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

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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).

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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,

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

1

(cosy
(cos 2
y ± sin y)2 =

1 2

= 1 + sin y,

and hence

6kt

1

1

3

2

8

(r + r') 2
(r+r')

= 6 cos2 — y sin 1⁄2 y + 2 sin3

2

= (cos y + sin y) - (cos y sin y)"

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1

2

1

2

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3

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

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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 This large reflecting telescope was originally made by Calver; it was for some time used by Dr.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

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Photographed with the Crossley Reflector of Lick Observatory.
Exposure 4 hours.

POPULAR ASTRONOMY, No. 71.

Photographic Efficiency of the Crossley Reflector.

5

ler with the Crossley reflector. A beautiful lantern slide of that photograph was sent to Dr. Wilson, of Goodsell Observatory, only a few days ago, and our illustration was made from the lantern slide. The size of our cut is one and one-half times that

of the picture which appears to be the size of the focal image of the nebula in the Crossley reflector. Of this, however, we are not sure for Professor Keeler has not yet given information on this point.

The picture of this Nebula by Mr. Roberts, of England, will be found in Knowledge Vol. 12, (1889) facing page 188. The Roberts' picture was taken April 21, 1889, with a 20-inch silveron-glass reflector with an exposure of 4 hours. The print is we think by the collotype process. By reading the following account any one will readily understand the advantages which Professor Keeler would have both in size of instrument and locality for work of this kind. Mr. Roberts' picture although enlarged 8 diameters is still a little smaller than that obtained at Mount Hamilton. The enlargement of a photograph so much has undoubtedly both diminished the sharpness of detail and occasioned a loss of the fine contrast and very minute features that may appear in the original negative. The exquisitely delicate mingling of shade and the soft effect of definite color in the Keeler picture is certainly charming to those who love to study good photographic work. Professor Keeler gives reasons for these results in the following article:

"The reflecting telescope has been so little used in this country, since the time of Henry Draper, that a few notes on the photographic efficiency of the Crossley reflector may be of interest to American readers.

The Crossley dome is built on the farther end of a long rocky spur, which extends from the principal summit of Mt. Hamilton toward the south, and is within a few minutes' walk (or climb) of the main Observatory. The mirror, which has a very fine figure, has an aperture of three feet, and a focal length of seventeen feet six inches. The mounting, as compared with the beautifully mounted large refractors of the Observatory, is undoubtedly a rude piece of mechanism, but with sufficient experience of its numerous idiosyncrasies, the observer can obtain negatives with exposures of four hours' duration, with only an occasional failure.

At present the Crossley telescope is being used for photographic observations of nebulæ. For such work the summer months at Mt. Hamilton present almost ideal conditions. The

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