METHOD PURSUED. IxXV 



and analogously, 



the stars observed apparent declination = d 



" true " " =d 



we shall have 



d d= 3. d,pr + R 



where R represents the difference of the refractions of the limb and the star, and has always 

 the same sign as 3 a d . 



If we denote by C the sum of the corrections in declination due to the assumed tabular 

 elements, (assumable as constant during any one night's work, and employ s, the mean error, 

 in the stead of the unknown error actually incurred in each individual comparison, we shall 

 have the observed difference of declination between planet and star 



J3 = 3 d + C s 



= 3.-d.pr + B + C* 



one equation of this form being furnished by every comparison. Indicating the time to which 

 the comparison belongs by a subjacent letter annexed to the respective symbols, we have for the 

 time, t, of any one comparison 



A3,= 8 at e2 ot Pt r,+ R, + <7 + s 

 and so for a given time near the middle of the series, T, 



AS, = J. T d p T r, + ^r + G + e. 



The difference of these two equations, when developed in series according to the powers of 

 t T=T is 



V> $* A* 



, r |rD 



Let us now take for the time T, the mean of the times of the several individual comparisons, 

 so that 



nT=t' + t" + t'" + ..... t (n} 

 arid put 



n / = M aV + JoV + Jo> + ..... ^.t w 



The mean of all the equations furnished by the individual comparisons will be identical with 

 the mean of all the observed equations after subtracting J3 oT . All the terms of the first order 

 will have disappeared ; and the resulting equation will present the form 



Of the two terms which constitute the first member of this equation, Jd otn is the (given) arith- 

 metical mean of the observed differences in declination, and J<J oT the desired true difference of 

 declinations between star and centre of planet at the time T. 



Taking the hour as unit of time, and putting f"(8),f"(p),f"(r),f"(If), for the second dif- 

 ferences of the numerical values of 3, p, r, R, for successive hours, we may be allowed to put also 



D*8,=f"(ff), D*p=f'(p), &c., &c., so that 

 J,), m - ^ OT = i2Vr | f(S) -f"(p) f'(r) +f'(R) j 



It is manifest that f"(R) can never become sensible. Of the three other quantities,/"((J) and 

 f"(r) are independent of the place of observation and may be obtained directly from the ephe- 

 meris, remembering that as this is computed for intervals of a day, its second differences are to 

 be divided by (24) 2 = 5T6. 



Before developing the term/"(p) it will be well to examine the maximum of the factor 

 2^ ^"r 2 , and to obtain an expression for this factor, more convenient even if not so rigorous, 

 by assuming that the times of the several individual comparisons are uniformly distributed 



