Washington University Studies

Certain comets for which periodic orbits were derived have not been found again in spite of the fact that search ephemerides have been computed for expected returns. In some cases a satisfactory explanation has been made, while in others the failure to find the comet has remained unexplained. The orbits from which the ephemerides are computed are in general definitive orbits based upon a least squares reduction of normal places formed from all the available observations. In the computation of a search ephemeris allowance should be made for uncertainties in the elements due to errors in the observations, limitations in the length of the arc over which the comet was observed, and partial theoretical indeterminateness. In some cases allowance has been made and still the comets were not found. The question arises whether perhaps because of the inaccuracies in the observations and the inconsistencies between them, it would have been advisable not to go to the laborious work of forming normal places and making a least squares reduction, but rather to compute a set of

* A dissertation submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy in the University of California.

elements based upon three reliable positions covering as long an arc as possible. Each of these three positions could be formed from three or four observations close together in time and shown by comparison with an ephemeris to agree fairly well with each other. In order to furnish criteria for determining what variations can be allowed in the elements because the inaccuracy of the observational material, the period can be varied arbitrarily and corresponding new orbits computed. A comparison of the changes in a and 8 for given observation dates due to this arbitrary change of the Period with the uncertainty of the given observations would furnish the criteria desired. This method of procedure will give a means of determining whether the search ephemeris gives sufficient range in which to search for the comet. If the range is larger than that allowed for in the ephemeris, the failure to find the comet would be explained by the fact that it probably lay in a region outside of the limits of the search ephemeris.

For example, a search ephemeris was computed for Brorsen's comet (1879 I) for its return in 1900. Only a small range of ± 4 days was allowed in the perihelion time. This small range produced a range in a of as much as 31° and in d of as much as 8°. A search was made by E. C. Pickering1 but the comet failed to appear. The range of the comet's position is probably much larger, a variation of only four days being rather small.

On the other hand Comet d 1913 (Delavan) which was later found to be identical with Comet 1852 IV (Westphal), was discovered in a region covered by Hnatek's ephemerides 2 for the return of Westphal's comet in 1913. These ephemerides are computed for periods 60.5, 60.6, etc. years up to 61.3 years. Thus according to these ephemerides on January 25 the comet could lie anywhere in a region 29° in right ascension by 37° in declination, and on September 22 anywhere in a region 87° by 118°. The positions of Delavan's comet were 1 A. N. 155, p. 247. 2 A. N. 193, p. 205.

found to correspond to positions of Westphal's comet for a period of 61.121 years. The computation of elements based upon this period and the 1913 observations resulted in elements closely agreeing with those of Westphal's comet and thus proved the identity."

In this paper the method of variation of period will be applied to Comets 1884 II (Barnard) and 1881 V (Denning). The work will be based upon three of the places used by the computer of what appears to be the best orbit. The period of this orbit will be arbitrarily varied, differential corrections found to the heliocentric velocities and coördmates, and the selected observations represented to obtain the residuals. These residuals will then be studied to obtain the range of solution.

The differential corrections will be obtained by means of Leuschner's formulae for a Differential Correction. In these formulae corrections are found to the distance from the earth (po) and the heliocentric velocities (x'o, y'o, z'o) at a given date as functions of the residuals of the initial orbit. The developments are made by means of both series and closed expressions, the former to be used for short arcs and the latter for long arcs for which the series are not convergent. Both the series and closed expressions are given for the case where it is desired to change from an initial parabolic orbit to another parabolic, a hyperbolic, or an elliptic orbit. In the case of an initial elliptic orbit the developments are limited to the computation of another ellipse, as it would so rarely be the case that one would wish to change an adopted ellipse into a parabola. Since the publication of Volume VII, however, Professor Leuschner has devised a Generalized Conditioned Solution whereby one may change any orbit of esmi-major axis a into any other of semi-major axis a. The formulae for this solution are very similar to those for the solution of a parabola except that in the auxiliary quantity P,

3 L. O. Bulletin No. 244.

4 Publ. L. O. Vol. VII.

5 P. 295.

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Washington University Studies, 第 9 卷