290 CATALOGUE OF POLAR STAES. 



Development of Ihe Funcliom a and d by Means of two or more Partial Series expressed in 



Terms of the Ascending Poicers of tlie Time. 



It has been shown that it is impossible to obtain the exact development of the 

 primary functions for stars within one degree of the pole when the time exceeds 

 forty years, even in the most favorable case which can occur. 



The time at which the values of the initial functions derived from the develop- 

 ment by Taylor's Theorem begin to deviate from those derived from equations (6) 

 may be extended many years by means of a secondary series which represents 

 tlie residuals between the exact co-ordinates and those obtained with any assumed 

 limit to the terms of the series. 



Let I'o ^ the values of the functions a or S derived from equations (6). 



Y^, Yi, 1^5 .. . Yio = the values of a or S derived from the development which termi- 

 nates with the 3d, 4th, 5th to the 12th term. 



We shall then have a series of residuals Yg — Y^, Jq — ^^4) i^o — ^sj • • ■ 

 Ig — I'jj, any one of which may be represented by a series similar in form to 

 the primary series from which the residuals have been obtained. 



It is to be remarked, however, that this second series will not be continuous 

 with respect to the first. 



If these residuals are obtained for the equidistant intervals to, 2 tr, 3 w, iw, &c. 

 years, we shall have a series of equations in which the number of the equations 

 is the same as the number of the unknown quantities. We shall thus obtain an 

 exact i-epresentation of the residuals for the intervals chosen, and a close apjjroxi- 

 mation to the true values for any intermediate interval which does not much 

 exceed twenty years. 



If the residuals really follow the law expressed by Taylor's Theorem, we shall 

 also be able to extend the agreement considerably beyond the limit for the largest 

 value of n iv employed, n being the coefficient of to. 



Let a = the value of Fo — ( Fg . 



b = the value of Fq — ( Fg . 

 c = the value of Y„ — { Fg . 

 d = the value of Fo — ( Fg . 

 e = the value of Fq — ( Fg . 

 /= the value of Fo - ( Fg . 



. F,o) for 70 A-ears, 

 . F,o) for 2 v; years, 

 . F12) foi- 3 w years, 

 . F,2) for 4 zv years, 

 . F,2 ) for 5 tv years, 

 . F,o) for 6 IV years, &c. 



