Empirical Mathematical Laws.
It is important to acquire a clear comprehension of the part which is played in scientific investigation by empirical formulæ and laws. If we have a table containing certain values of a variable and the corresponding values of the variant, there are mathematical processes by which we can infallibly discover a mathematical formula yielding numbers in more or less exact agreement with the table. We may generally assume that the quantities will approximately conform to a law of the form
y = A + Bx + Cx2,
in which x is the variable and y the variant. We can then select from the table three values of y, and the corresponding values of x; inserting them in the equation, we obtain three equations by the solution of which we gain the values of A, B, and C. It will be found as a 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; Regnault found that the results of his elaborate inquiry into the latent heat of steam at different pressures were represented with sufficient accuracy by the empirical formula
λ = 606·5 + 0·305 t,
in which λ is the total heat of the steam, and t the temperature.[400] 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
δt = at + bt2 + ct3,
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.[401] Theoretically speaking, this process of empirical representation 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 represented with any required degree of accuracy by formulæ involving the sines and cosines of angles and their multiples. The form of any tidal or other wave may thus be expressed, as Sir G. B. Airy has explained.[402] Almost all the phenomena registered by meteorologists are periodic in character, and when freed from disturbing causes may be embodied in empirical formulæ. 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,[403] I need not further enter into them.