Besides the Moon, we know that there must be countless much smaller bodies constantly circling about the Earth, in various directions and at various distances. Owing to collisions, and attractions of the Sun and Moon, some of these bodies are occasionally drawn from their normal orbits and precipitated upon the Earth as meteorites. Those that continue to circulate in direct closed orbits, nearly parallel to the plane of the ecliptic, must all, drawn by tides which they themselves produce, gradually recede from the Earth. When they have receded to something over one million miles their periods of revolution will be just one year, and as soon as they recede beyond that point they will be left behind by the Earth in her orbit, and lost to her as satellites. Henceforward they will be dependants upon the Sun. As they approach this limiting distance, however, their rate of recession will become slower, and accordingly a large proportion. of them will be found at about this distance from the Earth, and with periods of about a year.

Any meteorite whose period is just one year will remain at a constant difference of longitude from the Sun. As its period approaches this figure it will be much perturbed of the Sun, and rapidly change its longitude with regard to it, save in one position, that where it is in line with the Earth and Sun, and beyond the former. Here the perturbations will be slight, and consequently the change of position slow, and therefore there will be a greater number of the meteorites collected in this place. In this place, moreover, since they will appear full as seen from the Earth, they will appear at their brightest.

It is therefore suggested that we have here a possible explanation of the Gegenschein, which according to this hypothesis would be an actual body attendant upon the Earth; in short, a sort of cometary or meteoric satellite. Its mass would be small, but its bulk as judged from its angular dimensions would be great, being not far from that of the planet Jupiter.

No meteorite whose period of revolution was one year could remain in line between the Sun and Earth, as it would be drawn away by the former body. There seems to be no question but that some action such as that above described must take place, and therefore that the light of the Gegenschein must be due in part, at least, to this cause, the only question is whether it is wholly due to it; in other words, whether the suggested explanation is adequate to produce the effect observed. A suggestion that the Gegenschein might be of meteoric origin was made by Professor Searle in 1882 (A. N. C II. 266).

It may be pointed out if this hypothesis is correct, that the Moon should produce some effect upon the location of the Gegenschein. Thus, when the Moon, is full the Gegenschein should be slightly to the west of its mean position, and when the Moon is new, it should be somewhat to the east of it. In an article printed for private distribution, but taken in part from POPULAR ASTRONOMY for 1897, Vol. V., p. 178, Mr. Douglass gives the results of an examination of 254 observations of the Gegenschein made chiefly by himself, and concludes that the longitude of the Gegenschein does bear a definite relation to the lunar month. He finds "that observations made before new Moon have a tendency to place the Gegenschein farther east than those made after." Should these observations be confirmed, there would seem to be no doubt that the Gegenschein is a material object attending the Earth in its orbit, and not merely an electrical discharge or a phenomenon produced by remote bodies outside of the orbit of Mars.

HARVARD COLLEGE OBSERVATORY,

December 8, 1899.

THE EULER-LAMBERT EQUATION FOR PARABOLIC

MOTION.

ASAPH HALL.

FOR POPULAR ASTRONOMY.

It is well known that the important equation between the time of moving through the arc of a parabola, the two radii vectores, and the chord, was given first by Euler, although for many years it was known as Lambert's equation. This formula. was published by Euler in the Memoirs of the Berlin Academy, 1743. It is given in his work on the orbit of the Comet of

March, 1742. Euler gives two demonstrations, and finds in our modern notation the known equation,

He gives no plus sign to the second term, which is required when the heliocentric motion is more than 180°. Euler's method of determining the orbit of the comet is to make by trial the computed interval of time agree with the observed interval. He also tries the theory of an ellipse, assuming that the semi parameter is to the perihelion distance as 2-a is to 1 and finds a = 0.06047. Euler gives this value of the perihelion distance,

q=

(r-r'+c). (r-r'+c)

4 (r + r′) — 4. [ (r + r' − c) (r + r' + c) ]

I do not find that Euler ever made any further use of formula (1). It appears to have been forgotten by the author, and by every one, until it was rediscovered by Lambert, and published by him in his elegant treatise, "Insigniores orbitae Cometarum Proprietates," 1761. In article 83 Lambert gives formula. (1), with the negative sign only, as Euler had done. He expands (1) in a series according to the ratio : and this is the series employed by Encke for the solution of (1).

с

r+r

Equation (1) has been transformed in many ways. A useful method is to divide the equation by (r+r'), and write it in the form,

taking y less than 90°. Now it is evident that

the last equation gives by the forms for the sine of a triple arc,

We have been greatly interested in the recent photographic work of Professor J. E. Keeler, Director of Lick Observatory, Mount Hamilton, Cal., which is done by the aid of the Crossley Reflector. This large reflecting telescope was originally made by Calver; it was for some time used by Dr. A. A. Common, of England, who obtained with it some excellent celestial photographs for which he received the Gold Medal of the Royal Astronomical Society in 1884. Later it came into the hands of Mr. Crossley who, after making some improvements upon it, in 1895, presented it to the Lick Observatory while Professor E. S. Holden was Director.

Last May Professor Keeler published a brief account of some photographic work by the aid of this instrument in the Publications of the Astronomical Society of the Pacific. That account is so instructive and so useful to any one interested in celestial photography that we give below the article in full.

By permission of Professor Keeler we are able to reproduce one of the finest photographs of Earl Rosse's wonderful spiral (M. 51), sometimes called the "Whirlpool Nebula," we have ever seen. The reproduction in our cut, though carefully done, does not equal the exquisite photograph obtained by Professor Kee