CATALOGUE OF POLAR STARS. 



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



For w=l, C^» f- = U'^f 



U"{-lt + 1) 

 w = 2, CT" (« + 1)- = &"" (<■• +2^+1) 2 J^n 



^'" (2 < + 3) 

 M = 3, ?7" (« + 2)2 = Z7" (<= + 4 * + 4) 2 Z7" 



Z7" (2 < + 5) 

 w = 4, Z7° (< + 3)- = CT" (C^ -f 6 « + 9) 



265 



The second differences are therefore constant, and for w = 1, A, = 



value of to we have A, = »o'^ 



(It- 





For any 



In like manner we shall find 



d^ a 



A, = to' 



dUt 



d^a 



dt" ' 



^1 = w* "TT » A, = w^ ■ ^ 

 * dt* ' " f?<5 



&c. 



For the terms following the fourth, it will be better to employ the logarithms 

 of W, f7^S &c., in obtaining the products W'f', W'f, &c. 



We shall always have the check that for these terms the differences Jg, A^, &c., 

 will be constant. 



The values of U^t are found by adding the values of — successively to the 



initial function. 



Having obtained the products U^H- for three years, of U^^U^ for four years, 

 and of U^^ i* for &ve years, the succeeding terms will be obtained from the differ- 

 ences thereby derived. 



The following example will serve as an illustration. 



Given 



&c. 



