246 PARALLAX Mars undergo secular variations which increase from year to year, from century to century, and at last acquire very large values. These secular variations (on the assump tion that all the terms of the theories of the planets are mathematically accurate) have also a well-determined relation to the solar parallax, and therefore afford a means of determining that parallax with an accuracy which increases by the continuance of observation. (2) It has been shown (ASTRONOMY, vol. ii. p. 779 sq., and MECHANICS, vol. xv. p. 708) that the proportions of the interplanetary distances can be very accurately determined and tables be computed from observations of the apparent places of the planets, without any knowledge or assump tion as to absolute distances (although an accurate know ledge of the solar parallax is required for giving final perfection to the lunar and planetary tables). In astro nomical ephemerides therefore the distances of planets from the earth are accurately expressed in terms of the earth s mean distance from the sun, the latter being reckoned unity. Hence, to determine the solar parallax, it is only necessary to measure, at some favourable opportunity, the parallax of any planet, and to multiply the parallax so found by the number which expresses the relation of the distance of the planet from the earth to the earth s mean distance from the sun. (3) When Jupiter is in opposition he is nearer the earth by the diameter of the earth s orbit than when in conjunc tion ; hence, since light occupies a very sensible time to travel, eclipses of Jupiter s satellites will seem to occur too soon in the first case and too late in the latter, the differ ence between the extremes of acceleration and retardation being the time occupied by light in crossing the earth s orbit. This time is about 16 minutes for the mean diameter of the earth s orbit; hence, if the velocity of light can be independently determined, the diameter of the earth s orbit becomes known. The determination, by employing the velocity of light, is also arrived at in another way. The constant of aberration (see ASTRONOMY, p. 757), or the maximum apparent change of a star s true place due to the motion of the observer, depends on the relative velocity of the motion of the observer in space compared with the velocity of light. The angular velocity of the observer is perfectly known ; hence if his linear velocity is known his radius of motion is known. Thus, if the constant of aberration and the velocity of light are independently determined, the radius of motion (that is the sun s parallax) will be found. There are thus three distinctive typical methods:- (1) the gravitational method, depending on terms in the lunar and planetary theories, the constants of which are determined by observation ; (2) the geometrical, or direct observational, method; and (3) the physical method. 1. The Gravitational Method. The moon s parallactic inequality appears, at first sight, to furnish a very accurate method, as its constant is about 1 25", or fourteen times as great as the solar parallax, and the existing observations are very numerous. Unfortunately its determination is inextricably mixed up with the determination of the moon s diameter a diameter increased by irradiation, and therefore different for every telescope, and perhaps for every observer. But this is not all. The maximum and minimum effect of the parallactic inequality occur at first and last quarter, i.e., when the moon is half full. One half of the observations for parallactic inequality therefore are made when the sun is above the horizon, and a great portion of the other half during twilight ; whilst those on which the moon s diameter depend are made at midnight, when the irradiation is a totally different quantity from what it is in daylight or during twilight. Newcomb has attempted to determine the correction of the diameter by the errors in right ascension, derived by comparing Hansen s tables of the moon with observations made by daylight and at night ; but he confesses that the result is so mixed up with the correction of the coefficient of the variation (and, he might have added, with the observer s personality and the telescope employed) that it cannot be relied upon. The following are the most important discussions : Hanson, Mon. Notices R. A. 8., vol. xxiv. p. 8 result 8*92 Stone, Mon. Notices 11. A. S., vol. xxvi. p. 271 ,, 8 85 iS T evcoml>, Washington Observations, 1865 ,, 8 84 Neison, unpublished, probably to appear in Mem. R. A. ,V. , , 878 One cannot look with confidence upon a method which thus permits discordance of more than one per cent, in the discussion of the same observations by different astronomers. The result arrived at must depend on the adopted correc tions of the moon s diameter, and, since that diameter is not capable of determination under the same circumstances of illumination as those in which the observations for paral lactic inequality are made, the judgment of the theorist must step in and assign some more or less hypothetical grounds for the adoption of a particular diameter ; and upon this assumption will turn the whole of the quantity of which we are in search. It is, however, not impossible that the method of observ ing a spot near the centre of the moon, instead of the moon s limb, may lead to a more reliable result. But it will have to be shown by independent methods that the position of the selected spot is not systematically affected by phase. Attention was first called to the method which employs the secular variations of the elements of the orbits of Venus and Mars for determining the solar parallax by a most able and comprehensive paper communicated by Leverrier to the Paris Academy of Sciences, and published in their Comptes Rendus for 1872. July 22. The most important of these variations is that of the perihelion of Mars. The earth s attraction increases the heliocentric position of Mars at perihelion by about 50" in a century, and this change at a favourable opposition subtends an angle of 185" at the earth. On 1672, October 1, the star ^ Aquarii was occulted by Mars. The appulse was observed by Richer at Cayenne, by Picard near Beaufort, and by Homer at Paris. The separate comparisons diffor only 0" 5, 0" 8, and 0" - 3 respectively; and the star i// Aquarii was very frequently observed by Bradley. The increase in two centuries of the geocentric longitude, corresponding to the distance of the planet Mars from the earth on 1G72, October 1, is 294". Hence M. Leverrier concludes that (attributing an error not greater than 1" to the determination of the observed variation) the time has arrived when the solar parallax can be determined with a probable error not ex ceeding -y^ of its amount, or the concluded parallax will be exact to nearly 0"01. The value of the parallax so deduced M. Leverrier finds to be 8"-866 . Similarly he finds from the latitude of Venus, determined by the transits of Venus in 1761 and 1769, combined with the latitude determined by meridian observations of the present day 8"-853. From the discussion of the meridional observations of Venus in an interval of one hundred and six years, he finds 8" -859. These values from the theories of Venus and Mars accord in a wonderful manner, and would appear at first sight to justify considerable confidence in the result. But it is impossible to forget the extraordinary intricacy of the
