THE VALUES OF -7- I e" 



S/TTJ 





Vfc. 



207 



from which the differences used are derived.* It appears that if the intervals between 

 a series of values be sufficiently small and their number so large that the last difference 

 is practically zero, then the results will usually be about equally correct along the whole 

 series, — for the first interpolated value is affected by the last difference. 



14. In the computation of the values of any function to be tabulated with equi- 

 different arguments, the two usual formulas are — 



V„ = V + an + bn 2 + en 2, + dn 4 + en 5 +fn 6 + gn 7 + etc. 



and V„ = V + nA + 



n.n — 1 

 1.2 



A,+ 



n.n — l.n — 2 

 1.2.3 



A 3 + 



n.n — l.n — 2.n — 3 



(19) 

 A 4 + etc. (20) 



By the first each value has to be computed separately ; by the second, if we 

 determine the values of A', A' 2t A3, etc., for the intervals to be adopted, the process is 

 reduced to one of continuous addition and subtraction, according as the signs of the 

 differences require. Now the conversion of the one formula into the other is readily 

 effected by means of the numerical values of A n m .t The following table, rearranged 

 and extended to A 12 12 , will suffice for all purposes : — 





01 



O 2 



3 



4 



O 5 



8 



7 



8 



9 



10 



11 



O12 



A 1 



1 



1 



1 



1 



1 



1 



1 



1 



1 



1 



1 



1 



A 2 





2 



6 



14 



30 



62 



126 



254 



510 



1022 



2046 



4094 



A 3 







6 



36 



150 



540 



1806 



5796 



18150 



55980 



171006 



519156 



A 4 









24 



240 



1560 



8400 



40824 



186480 



818520 



3498000 



14676024 



A 5 











120 



1800 



16800 



126000 



834120 



5103000 



29607600 



165528000 



A c 













720 



15120 



191520 



1905120 



16435440 



129230640 



953029440 



A 7 















5040 



141120 



2328480 



29635200 



322494480 



3162075840 



A 8 

















40320 



1451520 



30240000 



479001600 



6411968640 



A 

 A 10 



— 

















362880 



16329600 



419126400 



8083152000 





















3628800 



199584000 



6187104000 



A 11 





















39916800 



2634508800 



A 12 

























479001600 



* Conf., e.g., De Morgan's Biff, and Integ. Gale, pp. 544, 545 ; and Woolhodse in Assur. Mag., vol. xi, p. 73, note, 

 t Herschel, Examp. of Calculus of Finite Bifferences, p. 9. His table extends to A 10 10 (conf. De Morgan, 

 Mft. and Int. Calc., p. 253.) This table is readily computed by the formula— 



A n+1 O m + 1 = (n + l) (A n ra + A n+1 m ). (21) 



That is, the sum of the quantities in the two lines for A" aii'l A" +1 , in the preceding column for O m , multiplied by 



