488 THE PRINCIPLES OF SCIENCE. [CHAP 



perature. 1 In other cases it may be requisite to include 

 the third power of the variable. Thus physicists assume 

 the law of the dilatation of liquids to be of the form 



8 t = a t + b t z + c P, 



and they calculate from results of observation the values 

 of the three constants a, b, c, which are usually small 

 quantities not exceeding one-hundredth part of a unit, 

 but requiring to be determined with great accuracy. 2 

 Theoretically speaking, this process of empirical repre- 

 sentation might be applied with any degree of accuracy ; 

 we might include still higher powers in the formula, and 

 with sufficient labour obtain the values of the constants, 

 by using an equal number of experimental results. The 

 method of least squares may also be employed to obtain 

 the most probable values of the constants. 



In a similar manner all periodic variations may be repre- 

 sented with any required degree of accuracy by formula; 

 involving the sines and cosines of angles and their mul- 

 tiples. The form of any tidal or other wave may thus be 

 expressed, as Sir G. B. Airy has explained. 3 Almost all 

 the phenomena registered by meteorologists are periodic 

 in character, and when freed from disturbing causes may 

 be embodied in empirical formulae. Bessel has given a 

 rule by which from any regular series of observations we 

 may, on the principle of the method of least squares, 

 calculate out with a moderate amount of labour a formula 

 expressing the variation of the quantity observed, in the 

 most probable manner. In meteorology three or four 

 terms are usually sufficient for representing any periodic 

 phenomenon, but the calculation might be carried to any 

 higher degree of accuracy. As the details of the process 

 have been described by Herschel in his treatise on 

 Meteorology, 4 I need not further enter into them. 



The reader might be tempted to think that in these 

 processes of calculation we have an infallible method of 

 discovering inductive laws, and that my previous state- 

 ments (Chap. VII.) as to the purely tentative and inverse 

 character of the inductive process are negatived. Were 



1 Chemical Reports and Memoirs, Cavendish Society, p. 294. 

 a Jamiii, Cours de Physique, vol. ii. p. 38. 



3 On Tides-and Waves, Encyclopaedia Metropolitan*, p. 366*. 



4 Encyclopedia Britannica, art. Meteorology. Reprint, f 152156. 



