272 Prof. Bessel's Additions to the Theory of Eclipses, 



(2) ... (aV - a' bY + (ad - a' cf (be' - V cf = 

 (a! sin q + a sin R) 3 + (V sin g + 6 sin R) 2 + (c' sin g + c sin R)~. 



This equation is the proper foundation of the analysis of 

 eclipses ; it is perfectly developed, as all the quantities which 

 it contains refer to the centre of the earth only. It is suscep- 

 tible of innumerable transformations ; for either the position 

 of the pole and the point of commencement of the angle at the 

 same, may be arbitrarily assumed, or the sum of the three 

 squares e~ +f 2 + g 2 may be transformed into the sum of three 

 other squares, (e cos a +f sin u . cost; — g sin u sin v) 2 

 + (e sin u —fcos it. cos v + g cos u sin v)* + (f + sin v 

 -f g cos v)% where u and v may be assumed arbitrarily. 



[4.] The most simple case of an eclipse is that in which 7r' 

 and R are = 0, or where the body eclipsed is at an infinite 

 distance and appears as a point ; this is the case of fixed stars. 

 I shall first develop this case, and use for this purpose the 

 5th equation of my former paper (Astr. Nachr. vol. vi. No. 145. 

 Phil. Mag. Nov. 1829, p. 338). This equation maybe de- 

 rived from the general equation (2) either by transferring the 

 pole to which 8, D ... refer to the circle of declination of the 

 fixed star at the distance of 90° from the same, and counting 

 the angles at this pole from the star, or by writing D and A 

 in place of the arbitrary quantities u and v of the preceding 

 section. The latter substitution gives immediately (without 

 applying any formulas of spherical trigonometry), 



sin q" = [cos 8 sin (« — A) — r . cos <p' sin n sin (jtx, — A)] 2 + 

 [sin S cos D — cos S . sin D cos (« — A) — r sin % (sin $' cos D — 

 cos <p' sin D cos (]«. — A))] 3 , or making sin q = k sin ir (where 

 according to Burckhardt's tables k = 02725) 



fo\ 7° r cos S sin (a— A) " "., '■• , a \ "12 



(3)...Ar= I r-i ■ r cos <p' sin (jx— A) 



, rsin S cos D— cos S sin D cos (a,— A) / . r>. 



+ . ? — r ( sin <p cos D — 



L. sin t \ ' 



cos <p' sin D cos (ju, — A) ) I . 



The single parts of this equation I shall denote, in the order 

 of succession, by P, u s Q, v, so that the equation is written thus: 

 F = (P - uy + (Q - v)\ The latitude of the place of ob- 

 servation is always supposed to be known ; the right ascen- 

 sion of the zenith /x is likewise given by the observation of 

 the sidereal time of the immersion and emersion, and the 

 place of the star is likewise assumed as known. But the 

 place of the moon «, 8, her parallax tt, the ratio of the equa- 

 torial radius of the earth to the radius of the moon k, and the 

 square of the excentricity of the terrestrial meridians e 2 are 



not 



