In like manner, in the triangle bcd, we have Again, in the triangle acd, we have (art. 62. (e')) sin. dc cotan. acd tan. ad, and substituting the equivalents, or = sin. PC'C cotan. SCP tan. sc'P, In like manner, in the triangle bcd, we have The angle sc'T (fig. to art. 295.) being a right angle, the angles C'SP and C'TP are together equal to a right angle, and tan. C'TP = cotan. C'SP; therefore the denominator of the fraction is equal to unity (= rad.2), and cos.2 CPC'tan. CSP tan. CTP.... (6); that is, cos.2 ytan. (a—n) tan. (B—n). Next, since cc' is parallel to BB' and to zz' we have (Euc. 2. 6.) SC SC: CB: C'B', or SC: sc':: a: a', and CT CT: Cz: c'z', or CT : c'T :: b: b'; SC' tan. STC': C'T Now sc'T being a right angle, but in the plane triangles TPC', TPC, right angled at P, PC' PT tan. STC' and PC PT tan. STC; whence PC. PC':: tan. STC tan. STC'. Again, in the plane triangle PCC', right angled at C, Also in the triangle SCT, SC: CT :: sin. STC sin. CST: cos. (a-n) sin. (a—n); :: -tan. (a-n) cos. (B-n) sin. (B–n) : sin.2 (a−n); sin. 2 ß cos. 2 n cos. 2 B sin. 2n: cos. 2 a sin. 2 n -sin. 2 a cos. 2 n; :;: sin. 2 B-cos. 2 ẞ tan. 2n cos. 2 a tan. 2n-sin. 2a: Now, multiplying extremes and means, a2 a'2 (cos. 2 a tan. 2 n—sin. 2 a) = (sin. 2 B—cos. 2 ß tan. 2n); Thus n, or the angle nsм may be found; and its value being substituted in that of cos.2 y above, the inclination of the apparent to the real orbit will be obtained. 297. The distances of the fixed stars from any part of the solar system are so great, that it is even yet doubtful whether, at the nearest of them, the diameter of the earth's orbit subtends a sensible angle; but, as efforts are now being frequently made with a view of ascertaining the existence and amount of such angle, which is designated the annual parallax of a star, it will be proper to notice here the nature of the observations which are required for the purpose, and the manner of determining the parallax from them. The method most generally put in practice hitherto consists in comparing the observed geocentric right ascensions and declinations of the brightest fixed stars (which are presumed on that account to be the nearest, and which should consequently have the greatest parallax), with their computed heliocentric right ascensions and declinations: and it is evident that, after all the known causes of error in the observations have been removed, if any differences could be found between the elements so compared, the circumstance would indicate the existence, and serve to express the value of the parallax. 298. The right ascension and declination of the star should be observed at the time that the star is in opposition to the sun in longitude; and, from these, with the known obliquity of the ecliptic, the longitude of the star must be computed by trigonometry; the right ascension and longitude so obtained are geocentric, but at the instant of opposition, the latter is evidently equal to the heliocentric longitude (l') of the star. The right ascension and declination of the star should also be observed at a time when the star is 90° from the state of conjunction or opposition; and from these the star's latitude should be computed; the declination and latitude so obtained are geocentric, but at the instant named the latter is equal to the heliocentric latitude ('). The geocentric right ascension (a) and declination (d) of the star are then to be observed as often as possible during the course of a year; and having corrected them for the effects of precession, nutation, and aberration, the corresponding values of the geocentric longitude (7) and latitude (λ) must be found by computation. Now, it is evident, that if the parallax of the star in longitude or latitude, in right ascension or declination, be insensible, we should have l'1, Nλ, &c.; and if the differences between the values of l' and 1, X' and λ, &c. deduced from the observations should be found to vary, it would follow that the parallax may be appreciated. S Let s represent the sun, s a fixed star, E the earth, and EF the orbit which it describes about the sun. Again, imagine sx, SY, to be rectangular co-ordinates in the plane of the ecliptic, the former passing through the equinoctial point; let fall EX' EY' perpendicularly on sx and SY, and produce Es to s': also let fall SM perpendicularly on the plane of the ecliptic, and draw MX, MY perpendicular to r SX, SY. X F X/ P M Y! Y Then, as in art. 184., sx' and SY' being represented by x and Y, SE by r, and the sun's longitude (=YSE + 180°) by L, we have, x=--r cos. L, Y=-r sin. L. Also, representing SX, SY, SM by x, y, z, and ss by r', we have as in the article just quoted, putting accents on λ and 7, x=r' cos. cos. l' y=r' cos. ' sin. l' and, joining the points s and M, SM=r' cos. N'. Again, drawing E T' parallel to sx, and joining the points E and M, it is evident that T'EM will express the geocentric longitude of s, and SEM its geocentric latitude: also XSM is the heliocentric longitude, and SSM the heliocentric latitude. Now, by trigonometry, y r' cos. ' sin. l'+r sin. L tan. Y'EM (=x)=cos. N' cos. l'+r cos. L' COS. multiplying each term in the numerator of the value of tan. T'EM by we have, cos. l' sin. l' which is equivalent to dividing it by tan. cos. I' r'cos. ' cos. l'+r sin. L sin. l' r'cos. cos. l'+r cos. L ; tan. Y'EM = tan. l' or dividing by r' cos. X' cos. l', and putting p sin. 1" (p being r expressed in seconds of a degree) for,which, when s E is perpendicular to SE, denotes an arc equivalent to the sine of the very small angle ssE, or of the annual parallax in its maximum state, we get 1+p sin. 1′′ tan. T'EM tan. l' sin. L cos. X'sin. l' 1+p sin. 1" COS. L Then, dividing the numerator of the second member by the denominator as far as two terms only (that is, neglecting powers of p above the first) after bringing the fractions in both to a common denominator, tan. T'EM=tan. l'{1+p sin.1" or, tan. T'EM-tan. l'tan. l' p sin. 1". sin. (L-') cos. X' sin. l' cos. l' but (Pl. Trigo. art. 38.), tan. Y'EM-tan. l'= sin. (T'EM-1') cos. T'EM cos. l'° and since p is extremely small, it is evident from the preceding equation, that tan. T'EM-tan. l', and consequently sin. ('EM-') must be extremely small; therefore, putting sin. l' (Y'EM—l') sin. 1′′ for sin. (T'EM-'); also putting COS. for tan. l', and considering cos. T'EM as equal to cos. l', the last equation becomes (in seconds of a degree), T'EM-' (the difference between the geocentric and heliocentric longitudes of the star)-P sin. (L-1') cos.X By corresponding substitutions the value of sin. sEM above becomes, on putting, as before, p sin. 1" for sin. X equal to {1—2 p sin. 1′′cos. X' cos.(L-1')}; or, developing the radical as far as two terms, we have, sin. sEM sin. '{1+p sin. 1" cos. 'cos. (L∙1')}, or again, sin. sEM-sin. X'=p sin. 1"sin. X' cos. A'cos. (L-l'): the second member containing p as a multiplier is evidently extremely small, therefore the first is also extremely small; and since (Pl. Trigo. art. 41.) sin. sE M-sin. X2 sin. (SEM-X) cos. (SEM+λ), on putting arcs for their sines, and considering SEM as equal to X', so that cos. ' may be put for cos. (SEM+λ), we have finally in seconds of a degree, sEM-X' (the difference between the geocentric and heliocentric latitudes of the star) =p sin. cos. (L-1'). Р On comparing these values with the formulæ for aberration in longitude and latitude (arts. 232, 233.) a certain correspondence will be found to subsist between them, and the latter may be made to agree with the former on substituting for the constant of aberration (=20" 36) and changing L (the longitude of the sun) into L+90°, in which case cos. L-7') becomes sin. (L'), and sin. (L-l') becomes cos. (L −1 ). Therefore making these substitutions in the formulæ for MN (art. 234.) the result will be the difference between the geocentric and heliocentric right ascensions of the star, or its parallax in right ascension: also making the |