these quantities, as determined by the equations T / T--T"" = 1-i-ps-), G"=3""', will not differ much from the correct values, and two or three hypotheses, or even less, will be sufficient. But when the intervals are large, and especially if the eccentricity is also considerable, several hypotheses may be required, the last of which will be facilitated by using the equations (82) The application of the formula? for the determination of an orbit from four observations, is not confined to orbits whose inclination to the ecliptic is very small, corresponding to the cases in which the method of finding the elements by means of three observations fails, or at least becomes very uncertain. On the contrary, these formula apply equally well in the case of orbits of any inclination whatever, and since the labor of computing an orbit from four observations does not much exceed that required when, only three observed places are used, while the results must evidently be more approximate, it will be expedient, in very many cases, to use th...