286 CATALOGUE OF POLAR STARS. 



For 3 equidistant intervals a = «„ + i IPi + ^Pi + PJ < = «o + ^i ' (^■^) 



For 4 equidistant intervals a = a^ + \ \_2h + ^P«- + ^Pi + P^ t = a^ + Pj (55) 



For 5 equidistant intervals a = u^ + ^ [1 p^ + 'i'^Pi + l^i^a + '^-Pi + " ^^5] t = u^ + I'J (56) 



And shnilarly for S. 



The values of jh . ■ ■ 2h in the application of this method should, in practice, 

 be computed from the co-ordinates derived directly from the observations. If, 

 however, there are not a sufficient number of observations available for the for- 

 mation of the equal intervals, the first thi'ee or four values of a derived by the 

 method already described may be used for the computation of the corresponding 



values of p. 



After this the values of p can be carried forward by differences Avith suffi- 

 cient accuracy for the computation of the values of a and 8 in advance of the last 

 value computed. It will be necessary to carry the reductions for a and S along 

 tog-ether. 



For the computation of the values pi, p^, p^, &c., equations (IG) and (20) 



become : 



— ^ — (w + m' t) + (rt + n' t) sin a tan 8 (57) 



d t 



= (n + n' t) cos a (^8) 



dt ^ ' 



The values of {m + in t) and log(M + ?/0 can be conveniently written in 

 tabular form. The values for intervals of four years from 1S75 to 1899 are as 

 follows : — 



Epoch. [m + m'O [log « + «'<] \og[n + n't] 



ir 



1875 3.0722450 0.1261147 1.3022060 



1879 3.0723209 0.1261075 1.3021988 



1883 3.0723969 0.1261003 1.3021916 



1887 3.0724729 0.1260931 1.3021844 



1891 3.072.5488 0.1260859 1.3021772 



1895 3.0726248 0.1260787 1.3021700 



1899 3.0727008 0.1260715 1.3021628 



&c. &c. &c- 



In illustration of the application of this method of transferring the co-ordinates 

 a and 8 for any epocli t^ to any epoch t' , we select as provisional values of a and 8 



those derived by computation from the differential coefficients to the fourth term 



inclusive, from 1875 to 1899. They are given on page 268. The corresponding 

 values of ^jj, p^, p^, have been computed from equations (57) and (58). 



