xxii.] QUANTITATIVE INDUCTION. 497 



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 Kaemtz's Meteorology. 1 



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 i 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 Ap, A 2 p> A 3 ??, be the first num- 

 ber in each column of differences, then the mth term of 

 the series of values will be 



m i 

 p+m&p+m 



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 inter- 

 polation, 2 which is intended to avoid much that is 

 arbitrary in the graphical method. His process will yield 

 the same results in all hands. 



So far as we can infer the results likely to be obtained 

 by variations beyond the limits of experiment, we must 



1 Cows complet de Meteorologie, Note A, p. 449. 



2 On the Calculation of Empirical Formula. The Messenger of 

 Mathematics, New Series, No. 17, 1872. 



K K 



