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'.56; but that when the fourth term is included, 

 the place by equation (48) is reproduced within about O'.Ol. 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 (G). 



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



Functions. 



Let Ai, Ao, A3, &c., represent the successive orders of differences of the com- 

 puted functions. 



Let A^, A", A'", &c., represent the differences opposite the initial function. 

 For the odd differences, 



A' = ^ , A _ 2 , A _ ^ , A.C. 



For the even differences, 



A° = A,, Ai^ = A,, A^ = A„, &c. 



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

 da 



^"- = ^ [a- - i A^ + 3V- A^ - tJ^A-« + ^i, A-^-Ac] 

 = - rA" --.VjA^ + ,V A^ - 5*.A^™ + ^tV^A^ _&C.] 



d'^a 



dt^ w' 



d^ a 1 



^ = - r A^ - i A^ + ^j, A^ - ^gk A'^ + &C.1 



dt' w' L -■ 



^ = -^ r A'^ - i A-' + ,U A-^n _ ^^^^ ^x _f. &e.] 



f^ = ii[^^ -iA-+A3^A- _&c.] 



J?- = ^^[^^ -iA-+^3^A- _&e.] 



(50) 



df to'' 



.^=1 [a^hI- J A^ +&C.1 

 dt'" m" L J 



(?<» TO 



fZ«' W' 



