CHAPTER V. ON THE CONVERSION OF A GIVEN CHRONOMETER TIME TO THE CORRESPONDING MEAN SOLAR AND SIDERIAL TIME, AND VICE VERSA. It will have already appeared to the reader of the foregoing pages, that whether the error of a Chronometer is determined from a meridional or an extra meridional observation, the general principle of deduction is the same in both cases. For instance in either case, the mean solar time of the observation is first computed. It is then compared with the observed time, and the difference between the two is taken as the error of the Chronometer. It is convenient to have this process expressed in Algebraical symbols. For this purpose, let s designate the mean solar time of an observation, and c the Chronometer time corresponding thereto, then sce is the Chronometer error at the instant of that observation. Again, calling s' and ' analogous elements to s and c for a subsequent observation, there will arises' c e' the Chronometer error at the time of the = second observation. Suppose t to be the Chronometer time lying between c and c', which is required to be converted to mean solar time. It is clear that if the Chronometer error & at the time t, were known, then t± ɛ would be the mean solar time sought, the upper sign being used when s>c, and the lower sign when sc. ε The term may be computed in the following manner: subtract c from c' and t, the resulting terms (c'-c) and (t-c) will represent Chronometer intervals, the former between c' and c, and the latter between t and c. Again e e stands for the rate, or the increment or decrement (as the case may be) of the Chronometer error engendered during the interval (c-c). Now assuming rates to be proportional to the intervals, during which they are produced, we shall have (c—c′): (e′ un e) : : (t—c) : The fourth term when brought out, will represent the rate produced during the interval (t-c.) When the first error e has been corrected by this rate, the resulting term will obviously be or the error at the time t. EXAMPLE. Suppose at Kaliana, 5th October 1836, the given Chronometer time is 5h 58m 27 it is required to compute the mean solar time corresponding thereto. On reference to page 667 of this work, it will be seen that the numerical values of s, c, e.. for the 5th and 6th October are as follows: This rate is positive, because the error is increasing, e being Hence & = and (t-ε) = h. m. S. S. h. m. S. 5 42 47.21 Mean solar time, corresponding to t. After having reduced a given Chronometer time to the corresponding mean solar time, the latter may now be converted to siderial time, in the following manner. Refer to the Nautical Almanac and take out the mean time of the transit of the first point of Aries immediately prior to the given mean time. Now deduct this transit time from the given mean time, the difference converted to siderial interval will be the siderial time sought. EXAMPLE. Take the mean solar time given above for Kaliana, 5th October 1836. The transit of the first point of Aries used in this computation belongs to the 4th October, because it is that which immediately precedes the given mean time, the same point transiting on the 5th which is after the given time. In the foregoing part of this chapter, the Chronometer errors e and e' are taken with reference to the mean solar time. This however is a circumstance which cannot always be expected to obtain in practice. For instance, it may sometimes happen that the terms e and e' are known with reference to the siderial time only. When this is the case, the given Chronometer time t cannot at once be reduced to the corresponding mean solar time. It must, in the first place, be converted to siderial, and then if required, to mean solar time. With respect to the former of these reductions, the process to be followed is exactly similar to that used for the conversion of any given Chronometer, to the corresponding mean solar time, the given terms s, e, s', e' in this instance, being taken in terms of the siderial, in lieu of the mean solar time. When the siderial time is computed, the corresponding mean solar time may be determined in the following manner: Take out from the Nautical Almanac the siderial time of the mean noon, immediately preceding the given siderial time. Subtract the mean noon time so found from the given siderial time, the difference converted to mean solar interval, will be the mean time sought. EXAMPLE. In illustration of this computation take the same example as that given before, viz., the Chronometer time t =5h 58m 275 as observed at Kaliana, 5th October 1836, and it is required to reduce it successively to the corresponding siderial and mean solar times. The siderial values of s, e, s', and e' as given at page 669, are as follows: Hence ε = 12 40 20-66 + 50·33 = 12 41 10-99 And (t + ) 18 39 37.99 siderial time corresponding to t. In this case, ε = is made because sc. To compute the mean solar time answering to Chronometer time t. Before the chapter is concluded, it is necessary to shew the method of working the following problem: Given a mean solar or a siderial time σ, to compute the Chronometer time t corresponding thereto. Retaining the characters we have already used, we will represent by s and s' the mean solar or the siderial times (as the case may be) of the two time observations, one taken before and the other after 。. c and c' being the Chronometer times corresponding to s and s and e and e' the Chronometer errors derived therefrom. Now deduct s first from s', and then from ; the resulting differences (s'—s) and (σ—s) stand for the mean solar or siderial intervals, the former between s' and s and the latter between and s. Again the rate produced during the interval (s'—s) is e' ce, whence the rate for (-s) will be the fourth term to the following proportion. (s'—s) : (e' cs e) : : (σ—s) :· The fourth term being computed and applied to e, will furnish the Chronometer error & at the given time σ. Correct ing σ by the error so found, there will result the required Chronometer time t. In the foregoing explanation the given terms s and s' and that required to be reduced σ, are supposed to be of the same denomination. But in practice this may not always be the case. For instances and s' may be mean solar and σ a siderial time or vice versâ; that is s and s being siderial and σ a mean solar time. In the former case, a must be converted to mean solar and in the latter to siderial time, after which the necessary reduction may be made as directed above. |