OF ARTS AND SCIENCES : SEPTEMBER 12, 1865. 6 



it is so arranged that there are only four multii^lications of the con- 

 stants to be performed by numbers which are small and convenient for 

 that purpose, in order to obtain any one of the interpolated values of 

 F,^ whereas in the preceding form there are for the most part ten, the 

 number being equal to the sum of the exponents of x in (3). This 

 formula is accurate for all cases in which it is necessary to use eight 

 orders of differences, and in all ordinary cases in which it is necessary 

 to use only four or five orders of differences it is quite simple. In 

 general, it is only necessary to compute the three constants A., B, and 

 (7, using ^ B for B', for even then the maximum possible error is only 

 of the order y^jxy A^. This formula is applicable in all cases in which 

 the number of interpolations does not exceed twelve. If we wish to 

 interpolate to twelfths, « in the expression of the preceding constants 

 A, B, C, &c. must be put equal to 12 ; if to tenths, equal to 10 ; and 

 so for any other number. If we interpolate to twelfths, we must use 

 (6) from F_Q to -^+6 5 if to tenths, from F_5 to i^+g ; and so on. In 

 this way we get the middle interpolated number from two sets of con- 

 stants ; first, by going forward from Fq, and secondly, by going back 

 from F^, which is the F^ in the formula belonging to the next set of 

 constants. This furnishes a very good check for the accuracy of the 

 interpolations in addition to that of the regularity of the differences. 

 In cases in which a is less than twelve, the formula from F_ ^tx) F^^ 

 would give several of the middle interpolations in duplicate, but it is 

 unnecessary to take it so as to have mox'e than one. 



In interpolating to sixths, it is evident that, instead of putting « 

 equal to six, and using the formula from F_s to F^^, we can put it 

 equal to twelve, and use only the functions of F with even subscript 

 numbers from F_q to F^q. By so doing, we have the advantage of 

 using the functions i^qi 4 and F:f 2 » which are very simple, since three 

 of numerical coefficients are ciphers in Fzf^, and two of them in Fz^2- 

 By putting co = 10, and using these same functions of x, we have a 

 very convenient formula for interpolating to fifths ; but it does not give 

 any one of the interpolations in duplicate as a check, which, perhaps, 

 is always unnecessary where the number of interpolations is so small, 

 the regularity of the differences being a sufficient check. Also, in 

 interpolating to fourths, instead of putting &> = 4 and using F^: 2 and 

 i^'qii, we can put w = 12, and use i^'qio and i^'ips; but it is much 

 better to put w :=: 8, and use -^q:4 and Fz^^j which, for reasons al- 

 ready stated, are much more simple. In interpolating to thirds we can 



