282 CATALOGUE OF POLAR STARS. 



The following conclusions are drawn from an examination of these residuals. 

 They relate strictly to the star Groombridge 1119, but they will apply in a general 

 way to all stars having nearly the same declination. 



(a) Between ^ = and if — 32 the correspondence between the results obtained 

 from the Bohnenberger equations and those found by the development by Taylor's 

 Theorem is sufficiently exact, whatever quantity is taken as the initial function in 

 the computation of the differential coefficients, and whatever value is given to iv, 

 except when the initial function is a. 



{b) For any date earlier than 1835 the development fails when tv = 1, while 

 the limit when tv = 8 may be placed at about 1825. 



(c) Between the limits tv — 1 and tv — 16 every form of development fails 

 when f exceeds fifty years. From this point the magnitude of the residuals varies 

 with the choice of the quantity taken as the initial function, and with the value 

 of tv. 



{(J) Variations in the value of to produce the least effect when ./= — , and 



the greatest effect when J =. a. 



(e) Whatever quantity is taken as the initial function in the computation of 

 the differential coefficients, there is a substantial agreement in the values of the 

 residuals between < = and ^ = 50 when tv = 16. 



( f) When a is taken as the initial function, tv should never be taken less than 

 16 when t exceeds 20 years, and a new equinox should be chosen when t exceeds 

 40 years. 



{(/) Between I — and t = SO, j- may be advantageously taken as the initial 



function in the computation of the differential coefficients, both on account of the 

 smallness of the residuals for to = IQ, and on account of the comparatively trifling 

 labor involved in the computation. 



(h) Notwithstanding the general increase in the accuracy of the development 

 with an increase in the number of the terms of the series, the gain when t exceeds 

 40 years is so slight that it will be better in any case to change the equinox at 



intervals of 30 years. In computing the new coefficients, — may be advanta- 

 geously selected as the initial function. It will be sufficient to carry the computa- 

 tion to the sixth term inclusive if ef = 16. 



(/) It will be seen, therefore, that the advantage gained by the increase in 

 the number of terms of the series may be counterbalanced by the effect of the 



