CATALOGUE OF POLAR STARS. 



20; 



) 



For the intervals w, 2tv, 3w, 4:tv, &c., we find : 



For w=l, U n t* 



?o 



w 



w 



II tl 



U u t 



2, TP l (t + 1) 2 = U n {t?+ 2t+ 1) 



3, U n {t + 2) 2 = U n (t* + At + 4) 



4, CT n (* + 3) 2 = CT" (* 2 + G t + 9) 



A 



U n Clt + l) 



r n c_v + 3) 



r n (-2 < + 5) 



A. 



2/- 



m 



2*7 U 



The second differences 



therefore constant, and for 



value of w Ave have a 



w 



d 2 a 

 df 



In like manner we shall find 



l, a, 



d*a 



df 



For any 



A 



w 



(Pa 



~dl* ' 



A 



J 4 a 



iC 



(It 



4 ' 



A 



w 



(Pa 



ttc. 



For the 



folio 



the fourth, it will be better 



°y 



- 



thms 



of U Y , U VI , &c, in obtaining the products £P7'\ U yi f, &c. 



We shall always have the check that for these terms the differences J 9 , J c , &c., 



will bo constant. 



tf 



r« 



The values of U l t are found by adding the values of 777 successively to the 



o 



(It 



initial function. 



Having obtained the products U n f for three years, of U lu i 3 for four years, 



and of U IV t 4 for five years, the succeeding 



terms will be obtained from the differ- 



ences thereby derived. 



The following example will serve as an illustration. 



Given 



U u 



S. 



0.1511894 



U m 



S. 



0. 



and 



8. 



jps- 



+ 0.000003472-. 



we have 



1876 



1877 



1878 



1879 



1880 



U*t 



8. 



0.1511894 1 



0.G047576 



1.3607046 J 



2.4190304 



3.7797350 



&c. 



Ai 



0.4535682 



( (.7559470 



•1.0583258 



1.3607046 



A 



0.3023788 



0.3023788 



0.3(1-23788 



