in comparison with their relative distances, a function, to be determined separately for each given case from their dimensions and their mean distance, must be substituted for the product of the magnitudes of the two elements, and which we will designate where it is employed by .
5. Hitherto we have taken no notice of the influence of the mutual distance of the elements between which an equalization of their electric state takes place, because as yet we have only considered such elements as always retained the same relative distance. But now the question arises, whether this exchange is directly effected only between adjacent elements, or if it extends to others more distant, and how on the one or the other supposition is its magnitude modified by the distance? Following the example of Laplace, it is customary in cases where molecular actions at the least distance come into play, to employ a particular mode of representation, according to which a direct mutual action between two elements separated by others, still occurs at finite distances, which action, however, decreases so rapidly, that even at any perceptible distance, be it ever so minute, it has to be considered as perfectly evanescent. Laplace was led to this hypothesis, because the supposition that the direct action only extended to the next element produced equations, the individual members of which were not of the same dimension relatively to the differentials of the variable quantities[1],—a non-uniformity which is opposed to the spirit of the differential calculus. This apparent unavoidable
- ↑ Poisson, in his Mémoire sur la Distribution de la Chaleur, Journ. de l'Ecole Polytechn. cah. xix. expresses himself on this subject thus:—
"If a bar be divided, by sections perpendicular to the axis, into an infinite number of infinitely small elements, and if we consider the mutual action of three consecutive elements, that is to say, the quantity of heat that the intermediate element at each instant communicates to and abstracts from the two others, in proportion to the positive or negative excess of its temperature over that of each of them, we may thence easily determine the augmentation of temperature of this element during an infinitely small instant; assuming therefore this quantity equal to the differential of its temperature taken with respect to the time, the equation of the propagation of heat according to the length of the bar is formed; but on examining the question more attentively, it is seen without difficulty that this equation would be founded on the comparison of two infinitely small non-homogeneous quantities, or of different orders, which would be contrary to the first principles of the differential calculus. This difficulty can only be made to disappear by supposing, as M. Laplace first remarked, (Mémoires de la 1re classe de l'Institut, année 1809,) that the action of each element of the bar extends itself beyond the contact, and that it exerts itself on all the elements contained within a finite space, as small as we please."
