248 



CATALOGUE OF POLAR STARS. 



It will be seen from this example, that, at the end of twenty years, equation (48) 

 to the third term inclusive fails by 0*.5G; but that when the fourth term is included, 

 the place by equation (48) is reproduced within about S .01. For any time greater 

 than twenty years, terms involving higher powers of t will be required in order to 

 obtain a correspondence with the results given by equations (6). 



Computation of the Differential Coefficients from Successive Orders of Differences of the given 



Functions. 



Let A 1? A 2 , A 3 , &c., represent the successive orders of differences of the com 

 puted functions. 



Let A 1 , A 11 , A m , &c, represent the differences opposite the initial function. 

 For the odd differences, 



A+1+ A- 1 A +i + A ~ * A +i 4- A ~ * 



2 2 2 



For the even differences, 



A n = A 2 , A IV =A 4 , A^^A,, &C 



Then (see BrUnnow, Spher. Astron., 1865, p. 28), 



da 1 



dt v> 



J 2 « 1 



[A 1 - J A^ + ^ A^ _ T ^ A™ + „J„ A" -&c] 



dt 2 w 

 d 3 a 1 



2 



[A" - ^ A iv + 8 V A™ - ^ A™ + ^ A* - &c 



J« 3 w z L 



[ A m _ i A v + ^ A vn _ rfK A ix + &c /j 



c? 4 « 1 



t? * 4 w* 



d 5 a 1 



<?« 5 to 5 

 d*a 1 



tf« 6 w 6 L 



[a v — £ A™ + ^ A ra — &c] 

 [ A ^ _ i A™+JftA* _ &C J 



(50) 



c? 7 « 1 



dt 1 v? L 



[ A™ - ^ A K + &c] 



f? 8 « 1 



[a™ — J A x +&c] 



dt* w* L 



</ 9 « 1 r 



= — A 1 * &c 



d 10 n_ 1 r -, 





