Ixxvi llioi' rilRSl'ED. 



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

 fur our purposes. 



I.t-t ft be the interval of time (in hours) between each pair of successive comparisons, and our 

 tiu-t.>r will become 



or since __ t ( " } 



n 1 



' v \a__I /jM_j, 

 *,-' -24 ' 



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



may almost uniformly substitute unity for re ' and consequently 



n 1 



2/11 -2* 



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

 over about five hours) the factor is very nearly =1 ; 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 



Man I. Man 11. Vemu I. I'eniu II. 



/"(<*) 9".6 14".9 49".9 29".3 



,/"() .02 .03 .09 .05 



/"(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 Fewwf-series. 



Therefore J8. r J<: = 2 



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) = Z"* 1 p cosy 1 sin d cos(0 a) 4 sin 9 ^ 15' 

 _ 8".57116 



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



L 



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 d, 6 a, and A; 

 the distance from the center of the terrestrial spheroid and the corrected latitude by (> and <p'\ 

 and the times of the first and last comparisons by I' and < <0) respectively. 



If we represent that part of the expression which is constant for the same place by 7-, 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, (Berl. Atr. JaUrb. 1852, p. 325,) 



o'=ElL: ; "^- 



1 c" sin-' 



,, COR ^ = __ 



V 1 e 8 sin 9 



