122 THE PRINCIPLES OF SCIENCE. 



ing 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 must 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 readily determined from two results. 

 Thus, if the two values for x, 7 and n, give the values 

 for 2/, 35 and 53, the solution of two simple equations 

 gives 2/ = 4 < 5 xx + 3*5 as the equation, and for any other 

 value of x, for instance 10, we get a value of y, 48-5. 

 When we take an exactly intermediate value of x, namely 

 9, this process yields a simple mean result, namely 44. 

 Three experimental results being given, we may assume 

 that they fall upon a portion of a parabola, and simple 

 algebraic calculation readily 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 Ksemtz' Meteorology?. 



When we have, either directly by experiment or by 

 the use of a curve, a series of values of the variant for 

 exactly equidistant values of the variable, it is often very 

 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 em- 

 pirical formula ; if the nth differences are zero, then the 

 formula will contain only the first n-i powers of the 

 variable. Indeed we may sometimes obtain by the Cal- 



P ' Cours complet cle Met^orologie/ traduit par Martina, Note A, du 

 Traducteur, p. 449. 



