CATALOGUE OF POLAR STARS. 



283 



uricavoidable errors in the logarithmic computation of the initial functions. Hence, 

 it may happen that it will be a positive disadvantage to extend the development 

 beyond a certain limit. This limit appears, in the present case, to be near the 

 seventh term when w = IQ and t = 50. 



Develoj^ment of the Functions a and 8 by Mechanical Quadratures. 



The differential coefficients already computed hold true only for the instant of 

 time at which the initial functions a and 8 are assumed to be true. In the series 

 of differences obtained from these functions, the differential coefficients all fiill 

 upon the same horizontal line. 



In order to obtain the summed series whicli will represent the values of a and 

 8 at any assumed epoch of the constants of the precession, it will be necessarj'^ to 

 find the values of the differential coefficients which correspond to the instant t -}- I 

 or t — |. 



If the series does not extend beyond 24 years, the fourth term may be taken 

 as a constant. The coefficients may be converted into differences from which the 

 summation may be directly made by the following relations: 



A,= 



da 

 dt 



Equivalent Fnnclions for the Instant ( i J. 



A^ = A'^ = a constant 



A3 = A'» 



A2 = A" + tIj A'^ 



Aj = A' + I A"' 

 A, + i = A, + ^ A, 



From the example given on page 247 we have tlie following data: 



Epoch. 



rfa 



dt 



d»a 

 dp 



d>a 

 dfi 



dt* 



h. m. 



1875.0 7 20 5.631000 73.51432400 -0.30237888 -0.00203746 +O.OO0O8334 



Whence for the instant t ± i 



