before remarked, observations at the dark limb can be made with much greater accuracy than at the bright limb (except perhaps in case of a few of the brighter stars). If it is thought desirable, therefore, only observations made at the dark limb need be used in the equations, especially so if stars are observed both north and south of the moon's equator.

On account of the advantages offered by the Pleiades for this purpose, Prof. Peirce developed the equations in a form especially adapted to this group, for use in the longitude. work of the U. S. Coast Survey. The reader who is sufficiently interested in the subject may refer to the reports of the U. S. Coast Survey, 1855-56–57–61, in the latter of which is given a numerical example of the application of the method.

Correction for Refraction and for Elevation above Mean Sea Level.

268. The fundamental equation which has been used as the basis of our analysis expresses the condition that the point from which the immersion or emersion is observed is situated in the surface of a right cylinder enveloping the moon and star. At the same time it has been supposed to be in the spheroidal surface of the earth.

The refraction which the ray suffers in passing through the atmosphere causes the elements of this cylinder to be curved lines instead of right lines; or, more correctly, the surface is not that of a cylinder. Further, it follows from the irregularities of the earth's surface that the point from which the observation is made will not in general be in the surface of the mean ellipsoid. Neither of our surfaces therefore conforms exactly to the mathematical form assumed. The effect upon the observed time of an occultation will

always be small, but in extreme cases must be taken into account in an accurate investigation.

If we consider a ray of light as it comes to the eye at the instant when the star is apparently in contact with the moon's limb, this ray will form a curved line, the asymptote of which will cut the vertical line of the observer at a point where the contact would be seen at the same instant as that observed if no refraction existed. The effect of refraction will then be taken into account if we substitute this point for the point occupied by the observer.

Let

h' the altitude of this fictitious point above the observer's position;

h = the altitude of the observer's position above the mean sea level.

Then h+h the altitude of the fictitious point above the mean sea level.

Let us then suppose the observation to be made from a point at this elevation above the surface of the mean ellipsoid. The necessary transformation will be accomplished by changing cos p' and p sin ' into p cos p' + (h+h') cos p and p sin o' + (h+h') sin p; or, by formulæ 446,

h and h' will always be very small fractions when expressed in parts of the earth's radius; therefore no appreciable error will result from neglecting the products of these

quantities by ee. Also (1+h+h') will be practically equal to (1 + h) (1 + h'), the small term hh' being of no account. The necessary correction for elevation above the mean sea level will therefore be obtained by adding to log p log (1h), and the correction for refraction by adding log (1 + h).

Expanding log (1 + h), we have

M = .43429448 is the modulus of the common system of logarithms.

h is here expressed in terms of the earth's radius. If it is given in feet we shall have, instead of the above,

h

20923597

Therefore, neglecting squares and higher powers of h,

log (1 + h) = h(:000 000 02076).

(462)

If, for instance, the elevation is 1000 feet, the correction to be applied to log & and log" will be .000 0208.

The factor (1+h') will now be considered.

In the general theory of refraction the atmosphere is regarded as composed of concentric strata the thickness of which is uniform and may be regarded as infinitesimal. If the distance of any point in a ray of light from the earth's centre be r, i the angle between the tangent and normal at the point to which is drawn, then it is shown by the theory of refraction that ur sin i is a constant, u'being the index of refraction for the infinitesimal stratum at the point under consideration.

For the point where the ray enters the eye let r., M., and z' be the special values of r, μ, and i. Then 'will be the apparent zenith distance of the star, and from the foregoing

If the first point is taken so far away as to be beyond the limit of the earth's atmosphere, then the refraction at this point is zero and u becomes unity.

The above equation then becomes

Mr, sin z'r sin i.. (464)

0

In the figure,

Z

r

P

r, will not differ appreciably for this purpose from the

equatorial radius of the earth; so that if we regard has expressed in terms of this quantity we have

log (1+ h) = log

sing' sin z

+log M...

(465)

The mean value of μ, is 1.000 2800.

A table is readily arranged for log (1 + '), with the argument, the zenith distance of the star. By referring to the value of 2-equations (411)-we see that & is very nearly equal to cos z. For this purpose we may consider it the same.

The following is Bessel's table for log (1+ h). . In addition to the argument z we have given cos z, for which we may use log 2 without appreciable error.