58′′ tan. zs (art. 143.), and expressed in seconds of a degree, we have ss' cos. ss't s't, and ss' sin. ss't = st.

The value of s't is the required variation of the polar distance

st
15 sin. PS

or

ss' sin. ss't 15 sin. PS

Ps in seconds of a degree, and the variation, in seconds of time, of the hour angle ZPS, that is, of the star's right ascension.

148. Since a celestial body when in the horizon of a spectator is elevated by the effects of refraction so as to appear about 32 minutes of a degree above that plane, and since the refractions vary by about 10.5 seconds for each minute of altitude within the limits of half a degree from thence; it follows that the vertical diameter of the sun, or moon, when the luminary touches the horizon, must, from the excess of the refraction of the lower limb above that of the upper limb, appear to be diminished by a quantity which, from the tables, will be found equal to about 4′ 55′′. A certain diminution of the horizontal diameter of the sun and moon also takes place, but this is so small that it may be disregarded. The figure of the disks must therefore have the appearance of an ellipse, but the difference between the vertical and horizontal diameters is less than one second when the luminaries are elevated more than 45 degrees above the horizon.

149. All the oblique diameters of the sun and moon must evidently also suffer an apparent diminution when compared with the horizontal diameters; and if it be supposed that the disks of the luminaries are exactly elliptical, the diminution of any oblique semi-diameter may readily be ascertained, the difference between the horizontal and vertical semi-diameters being found from the tables of refractions, and the inclination of the oblique diameter to the horizon being determined or estimated.

B

M

Thus, let CA be the horizontal, CB the vertical, CM any oblique semi-diameter whose inclination to CA may be represented by 0; then, by conic sections, CA being unity, CB = c, and e representing the excentricity, we have CM=

✓ {1—e2 cos. 20}

:

2

C

2

by the binomial theorem, CM=c (1+ e2 cos. 20), which may be put in the form CM = c (1 + e2 — e2 sin. 20). Finally, considering c (1+e2) as equal to CA or unity, we get CM=1ce2 sin. 20: therefore CA-CM = ce2 sin. 20, or the decrements of CA vary with sin.20.

But when = 90°, the decrement (CA-CB) is equal to the difference between the refraction of c and the refraction of B: let this be found from the table of refractions, and represented in seconds by a; then a sin. 20 will express the excess of CA above CM, or the quantity which must be subtracted from the horizontal semi-diameter of the sun or moon in order to have the inclined semi-diameter. The values of the said excess for every fifth or tenth degree of altitude, and every fifteenth degree of 0 are given in the following table.

5° 10°

Altitudes of the Sun or Moon.

15° 20° 25° 30 40° 50° 60° 70° 80° 90° Decrements of the oblique semidiameters of the Sun or Moon.

150. The latitude of an observatory or station is an element of practical astronomy which can be determined without any knowledge of the movements of the celestial bodies beyond the fact of the diurnal rotation, and, except the effects of refraction, without any data from astronomical tables: it is merely necessary to be provided with a mural, or any circle which can be placed in the plane of the meridian; or, for the ordinary purposes of geography, a sextant with an artificial horizon.

The observations required for determining this element, are. the altitudes or zenith distances of any circumpolar star at the times when it culminates, or comes to the meridian of the station above or below the pole: and, if a circle which is capable of being reversed in azimuth be used, the zenith distance in each position may be taken; that is, with the graduated face of the circle towards the east, and again with that face towards the west, by which double process the error of collimation may be eliminated. The pole star is one which may be advantageously employed in the northern hemisphere, for this purpose, as the slowness of its motion will allow it to be observed on the same night, by direct view and also by reflexion.

Now, let c be the centre of the earth, A the place of an observer on its surface, and HPQ the circumference of the

observer's meridian in the celestial sphere: also let h A, a tangent to the earth's surface at A, in the plane of the meridian, represent the position of the observer's horizon; and let s and s' be the apparent places of a fixed star

at its lower and up

per culmination. The distance of the star being so great that the effect of parallax is insensible, so that ha may be conceived to coincide with HC, drawn parallel to it through the centre of the earth, the apparent altitudes will be нs and Hs'; and the effects of refraction at the lower and upper culmination being represented by ss and s's', s and s' become the true places of the star, and Hs, Hs' the true altitudes: then Ps being equal to Ps', half the sum of the true altitudes is equal to HP.

151. If the earth were a sphere, a plumb line suspended at A would take the direction AC, passing through c, the centre of the sphere, and if produced upwards it would meet the heavens in z: this line ZAC would be perpendicular to ha or HC, and the angle HCP would be equal to zcQ, which expresses the latitude of A. But, if it be assumed that the earth is a spheroid, and if eaq be an elliptical meridian, a plumb line at A would, on account of the equality of the attractions on all sides of a normal line AN, take the direction of that line, and NA being produced would meet the celestial sphere in z' the latitude obtained from the observation will in this case be expressed by the angle z'Nq, or the arc z'Q. This is called the nautical, or geographical latitude; while, z being the geocentric zenith, the angle ZCQ, or the arc zq, is called the geocentric latitude.

:

152. To investigate a formula for reducing one of these kinds of latitude to the other, the following process may be used. Draw the ordinate AR; then by conic sections, we have Cq2 Cp2 CR NR;

2

and in the right-angled triangles ACR, ANR,

CR

NR tang. CAR tang. NAR,

or CR NR:: cotan. ACR cotan. ACR).

ANR (tan. ANR: tan.

Hence Cq2 cp2 tang. ANR tang. ACR.

Let the equatorial and polar semiaxes, cq and cp, be to one another as 305 to 304, which from geodetical determinations (arts. 415. 418.) is the ratio adopted by Mr. Woolhouse in the appendix to the Nautical Almanac for 1836 (p. 58.): then we shall have Cp =.9934; therefore if the geographical latitude be represented by L, and the geocentric latitude by 1, we have .9934 tang. Ltang. l.

Cq2

The second line in the following table shows, for every tenth degree of geographical latitude, the number of minutes, &c., which should be subtracted from that latitude in order to obtain the corresponding geocentric latitude.

153. The word parallax, is used to express the angle at any celestial body between two lines drawn from its centre to the points from whence it may be supposed to be viewed: or it is the arc of the celestial sphere between the two places which a body would, at the same instant, appear to occupy if it were observed at two different stations. The celestial arc between the sun and moon, or between the moon and a star, at any instant, will evidently subtend different angles at the eyes of two observers at different stations on the surface of the earth; and in order that a common angle may measure the same arc, it is necessary that each observed angle should be reduced to that which would be subtended by the same arc at the centre of the earth. When the arc measures the altitude of a celestial body above the visible horizon, the correction which must be applied in order to convert the observed angle of elevation to that which would have been obtained if the angular point had been at the centre of the earth, is called the diurnal parallax. This name has been given to it because it goes through all its variations between the times at which the body rises and sets, being greatest when the latter is in the horizon, and least when in the plane of the meridian; and since every parallax is necessarily in a plane passing through the two points of observation and the object, it is evident that the diurnal parallax will be in a plane passing through the spectator and the centre of the earth; that is, in a vertical plane.

154. The angular distance between two places to which, in the heavens, a celestial body is referred when it is supposed to be viewed from the sun and from the earth, or from the earth at two different points in its orbit, is called the annual

parallax. This will be considered further on (art. 256.); the investigations which immediately follow relate only to the diurnal parallax.

Let s be the place of a spectator on the surface of the earth, supposed at present to be a sphere, and let SB be a section through s, and through c the centre. Let MM' be part of the circumference of a vertical circle passing through the sun, the moon, or any planet; and let zn be a quarter of the circumference of the circle when produced to the celestial sphere.

S

C

M

E

M

B

n

H

Again, let м be the place of the celestial body in the sensible horizon of the spectator, and M' its place when at any altitude; also let m and n, p and q be the points in the heavens to which м and M' are referred when seen from s and c: then the angle SMC, which may be considered as equal to nsm, is called the horizontal parallax, and the angle SM'c or qsp is the parallax in altitude. Let M's be produced till in A it meets a line CA, let fall perpendicularly on it from c; then, in the right-angled triangles MSC, M'AC, the hypotenuses CM and CM' are equal to one another, and Cs, CA represent the sines of the parallaxes: therefore CS: CA: sin. horizontal parallax : sin parallax in altitude. But in the right-angled triangle CAS, by Trigonometry,

CS: CA radius (= 1) : sin. ASC (= cos. M'SM): the angle M'SM expresses the apparent altitude of M'; therefore, rad. : cos. app. alt. of M' :: sin. hor. parallax : sin. par. in alt.

Hence, if the horizontal parallax be given (it may be found in the Nautical Almanac) the parallax in altitude may be computed and as the parallaxes are small, the arcs which subtend them (in seconds) may be substituted for their sines.

155. Since the true altitude of a celestial body M' is expressed by the angle м'CH or (the lines SM and CH being parallel to one another) by its equal M'EM, and since the exterior angle M'EM of the triangle SM'E is equal to the sum of the interior and opposite angles; it follows that the parallax in altitude must be added to the observed altitude of a celestial body in order to obtain the corrected altitude. Let p be the horizontal parallax, p the parallax in altitude, and z the apparent zenith distance; then from the above proportion we have sin. p = sin. P sin. z;