270 



DR JAS. BURGESS ON 



Then, by equation (24), we readily deduce the values of A[. A£, etc., from the above 

 by dividing successively upwards each quantity in the column, except the lowest, by 

 /-. ?r, n 3 , etc., adding the quotients to the value of A, B, or C, etc., and lastly dividing 

 the sum by the coefficient of A'. When ?i = 10, this can be done by mere inspection. 

 Thus, for example — 



10A ' =/;+ f0 + 67l0s + 3.103 + 



+ 



6821JT 

 30240.1 06 ' 



And the quantities in the same horizontal lines may be computed by the fractional 

 coefficients ; or, k, i, h, etc., being first found directly from A', /, II, etc., we may use the 

 integral coefficients in eq. (22). 



16. Again, if in a series of values of a function, the first differences before and after 

 any value V be A_ t and A ; A 2 = A_ x — A ; the third differences, before and after A-' be 

 A 3 _j and A 3 ; A 4 - A 3 .!- A 3 ; the fifth differences, before and after A 4 , be A 5 .,-^, 

 and so on, — 



Putting AJ = J(A-i +A) =A -|A 2 > \ 



A^ = KA 3 .i+A 3 ) = A 3 -iA 4 , 



A^KAi 1 + A^) = A 5 -^A 6 , } (26) 



A 7 o = KA 7 . 1 + A 7 ) = A 7 - 1 T A 8 , 

 A* = KA 9 . 1 + A 9 ) = A 9 - , jA 10 . j 



Then, as before, expressing the values of A n , A 2 , A 3 , etc., in terms of the coefficient 

 a, b, c . . . in the formula (19), we have — 



Al = a + c+e+[/ + i, 



A 3 = 3 



a:=5 



A 7 = 7 

 A" = 9 



(c + 5e + 21g + 85i), 

 (e+Uff + Wi), 

 (g+SOi), 



•i 



From these we deduce — 



A 3 . 4A* 36AT 576A* 



7! 



+ 



9! 



,_A?_Aj,7A]_.S20Aj 

 ~3! 4T 6! 9! ' 



s M 2A 7 273A° 

 5! 6!"*" 9! ' 



_A]_3_0Aj 

 °~7\ 9! ' 



tf =2(b + d+f+h + k), 

 A 4 =±\(d + 5f+21h + 85k), 



A 6 =6!(/+14A + 147A;), 

 A 8 = 8!(A + 30£), 

 and A 10 = 10! .k. 



, , A2 A 4 , 4A 6 36A 8 , 576A 10 

 b = ^ A —4! + in— 8T + ToT' 



A^_5A 6 49 A 8 82 A 10 



4i gT + T\ W 



A 6 _14A 8 273A 10 

 J~6l 8! + 



(27) 



,(28) 



, A s 3A 10 



i=4i 



9!' 



and k 



10! ' 



A 10 



io r 



Substituting these values in the general form of the function (19), and simplifying, we 

 have — 



V n = V+7i(AJ+^A 2 )+ 



^ 2 -l),A3^ A4 . n(rc2-l)(H 2 -4) 



3! 



'(A 3 + 7 A 4 ) + 



O! 



(A*+ g A 6 )+etc.< 



(29) 



* This is only an altered mode of writing the formula given in De Morgan's IHff. and Intey. Calculus, p. 546 

 conf. Woolhou.sk, A mir, Mag., vol. xi, (1803), p. 68. 



