ON QUADEATURES AND INTERPOLATION. 333 



to ?^. "When the values are equidistant, r h ov rAx must be used instead 

 of x^. 



There is usually an advantage, both in the symmetry of the formula 

 and in the probability of an accurate result, in taking ordinates on both 

 sides of the origin of interpolation. On any reasonable hypothesis, a mean 

 result is generally better than one near an extreme, and this remark is 

 verified by the greater tendency to convergence of the formulae when the 

 interpolated value lies near the origin of interpolation than when it lies 

 away from it. As a general rule, where extreme accuracy is required, it 

 should not lie farther off than half the equidistant interval! ' There are 

 then two kinds of symmetry to consider : symmetry to a central ordinate, 

 involving an odd number of ordinates and an even number of intervals ; 

 and symmetry to an interval, involving an even number of ordinates 

 and an odd number of intervals. 



Taking the former case, of symmetry to an ordinate, we may write for 

 the ordinates 



and the general formula of interpolation will be given by 



«^ = C-„ «_„ + C_i ti_i + Co ?(o + • • • . + c„ M„, 



* 



where the coefficients c are functions of x, determined by the condition 

 that x^rh shall make c,.=l, and all the others vanish. The simplest way 

 in which thi^ can be done rationally and integrally is by writing 



_ x ill' - x"^) (4 h^-x"^) (Tt^ 7^2 - a;2) 



{ 



X X — h X + h 



X — nh X + nh i 



where v,. is the coefficient of a;'" in the binomial expansion (1 + xY". 

 This gives for 



n = 0, %. = «o (3'S it ought) 



, X (/i-^ — X^) r 2 Up _ t(i «_!_ \ 



n - i, %- ^^-^ <^^ ^_j^ x+hi 

 _n X (li^ — x^) (4 h^ — x^) ^ 6Uq 4Mj _ 4 ?6_i ' 



U2 _ M_2 1 



X — 2h x + 2hi 



In the second case, of symmetry to an interval, let any even number 

 of ordinates be 



N,M, ....B,A,Ai,B„ M„N, 



and let the variable x = ^ z, or x be the independent variable measured 

 from the middle point of the middle interval A Ai = /t = 2 /c. Then if v'' 



* Boole writes this in a. slightly different form ; see his Finite Differences 

 2nd edition, p. 50. The formulse themselves are due to Newton ; see his Metkodits 

 Bifferentialis (London, 1711, published by W. Jones as part of the Analysis per 

 qvantitatum series, Jiitxiones, &c.) pp. 93-101. They are also to be fo^ind in vol. i. 

 of Horsley's edition of Newton's works. 



