The physical law expressed by the formula just spoken of is this:--that when a body is cooling in an empty inclosure which is kept at a constant temperature, the quickness of the cooling, for excesses of temperature in arithmetical progression, increases as the terms of a geometrical progression, diminished by a constant number. 4. In the actual investigation of Dulong and Petit, however, the formula was not obtained in precisely the manner just described. For the quickness of cooling depends upon two elements, the temperature of the hot body and the temperature of the inclosure; not merely upon the _excess_ of one of these over the other. And it was found most convenient, first, to make such experiments as should exhibit the dependence of the velocity of cooling upon the temperature of the enclosure; which dependence is contained in the following law:--The quickness of cooling of a thermometer in vacuo for a constant excess of temperature, increases in geometric progression, when the temperature of the inclosure increases in arithmetic progression. From this law the preceding one follows by necessary consequence[27\3]. [Note 27\3: For if _θ_ be the temperature of the inclosure, and _t_ the excess of temperature of the hot body, it appears, by this law, that the radiation of heat is as _a^θ_. And hence the quickness of cooling, which is as the excess of radiation, is as _a^θ+t_ - _a^θ_; that is, as _a^θ_(_a^t_ - 1) which agrees with the formula given in the last note. The whole of this series of researches of Dulong and Petit is full of the most beautiful and instructive artifices for the construction of the proper formulæ in physical research.] This example may serve to show the nature of the artifices which may be used for the construction of formulæ, when we have a constantly progressive series of numbers to represent. We must not only endeavour by trial to contrive a formula which will answer the conditions, but we must vary our experiments so as to determine, first one factor or portion of the formula, and then the other; and we must use the most {199} probable hypothesis as means of suggestion for our formulæ.