Regarded in this light, interpolation is in reality an indeterminate problem. From given values of a function it is impossible to determine that function; for we can invent an infinite number of functions which will give those values if we are not restricted by any conditions, just as through a given series of points we can draw an infinite number of curves, if we may diverge between or beyond the points into bends and cusps as we think fit.13 In interpolation we must in fact be guided more or less by à priori considerations; we must know, for instance, whether or not periodical fluctuations are to be expected. Supposing that the phenomenon is non-periodic, we proceed to assume that the function can be expressed in a limited series of the powers of the variable. The number of powers which can be included depends upon the number of experimental results available, and must be at least one less than this number. By processes of calculation, which have been already alluded to in the section on empirical formulæ, we then calculate the coefficients of the powers, and obtain an empirical formula which will give the required intermediate results. In reality, then, we return to the methods treated under the head of approximation and empirical formulæ; and interpolation, as commonly understood, consists in assuming that a curve of simple character is to pass through certain determined points. If we have, for instance, two experimental results, and only two, we assume that the curve is a straight line; for the parabolas which can be passed through two points are infinitely various in magnitude, and quite indeterminate. One straight line alone can pass through two points, and it will have an equation of the form, y = mx + n, the constant quantities of which can be determined from two results. Thus, if the two values for x, 7 and 11, give the values for y, 35 and 53, the solution of two equations gives y = 4·5 × x + 3·5 as the equation, and for any other value of x, for instance 10, we get a value of y, that is 48·5. When we take a mean value of x, namely 9, this process yields a simple mean result, namely 44. Three experimental results being given, we assume that they fall upon a portion of a parabola and algebraic calculation gives the position of any intermediate point upon the parabola. Concerning the process of interpolation as practised in the science of meteorology the reader will find some directions in the French edition of Kaëmtz’s Meteorology.14
When we have, either by direct experiment or by the use of a curve, a series of values of the variant for equidistant values of the variable, it is instructive to take the differences between each value of the variant and the next, and then the differences between those differences, and so on. If any series of differences approaches closely to zero it is an indication that the numbers may be correctly represented by a finite empirical formula; if the nth differences are zero, then the formula will contain only the first n - 1 powers of the variable. Indeed we may sometimes obtain by the calculus of differences a correct empirical formula; for if p be the first term of the series of values, and Δp, Δ2p, Δ3p, be the first number in each column of differences, then the mth term of the series of values will be
A closely equivalent but more practicable formula for interpolation by differences, as devised by Lagrange, will be found in Thomson and Tait’s Elements of Natural Philosophy, p. 115.
If no column of differences shows any tendency to become zero throughout, it is an indication that the law is of a more complicated, for instance of an exponential character, so that it requires different treatment. Dr. J. Hopkinson has suggested a method of arithmetical interpolation,15 which is intended to avoid much that is arbitrary in the graphical method. His process will yield the same results in all hands.