QUANTITATIVE INDUCTION. Ill 



general rule that the formula thus obtained yields the 

 other numbers of the table to a considerable degree of 

 approximation. 



In many cases even the second power of the variable 

 will be unnecessary ; thus Regnault found that the results 

 of his elaborate inquiry into the latent heat of steam at 

 different pressures were represented with sufficient ac- 

 curacy by the empirical formula 



X = 606-5 4- 0-305 t, 



in which X is the total heat of the steam, and t the tem- 

 perature 6 . 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 



i t = a t 4- b t 2 + c t 3 , 



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 d . 

 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. 



In a similar manner all periodic variations may be repre- 

 sented with any required degree of accuracy by formulas 

 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 6 . Almost all 

 the phenomena registered by meteorologists are periodic 

 in character, and when freed from disturbing causes may 

 be embodied in empirical formulas. Bessel has given a 





c Chemical Reports and Memoirs,' Cavendish Society, p. 294. 



d Jamin, ' Cours de Physique,' vol. ii. p. 38. 



e ' On Tides and Waves/ Encyclopaedia Metropolitan^ p. 366*. 



