and each of the other analogous functions or combinations (Y2.) must satisfy an analogous equation: if then we change G to G1 + G2, and neglect the squares and products of the coefficients of the small correction G2, G1 being a first approximation such as that already found, we are conducted, as a second approximation, on principles already explained, to the following expression for this correction G2: which may be continually and indefinitely improved by a repetition of the same process of correction. We may therefore, theoretically, consider the problem as solved; but it must remain for future consideration, and perhaps for actual trial, to determine which of all these various processes of successive and indefinite approximation, deduced in the present Essay and in the former, as corollaries of one general Method, and as consequences of one central Idea, is best adapted for numeric application, and for the mathematical study of phenomena. VIII. Continuation of a former Paper on the Twenty-five Feet Zenith Telescope lately erected at the Royal Observatory. By JOHN POND, Esq. A.R. F.R.S. Received March 11,-Read March 12, 1835. DURING the last summer I had the honour of submitting to this Society a short paper on the subject of the large zenith telescope lately erected at this Observatory. It is now nearly twenty years since the erection of such an instrument was first suggested to the President and Council of this Society; at that time the Royal Observatory was in a very inefficient state compared to what it is at present. We had only one circle; and there existed doubts as to the excellence of this instrument, though not any were ever entertained by me. The erection of a second circle put this question at rest; it has been abundantly shown in various volumes of the Greenwich Observations, by a series of more rigorous investigations than any instrument was ever submitted to before, that both the circles may be considered as perfect, their errors being less than their respective makers themselves assigned. This circumstance, though satisfactory to myself, a little diminished the importance of the new zenith telescope. It was hardly to be expected that any new instrument could throw light upon errors already reduced within such small limits; this, however, has been done, and the object of this paper is to explain the process I have employed for the purpose. Whoever is acquainted with the method of constructing the Greenwich Catalogue, must have perceived that the places of those stars which are observed by reflection are, according to all probability, more exactly determined than those which have been observed only by direct vision. y Draconis, a star which since the time of BRADLEY has been of first-rate importance in the Greenwich Observations, cannot be observed by reflection. The probability of error was therefore greater in the place of this star than in that of any other. The new instrument has shown that this error does not exceed a quarter of a second; a degree of accuracy scarcely credible, and no doubt requiring to be confirmed by future observations. The nature of the question to be determined in this case has happily produced a competition for excellence among the observers with the different instruments, which gives me an opportunity of showing the present state of practical astronomy at Greenwich. The new instrument has been employed during the last summer under very unfavourable circumstances, both the building and the instrument having been almost constantly under repair. It is not requisite on this occasion to enter into the details of these difficulties; I only wish to explain the nature of the experiments, the results of which I am now about to lay before the Society. We have now three distinct methods of determining the place of any star passing the meridian near the zenith. First, by means of the mural circles; secondly, by the zenith telescope used alternately east and west, as is usually done with similar instruments; and lastly, by means of a small subsidiary star, as described by me last year in a paper laid before this Society, and which I am inclined to think more exact than any other method. By the following computations it will be seen that the three methods give results nearly identical; and that when the observations with the two circles are numerous and made with sufficient care, a quarter of a second is the greatest error to be apprehended. Royal Observatory, March 10, 1835. Results of Observations on y Draconis and Bode 170 Draconis. Zenith distance of y Draconis determined by three different methods. Zenith distance of Bode 170 Draconis determined by three different methods. TABLE I. Containing 60 successive observations of the small auxiliary star, Bode 170 Draconis, divided into series of 10 each. From this it appears that the mean error of 10 observations = 0"-155, and that the mean error of 30 observations, as deduced from the next page, = 0"·067. * The accuracy of this quantity is of no importance, as the circles, according to our present mode of employing them, give, in fact, zenith distances, which are afterwards converted into polar distances by the application of the above co-latitude, and as such are registered in the Greenwich Catalogues. U 2 |