OF ARTS AND SCIENCES : SEPTEMBER 12, 1865. 3 



(3.) F x = F + B x x + B 2 x? + B 3 x s + B iX \ 



in which 



A = ^ (a/ - i a; + gv-^f of A) a,' - T |* a/ ), 



and in which the maximum possible error E of any interpolated num- 

 ber arising from neglected terms is only 



(5.) E ■=■ ± (u7jtr?T A, FS'A^TJ A T«TffV(J7J A, ). 



The constants B l , B. 2 , &c. are so determined as to make the two pre- 

 ceding expressions of F x correspond for the four values of x = =p J- <o, 

 and x = =F ^ © ; so that, corresponding at equal intervals of ^a>, 

 they cannot differ much for any intermediate value of x, as is shown 

 by (5). The advantage of this last formula over the preceding is, that 

 with only four terms containing the variable you have nearly all the 

 accuracy of seven or eight terms of the former. But it can only be 

 used within the limit of ^ m before and after F ; so that in interpolat- 

 ing it requires the constants B x , B 2 , &c. to be computed for every 

 interval or given value of F x . 



As the unit of x is arbitrary, when the interpolations are made at 

 equal intervals it can be taken equal to one of the equal parts of x 

 corresponding to the interpolated values of F x , and then a will repre- 

 sent the number of interpolated intervals contained in one of the origi- 

 nal intervals ; that is, w — 1 will represent the number of interpola- 

 tions in each original interval. In this case the value of x used in 

 interpolating is always one of the numbers ^ 1, ^ 2, ^ 3, &c. ; and 

 if the number of interpolations to each original interval is not too 

 great, the different terms in the expression of F x are readily obtained 

 after the constants B 1 , B. 2 , &c. have been computed. 



.For all cases in which the value of a: does not exceed 6, that is, in 

 which a> does not exceed 12, the preceding formula may be put into a 

 form still more convenient for interpolating. The preceding expression 

 of F x for all values of x from — G to -)- G gives 



