PARALLAX 247 processes through which these results have been evolved, and the liability to some systematic source of error, such, for example, as some neglected term producing a long inequality which may become mixed up with the secular variation. In 1874 the tabular errors of Venus, as determined by the planet s transit across the sun s disk, amounted to more than 5" of arc both in R.A. and declination, and the tabular errors of Mars amounted to more than 8" in R.A. and to about 3" in declination at the opposition of 1877, equivalent to an error of 2" 45 in heliocentric longitude (Mem. R. A. S., vol. xlvi. p. 172). Leverrier s planetary tables do not, therefore, possess the accuracy attributed to them by their distinguished author, and the conclusions at which he arrived probably require some further modi fication. Tisserand (Comptes Rendus, 1881, March 21) has continued the researches of Leverrier, and finds that they require modification, and are also subject to very considerable probable error. The later researches of Tisse rand appear to point to a value of the solar parallax smaller than that found by Leverrier, but his work has not yet been brought to final conclusion. 2. The Geometrical Method. The most favourable oppor tunities for the application of this method are afforded, in a geometrical sense, by the planets Venus and Mars, when the former is in conjunction and the latter in opposition. Of these Venus approaches the earth within one-fourth of the sun s mean distance, whilst Mars, in the most favour able circumstances, approaches only within one-third of the same distance. When Venus is near conjunction she is only visible as a slende* crescent in the neighbourhood of the sun, and at conjunction is only visible on the occasion of a transit across the sun s disk. It generally happens, therefore, that the only means of determining the apparent position of Venus near conjunction is to refer that position to the sun s limb or sun s centre. But the sun s place is also affected by parallax, so that when the position of Venus is referred to the sun the parallactic displacement is only the difference of the parallax of the sun and Venus. Mars, on the other hand, can be referred to stars of which the parallax is absolutely insensible ; thus it happens that the advantage of Venus in point of parallactic displacement is diminished till the geometrical conditions are only 5 per cent, in favour of Venus. Transits of Venus across the sun s disk have been observed for parallax in 1761, 1769, 1874, and 1882. 1 If an astronomer at each of two widely separated stations observes the absolute instant of apparent internal contact of Venus with the sun s limb, he is sure that the centres of the sun and Venus are separated by an angular distance equal to the " semidiameter of the sun minus the semidiameter of Venus." The difference of the absolute times at the two stations is due to parallactic displacement, and, the planet s tabular motion being accurately known, the amount of displacement becomes known. If instead of one contact only the two observers note the instants of internal contact both at ingress and egress, then they practically find the chords described by the planet as seen from both stations. The difference of length of these chords (in time) being known, as well as the approximate diameters of the sun and Venus, and their tabular motion, we have the data for computing the difference of least dis tance of centres of the sun and Venus at the two stations, and this distance being due to parallax, we have the means of computing the parallax of Venus and thence the solar parallax. This latter method (originally proposed by Halley in 1716) has the advantage of not requiring a 1 For conditions when a transit will occur, and past and future transits, see ASTRONOMY, vol. ii. p. 796 rigid determination of the absolute instant of each contact, but merely of the duration of the transit ; in other words, it involves no very rigid determination of the longitude or clock error, but only an exact knowledge of the clock rate. It was Halley s opinion that the instants of contact could be observed with an accuracy within two or three tenths of a second of time, but experience has gone to show that the actual errors are from ten to forty times this amount, and the causes of those errors can now be assigned with considerable certainty. These causes are (1) irradiation and diffraction; (2) disturbance of the image by irregular refraction in the earth s atmosphere; (3) the effect of the atmosphere of Venus in complicating the phenomena at the point of contact. (1) Irradiation increases the diameter of the sun and diminishes that of Venus. Its extent depends on the aper ture of the telescope, the perfection of its optical quality, and the perfection of the focal adjustment. Its amount is also changed by the brilliancy of the sun, i.e., is affected by the transparency of the sky and the density of the sun-shade employed. Also, when the space between the limbs of the sun and Venus becomes smaller than the diffraction disk of the object-glass employed, a greyness or shadow is perceived at the point of past or approaching contact; therefore, within a minute angle equal to the sepa rating power (the diameter of the diffraction disk) of the object-glass, the actual instant of contact can only be esti mated by changes in the diffraction phenomena. (2) When the images are thrown into rapid vibration by irregular re fraction in the earth s atmosphere, it becomes impossible to distinguish between the vibration of the image of the dark body of Venus across the sun s limb near the point of contact and the regular phenomena of irradiation, provided that the atmospheric vibrations are sufficiently rapid to produce a persistent image on the retina of the observer s eye. Thus at the transit of Venus in 1882 observers were instructed to note at ingress the time when there was " a well-marked and persistent discontinuity in the illumina tion of the apparent limb of the sun." Now it so happened that at the Royal Observatory, Cape of Good Hope, the definition was very bad a south-easter was blowing, the effect of which was, as is almost invariably the case, to create a rapid minute vibration in the images of celestial objects (see Sir John Herschel s Results of Observations at the Cape of Good Hope, p. xiv.). Thus "a well-marked and persistent discontinuity in the illumination of the apparent limb of the sun" was seen by all of five observers at the Royal Observatory from 10 to 20 seconds of time longer than at the adjoining stations in the Cape Colony, where the images were seen comparatively steady and well- defined. The instant of occurrence of the above-described phase is therefore a function of the state of the atmo spheric definition, and no accurate means exist of estimat ing such influence. (3) The observation is besides com plicated by the illuminated atmosphere of Venus, which forms an arc of light round the planet near the point of contact. In many cases this light has been confounded with the light of the sun, and has thus caused very con siderable errors of observation. From these various causes the apparent phenomena are different at different stations ; and probably also the same phenomena are described by different observers in very different language. The real difficulty of the discussion of the results arises when these different and differently described phenomena have to be combined. It is of no consequence whether a real or seeming contact has been observed ; it is only necessary to be certain that those observations are combined which represent the same phenomenon. The same phenomenon would correspond with the same apparent angular distance of centres of the
