2. We may remark, however, that the construction of the right Formula for any such case, and the determination of the Coefficients of such formula, which we have spoken of as two separate steps, are in practice almost necessarily simultaneous; for the near coincidence of the results of the theoretical rule with the observed facts confirms at the same time the Formula and its Coefficients. In this case also, the mode of arriving at truth is to try various hypotheses;—to modify the hypotheses so as to approximate to the facts, and to multiply the facts so as to test the hypotheses.
The Independent Variable, and the Formula which we would try, being once selected, mathematicians have devised certain special and technical processes by which the value of the coefficients may be determined. These we shall treat of in the [next] Chapter; but in the mean time we may note, in a more general manner, the mode in which, in physical researches, the proper formula may be obtained.
3. A person somewhat versed in mathematics, having before him a series of numbers, will generally be able to devise a formula which approaches near to those numbers. If, for instance, the series is constantly progressive, he will be able to see whether it more nearly resembles an arithmetical or a geometrical progression. For example, MM. Dulong and Petit, in their investigation of the law of cooling of bodies, obtained the following series of measures. A thermometer, made hot, was placed in an enclosure of which the temperature was 0 degrees, and the rapidity of 197 cooling of the thermometer was noted for many temperatures. It was found that
| For the temperature | 240 | the rapidity of cooling was | 10·69 |
| 〃 | 220 | 〃 | 8·81 |
| 〃 | 200 | 〃 | 7·40 |
| 〃 | 180 | 〃 | 6·10 |
| 〃 | 160 | 〃 | 4·89 |
| 〃 | 140 | 〃 | 3·88 |
and so on. Now this series of numbers manifestly increases with greater rapidity as we proceed from the lower to the higher parts of the scale. The numbers do not, however, form a geometrical series, as we may easily ascertain. But if we were to take the differences of the successive terms we should find them to be—
1·88, 1·41, 1·30, 1·21, 1·01, &c.
and these numbers are very nearly the terms of a geometric series. For if we divide each term by the succeeding one, we find these numbers,
1·33, 1·09, 1·07, 1·20, 1·27,
in which there does not appear to be any constant tendency to diminish or increase. And we shall find that a geometrical series in which the ratio is 1·165, may be made to approach very near to this series, the deviations from it being only such as may be accounted for by conceiving them as errours of observation. In this manner a certain formula[26] is obtained, giving results 198 which very nearly coincide with the observed facts, as may be seen in the margin.
[26] The formula is v = 2·037(at − 1) where v is the velocity of cooling, t the temperature of the thermometer expressed in degrees, and a is the quantity, 1·0077.
The degree of coincidence is as follows:—