364 . EEPOKT— 1880. 



wliicli, if a fractional value be given to n, becomes a formula of inter- 

 polation. 



Section 2.~r-Zagi*anffe's method, 



Lagrange's tbeorem of interpolation, althoagh identical in its results 

 with Newton's, is in a form analogous rather to the case of ordinates not 

 differenced, than to the use of differences, which is the analogy followed 

 by Newton. As is well known, it depends upon 



X — (^ ~ ^i) ( ^ ~ "2) (jg — g) 



" («o - «i) («o - «2) (^0 — ««) 



becoming unity for x = Uq, and vanishing when x is equated to any other 

 of the quantities aj Oj ^3 . . . . «„. Then interchanging Oq with a^ a^ &c. 

 in succession, so as to obtain a series of quantities Xg Xj Xj . . . X^, the 

 formula of interpolation is 



U^ = ^qUq + X, ■zt, + X2 «2 + X„«„ 



The proof consists in the observation that since X^ = 1, and all the other 

 Xs vanish when a' =: a,, it leaves u^ = ?tj. as it ought to do. Although this 

 formula usually bears Lagrange's name, it is said to be really due to 

 Buler. The chief advantage of it over Newton's method is, that the co- 

 eflScients are in a form adapted to logarithmic computation. 



' The differential coefficients of u^ with regard to x may be obtained by 

 actually differentiating the resulting formula, either in Newton's or in 

 Lagrange's method. But they may also be obtained, as in the common 

 theory of equations, by an obvious application of Maclaurin's theorem.* 



The quadrature can also be effected, as was pointed out by Newton, 

 by the application of the ordinary methods for the measurement of an 

 area of parabolic form. It does not appear that the details have as yet 

 received much attention, but it is evident that the final formula of either 

 Newton or Lagrange may be integrated between limits, by direct integ- 

 ration, and that the result of the integration may be expressed in terms of 

 the intervals and ordinates, either by direct substitution, or by the help of 

 indeterminate coefficients. Judging from the form in which Laplace has 

 placed the differential coefficients, it is probable that some interesting 

 results would be obtained by such an investigation ; but these results 

 would be likely to be more interesting as a matter of form, than useful as 

 a matter of arithmetic. 



Section 3. — Gauss's Method of Quadratwre. 



This method is of high analytical interest, as connecting the theory of 

 interpolation with what are known indifferently as Spherical Harmonics, 

 or as Lagrange's or Laplace's functions. It is also useful and in- 

 teresting from the point of view of their inventor, in making the best 

 possible use of a small number of ordinates, when the function subjected 

 to this mode'" oT quadrature is capable of exact expression in a parabolic 

 form. If this last condition be not assured, its arithmetical value also 

 becomes in a like degree uncertain. It has assuredly never been proved that 

 there is any general advantage in adapting the rules, indicated by the 

 parabolic theory as the most exact, to cases which are not known to fall 

 strictly within that theory. 



* See Boole's Finite Differences, 2nd edit. p. 41. 



