Cor. It is evident that parallax increases the zenith distance, and consequently diminishes the altitude. Hence to obtain the true zenith distance from the apparent, the parallax must be subtracted; and to obtain the true altitude. from the apparent, it must be added.

92. The sine of the parallax at any altitude is equal to the produc: of the sine of the horizontal parallax by the sine of the apparent zenith distance.

Since, sin CAB = sin ZAB = sin N, we have from the

As the parallax is always a small angle, that of the moon which is much the greatest being only about a degree, we may frequently take the parallax itself instead of its sine (App. 51). We then have,

=

Р P sin N

(C)

When the spheroidal figure of the earth is taken into view, the zenith distance must be taken in reference to the geocentric zenith, and r must be the radius of the earth at the place of observation.

93. Distance of a body in terms of the horizontal parallax and radius of the earth.

From (92 A), we have,

=

If R the equatorial radius of the earth and the equatorial parallax, then

(E)

From these two expressions for D, we have the following relation between the equatorial parallax and the horizontal parallax at a place, where r is the radius of the earth.

Rr ::

: P

(F)

It also follows that the parallaxes of different bodies, or of the same body at different distances, are, inversely as the distances.

For, let D and D' be the distances of two bodies from the earth, and and the corresponding equatorial parallaxes.

Then,

94. To find the equatorial parallax of a body.

(G)

Let B, Fig, 16, be the body, and A and A' two places situated remote from each other on the same meridian. Let the meridian zenith distances ZAB and Z'A'B be observed at the same time by two observers at A and A' and let them be corrected for refraction. Also let the meridian distances ZAS and Z'A'S of a star which passes the meridian at nearly the same time with the body, be observed and corrected for refraction. Then BAS which is the difference of the corrected values of ZAS and ZAB, is known; and also BA'S, the difference of Z'A'B and Z'A'S. Now ABA' + BAS BA'S + ASA', both sums being equal vertical angles at L. But

the supplements of the

the angle ASA' is entirely insensible (13). Hence ABA' + BAS = BA'S, or ABA' BA'S BAS. If the zenith of the body is greater at each place than that of the star as may sometimes occur when the zenith distances of the body and star are nearly the same, B falls between AS and A'S. In this case it will easily be seen that ABA' = BA'S + BAS. Hence ABA' the difference or sum of the known angles BA'S and BAS, is known.

From the latitude Zdq and Z'd'q of the places A and A', the geocentric latitude 2Cq and z'Cq may be found (76).

The difference between Zdq and 2Cq gives the angle ZAz, and this angle taken from the zenith distance ZAB leaves the geocentric zenith distance zAB. In like manner we find the geocentric zenith distance z'A'B. Put,

N, N'
P, P' = "

the app. geocen. zen. distances zAB and z'A'B, horizontal parallaxes at A and A', P, p',, parallaxes ABC and A'BC. r' = radii CA and CA',

r, r'

and let R and be as in the last article.

Then since ABC + A'BC ABA', we have p + p

=

= 7.

r sin Nr sin N'

R

r sin N + r' sin N

found from the latitudes Hence the quantities in

It is not essential that the two observers should be on exactly the same meridian; for if the meridian zenith distances of the body be observed on several consecutive days, its change of meridian zenith distance in a given time will become known. Then if the difference of longitude of the two places is known, the zenith distance of the body as observed at one of the meridians may be reduced to what it would have been found to be if the observations had been made in the same latitude at the other meridian.

95. Moon's parallax and distance. In the year 1751, La Caille and La Lande, two French Astronomers, made corresponding observations on the moon; the former at the Cape of Good Hope and the latter at Berlin. From these observations, others of a similar kind, which have since

been made, and from other methods, the moon's parallax has been ascertained with much greater precision than it was previously known. The parallax and consequently the distance (93) are found to vary considerably during a revolution of the moon round the earth. It is also ascertained that the least and greatest parallexes or greatest and least distances in one revolution of the moon differ materially from those in another. There is, however, a mean distance, a mean of the average greatest and least distances that is not subject to this change. The parallax corresponding to this mean distance is called the constant of the parallax. The constant of the moon's equatorial parallax is found to be 57' 1". The equatorial parallax when least, is about 53' 51" and when greatest, 61′ 29′′.

From tables that will be hereafter noticed, called lunar tables, the equatorial parallax of the moon may be obtained for any given time. The parallax computed from these is given in the Nautical Almanac* for every 12 hours throughout the year; whence it may easily be obtained for any intermediate time. From the equatorial parallax the horizontal parallax at a given place may be found by (93 F), or by a table computed for the purpose.

Taking the moon's parallax 57' 1", we have, (93 E), DR. R

=

206265
3421

=

=

Rx 60.3239000 miles, nearly,

Hence the moon's mean distance from the earth is about 60 times the equatorial radius of the earth or 239000 miles nearly. The least distance is about 56 times the equatorial radius and the greatest, 64 times that radius.

96. Sun's parallax and distance. By the preceding method (94), the sun's parallax may be ascertained to be about 9". By a method that will be noticed in a subse

* The Nautical Almanac is an astronomical ephemeris published annually at London and republished at New York. It contains a large amount of data of great importance to the mariner and also to the practical astronomer. It is usually published about three years prior to the year for which it is computed. The Connaissance Des Tems, published at Paris, the Astronomisches Jahrbuch, published at Berlin, and the Effemeridi Astronomiche, published at Milan, are ephemerides of a similar character.

quent chapter his mean equatorial parallax has been found to be 8".6. The parallax when least is about 8".5 and when greatest, about 8".7. From the mean parallax the mean distance is found to be 23984 times the equatorial radius of the earth or 95000000 miles nearly (93. E).

97. The apparent semidiameter or diameter of a body, seen at different distances is inversely proportional to the distance.

Let A and A' Fig. 17, be two positions from which a body whose centre is C, is viewed. Then AB and and A'B' being tangents to the body at B and B', the angle CAB is the semidiameter of the body as seen from A, and CA'B as seen from A'. Put CAB, CA'B, D = AC ♪ = and D' A'C. Then D sin CB = CB' = D' sin &, or ♪ DX D' x . Hence,

=

=

=

D: D'::::: 2:28.

=

Regarding CB, Fig. 15, the distance of a body from the centre of the earth, as constant, the distance AB from a place on the surface must diminish as the altitude increases; and consequently the apparent semidiameter of the body as seen from A, must increase. The apparent semidiameter when the body is in the horizon, is sometimes called the horizontal semidiameter, and when it is elevated, the augmented semidiameter. When the expression, apparent semidiameter, is used without reference to the altitude of the body, it implies that of the body when in the horizon.

98. The sine of the apparent zenith distance of a body is to the sine of the true zenith distance as the apparent diameter of the body at that zenith distance is to the horizontal diameter. Let be the horizontal semidiameter of the body, and the apparent diameter at B, Fig. 15. Then (97), AB′ : AB:::or since AB' may be regarded as sensibly equal to CB' or CB, we have, CB: AB. But CB: AB

:: sin ZAB: sin ZCB. Hence,

sin ZAB sin ZCB ::: ♪ :: 2♪' : 2♪.