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 -f 0-305 t, 



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

 perature . In other cases it maybe requisite to include 

 the third power of the variable. Thus physicists assume 

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



&amp;lt;J t = a t + It 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 &amp;lt;l . 

 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 formulae 

 involving the sines and cosines of angles and their mul 

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

 expressed, as Sir G. B. Airy has explained . 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 Metropolitans, p. 366*. 



