THE PRINCIPLES OF SCIENCE. [CHAP. 



which might have been expected independently of a the- 

 oretical investigation. 



Eegarded in this light, interpolation is in reality an inde- 

 terminate 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. 1 In inter- 

 polation we must in fact be guided more or less by a 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 formulae, 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 formulas ; 

 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 mag- 

 nitude, 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#, 7 and II, give the values for y, 35 and 53, 

 the solution of two equations gives y = 4*5 x x f 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 



1 Herschel : Lacroix' Differential Calculus, p. 551. 



