420 Transactions. — Chemistry and Physics. 



Art. LIV. — On the Use of the Standard Functions in In- 

 terpolation. 



By E. G. Brown. 



[Read before the Wellington Philosophical Society, 16th December, 



1902.'] 



There is a parallel between the expression of functions by 

 Taylor's Series expansion formulae, and of tables of numbers 

 by interpolation formulae, depending on finite differences. 

 In both cases there is usually a "remainder" which is 

 neglected as being immaterial. This parallel is seen to be 

 very close if closely examined ; but we need here only remark 

 that, just as we have found it possible to reduce the degree of 

 many Taylor's Series expansions by means of the standard 

 functions,* so also it is possible to reduce the number of dif- 

 ferences required to interpolate in a normal table of figures. 

 This, again, is a matter into which we need not go ; defining 

 the problem in hand as follows : Given a table of numbers, 

 and having differenced them, what is the best formula — or, in 

 other words, the formula of the least number of terms — that 

 can be found to perform the interpolation by finite differences 

 between two consecutive values of the numbers, the argu- 

 ments being, of course, at equal intervals? 



There are a number of ways of deducing formulae of inter- 

 polation, the chief of which result in what are called! New- 

 ton's, Lagrange's, and Bessel's methods, the latter being 

 generally employed when more than two differences are sig- 

 nificant. Each of these methods gives rise to a series of 

 functions analogous to the standard functions, and in cases 

 identical with them. Thus, in all cases the first function is 

 the linear (L), and the second the standard parabola, x(l— x). 

 In Newton's and Lagrange's methods the other formulae are 

 not the same, but diverge widely from the standard form — 

 less widely in Lagrange's method than in Newton's. Bessel's 

 method, however, brings in the standard cubic and then 

 diverges from the standard form. 



* Trans. N.Z. Inst., 1901, p. 519 et seq. 



t This nomenclature is drawn frorn that of F. G. Gauss (5-fig. Log. 

 and Trig. Tables, larger edition, 1900, p. 150), who gives Lagrange's and 

 Newton's methods for unequal increments, and then says, " Bei gleichen 

 Intervallen gehen diese Formeln in die Formeln (1) und (2) liber," i.e., 

 into the methods referred to (Bessel's being, of course, a modification of 

 Lagrange's method). 



