Fig. 108 shows the path of the Great Comet of 1882 during the period in which it was seen, from September 3, 1882, to May 26, 1883. These dates--IX, 3, and V, 26--are marked in the figure opposite the parts of the orbit in which the comet stood at those times. Similarly, the positions of the earth in its orbit at the beginning of September, October, November, etc., are marked by the Roman numerals IX, X, XI, etc. The line _S V_ shows the direction from the sun to the vernal equinox, and _S_ Ω is the line along which the plane of the comet's orbit intersects the plane of the earth's orbit--i. e., it is the line of nodes of the comet orbit. Since the comet approached the sun from the south side of the ecliptic, all of its orbit, save the little segment which falls to the left of _S_ Ω, lies below (south) of the plane of the earth's orbit, and the part which would be hidden if this plane were opaque is represented by a broken line.
[Illustration: FIG. 107.--Head of Donati's comet, September 30, October 2, 1858.--BOND.]
162. ELEMENTS OF A COMET'S ORBIT.--There is a theorem of geometry to the effect that through any three points not in the same straight line one circle, and only one, can be drawn. Corresponding to this there is a theorem of celestial mechanics, that through any three positions of a comet one conic section, and only one, can be passed along which the comet can move in accordance with the law of gravitation. This conic section is, of course, its orbit, and at the discovery of a comet astronomers always hasten to observe its position in the sky on different nights in order to obtain the three positions (right ascensions and declinations) necessary for determining the particular orbit in which it moves. The circle, to which reference was made above, is completely ascertained and defined when we know its radius and the position of its center. A parabola is not so simply defined, and five numbers, called the _elements_ of its orbit, are required to fix accurately a comet's path around the sun. Two of these relate to the position of the line of nodes and the angle which the orbit plane makes with the plane of the ecliptic; a third fixes the direction of the axis of the orbit in its plane, and the remaining two, which are of more interest to us, are the date at which the comet makes its nearest approach to the sun (_perihelion passage_) and its distance from the sun at that date (_perihelion distance_). The date, September 17th, placed near the center of Fig. 108, is the former of these elements, while the latter, which is too small to be accurately measured here, may be found from Fig. 109 to be 0.82 of the sun's diameter, or, in terms of the earth's distance from the sun, 0.008.