290 



CATALOGUE OF POLAK STARS. 



Development of Ihe Functions a and d by 3feans of two or more Partial Series expressed in 



Terms of the Ascending Powers of the 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 

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

 limit to the terms of the series. 



Let Y = the values of the functions a or 8 derived from equations (6). 



Y 3 , Y 4 , Y 5 , . . . Y 12 = 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 Y — Y s , Y — Y 4 , Y — Y 5 , . . . 

 Y 12 , 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 iv, 2 ?f, %iv, 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 representation of the residuals for the intervals chosen, and a close approxi- 

 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 w employed, n being the coefficient of w. 



Let a = the value of Y — (Y 3 . . . Y l2 ) for w years, 



h = the value of Y — ( Y s . . . Y u ) for 2 w years, 

 c — the value of Y — (Y 3 . . . Y 12 ) for 3 w years, 

 d = the value of Y — ( Y 3 . . . Y X2 ) for 4 iv years, 

 e = the value of Y — ( Y 3 . . . F I2 ) for 5 to years, 



the value of F — ( Y 3 . . . Y l2 ) for 6 w years, &c. 



