METHOD PURSUED. 



about the mean, or follow one another at equal intervals, which is sufficiently near the truth 

 for our purposes. 



Let & be the interval of time (in hours) between each pair of successive comparisons, and our 

 factor will become 



or since __ 



_ 

 = 2* 



n 1 



Since the correction sought is only sensible for the more extended series of comparisons, we 



may almost uniformly substitute unity for n ' and consequently 



n 1 



2^l -24 



This very simple expression shows that for the longest series of comparisons (which extends 

 over about five hours) the factor is very nearly=l ; but that in other cases it is a real fraction. 



By simple inspection of the ephemerides, we shall find the maximum values of the second dif- 

 ferences to be 



Mars I. Man II. Venus I. Venus II. 



f"(S) 9". 6 14".9 49".9 29". 3 



T*T/"(*) .02 .03 .09 .05 



f"(r) .01 .01 .04 .04 



The change in the apparent semi-diameter is thus seen to be certainly insensible, and it is also 

 evident that in declination the influence of these second differences can be possibly perceived 

 only in a few particular cases, occurring in the first FemMS-series. 



Therefore JJ oT J3. m = ^ (t (n) t'Jf'(p) 



The only quantity remaining for consideration is/"(p), which may be directly obtained by 

 differencing the formulas which represent the parallax in declination, as computed for successive 

 hours, and will easily be found to be 



f"(p) = 8 "^ P cos <f l sin 3 cos(0 ) 4 sin 2 15" 



R" 'I'Zllfi 



- [8.83346] p coB<f l sin<? cos(0 ) 



~ 



Substituting this value in the expression above, we obtain finally, 



sin 3 cos 



which gives the correction applicable to the mean of the observed differences for the mean time 

 T; the declination, hour-angle and distance from the earth being denoted by 8, 6 a, and A; 

 the distance from the center of the terrestrial spheroid and the corrected latitude by p and <p'; 

 and the times of the first and last comparisons by I' and <<"> respectively. 



If we represent that part of the expression which is constant for the same place by f, we may write 



Adopting Bessel's value of the eccentricity of the terrestrial spheroid, so that 



log. e = 8.9122052, 

 we have, by the ordinary formulas, (Be.rl. Astr. Jalirb. 1852, p. 325,) 



P sin f ' = gLiLriijgjigjL 



V 1 e a sin a if 



/*C08C,' = . aCOS <P 



V 1 -e a sin 2 c, 



