Excerpt from On the Convergency of the Series Used in the Determination of the Elements of Parabolic Orbits, and the Errors Introduced in the Elements by Imperfections of the Observations: A Dissertation Submitted to the Faculties of the Graduate Schools of Arts, Literature, and Science, in Candidacy for the Degree of Doctor of Philosophy

1. Introductory. - The elements of the orbit of a comet arc usually determined by means of the data obtained at three separate, complete observations, and it often becomes a question of importance to the astronomer as to the suitability, or perhaps we might say the sufficiency, of such a set of observations to determine with accuracy the required elements. He is confronted, on the one hand, with a set of formulae somewhat complex in their nature, and which are subject to limitations in their application owing to the properties of the various functions involved in their construction, on the other hand, the data of observation are subject to limitations owing to unavoidable inaccuracies in the construction of the telescope and the multitude of items which fall under the class known as errors. It is thus both a mathematical and a physical problem with which he has to deal, and it becomes important, first, that a careful analysis be made of the properties of the formulas and the conditions under which they may be applied, and, secondly, that the errors which present themselves in the observations, in spite of the greatest care and skill on the observer's part, may not be allowed to become obscured in the final results of the computation. It has been the purpose of the study of which this paper is a partial result to investigate the formulas for computing cometary orbits from each of these two standpoints. In pursuance of this plan, we have investigated, among other questions, the nature of the functions usually known as the "ratios of the triangles, " and have found the precise conditions under which they may be developed into converging power series of the time intervals between the observations. This discussion is given in the first part of this paper. In another part of the investigation we have found the effects of the errors of the observations upon the computed elements of the orbit of a comet, using Obers's method as a basis of the study. To this is added the results of a computation, by use of the formulae so deduced, of the differentials of error in an actual case.

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