Page:Encyclopædia Britannica, Ninth Edition, v. 18.djvu/269

PARALLAX 251 equations involving five unknown quantities, from which the most probable value of n can be eliminated with its weight by the method of least squares, in terms of K and K. Care, however, must be taken to confine the combination to such groups as depend on measures from the same stars, if it is desired to eliminate the effects of errors in the adopted star places. Also, since it is assumed that K and K vary proportionally with the time, it is necessary that only such observations should be com bined as have been made at epochs sufficiently near together to render this a safe assumption. Finally the absolute values of K and K for the various combina tions are deduced by developing the values of Aa and A3 from each combination in terms of the time, and thus the definitive values of n are obtained. The combination of these values of n, having regard to the weight of each, led to the result n= -0-209. Whence the value of the solar parallax was 8" 780"-012. It should be remarked that in these observations a reversing prism was so employed as to eliminate any systematic error on the part of the observer which might be due to astigmatism of his eye, or a habit of placing the image of the star otherwise than truly central on the image of Mars. The probable error of one observation of distance having weight unity was found to be0" 24. Twelve such observations were generally made (and often more) on each night, and complete combinations of observations were secured on twenty-five nights. This probable error does not exceed that of a single obser vation of contact on the occasion of a transit of Venus, and yet one hundred and ninety-six such observations were secured, as compared with two which is the utmost that can be secured as the result of any single observer s expedition to observe a transit of Venus. It is impossible, however, to say with certainty that the above result is entirely free from systematic error. There is one possible source of such error to be suspected, viz., the possible effect of the chromatic dispersion of the atmosphere which colours the limbs of Mars in the manner already described. In the case of heliometer observations the effect is certainly minimized from the fact that the star disk which is compared with the limb of Mars is coloured precisely in the same way as the limb but whether all error is so eliminated it is impossible to say. A detailed account of these observations and their reduc tions is given in Mem. R. A. S., vol. xlvi. pp. 1-172. If a minor planet, however, is observed in the above described manner, no suspicion of the error in question can attach to the final result ; and, so far as is known, that method affords the only geometrical means of arriving at an absolutely definitive value of the solar parallax. The following table represents the oppositions of minor planets that will be available for determining the solar parallax till the end of the present century. Date of Opposition. Number and Name of Planet. Approximate Horizontal Parallax at Opposition. Magnitude at Opposition. 1886 November. 8 Flora. 9" 84 1886 December. 79 Eurynome. 8 94 1888 September. 75 Eurydice. 10 94 1888 November. 7 Iris. 10 7 1889 July. 12 Victoria. 10 8 1889 August. 80 Sappho. 9 9 1890 January. 27 Euterpe. 8 84 1>90 June. 43 Ariadne. 10 84 1890 December. 20 Massilia. 8 84 1892 August. 192 Nausicaa. 8 84 1893 September. 6 Hebe. 9 74 1894 September. 84 Clio. 9 94 1897 July. 194 Procne. 8 9 1898 June. 25 Phocea. 8 9i 1899 November. 7 Iris. 9 74 1899 December. 8 Flora. 8 8 The results of many hundreds of observations for stellar parallax by Gill and Elkin (Mem. K. A. #., vol. xlviii. part 1) prove that the difference of two opposite angular distances each not greater than 2 can be measured by a small heliometer with a probable error not exceeding 0" 4 15 when the objects measured are points of light such as two stars (or a star and a minor planet). Hence it is easy to show, that a single observer at an equatorial station (furnished with a suitable heliometer] can determine the solar parallax by the careful observation of two or three of the more favourable of the above oppositions with a probable error not exceeding 0" 01, and with absolute freedom from systematic error. Such a result is not possible by any other known method. 3. The Physical Method. The determination of the velocity of light has recently been the subject of very refined and accurate measurement by the methods both of Fizeau and of Foucault (see LIGHT, vol. xiv. p, 585). The results of the most recent and best determinations of the velocity of light, expressed in kilometres per second, are the following (Sidereal Messenger, vol. ii. No. 6) : Cornu, by Fizeau s method 300,400 Michelson, by modification of Foucault s method 299,940 Newcomb, by still more powerful apparatus and modifica tion of Foucault s method 299,717 If we denote by k the interval required by light to cross the mean radius of the earth s orbit, any independent determination of k will obviously afford, when combined with the velocity of light, a determination of the sun s dis tance, i.e., of the solar parallax (see LIGHT, vol. xiv. p. 584). Such a determination of k is afforded by a discussion of the eclipses of Jupiter s satellites. Only two such discus sions that have any claim to acceptance exist: the first by Delambre in the early part of the present century, from a discussion of an immense mass of eclipses of the satellites of Jupiter comprising observations from 1662 to 1802 ; the second by Glasenapp, in a Russian thesis, in which there are discussed the observations of the first satellite of Jupiter from 1848 to 1873. Instead of Delambre s value of k = 493 8 2 Glasenapp finds &=500 8 81 8> 02. Todd, in calling attention to Glasenapp s results (Am. Journal of Science, vol. xix. p. 62), remarks on these two values as follows : "The former determination rests on a much greater number of observations than the latter; but it is difficult to form a just estimate of the work of an average last-century observation of an eclipse of a satellite of Jupiter. And, moreover, astronomers have no means of knowing the process which led the distinguished French astronomer to his result which was adopted in his own tables of the satellites, and which -was adopted by Damoiseau in his Tables Ecliptiques, published in 1836. The latter determina tion rests upon a mass of observations of definite excellence, ichich have been discussed after the modern fashion." Astronomers, however, whilst generally endorsing these remarks, will not be inclined to follow Todd in combining Dalambre s value with Glasenapp s by giving double weight to the latter. Having regard to those portions of Todd s remarks which we have printed in italics, astro nomers would generally be of opinion that only Glasenapp s value of k can be seriously considered at the present day. This value, combined with the above mean value of the velocity of light, leads to 8"760"-02 as the value of the solar parallax. The photometric observation of the eclipses of Jupiter s satellites as now being carried out at Cambridge, U. S., under Prof. Pickering, will probably ere long furnish the data for a much more accurate determination of k, and it is not impossible that very refined heliometric observa tions of the distance of the first satellite (when apparently