Page:Scientific Memoirs, Vol. 2 (1841).djvu/454

442 disproportion between the members of a differential equation, belonging nevertheless necessarily to one another, is too remarkable not to attract the attention of those to whom such inquiries are of any value; an attempt therefore to add something to the explanation of this ænigma will be the more proper in this place, as we gain the advantage of rendering thereby the subsequent considerations more simple and concise. We shall merely take as an instance the propagation of electricity, and it will not be difficult to transfer the obtained results to any other similar subject, as we shall subsequently have occasion to demonstrate in another example.

6. Above all, it is requisite that the term goodness of conduction be accurately defined. But we express the energy of conduction between two places by a magnitude which, under otherwise similar circumstances, is proportional to the quantity carried over in a certain time from one place to the other multiplied by the distance of the two places from each other. If the two places are extended, then we have to understand by their distance the straight line connecting the centres of the dimensions of the two places. If we transfer this idea to two electric elements, $$E$$ and $$E'$$, and call $$s$$ the mutual distance of their centres, $$\varsigma$$ the quantity of electricity, which under accurately determined and invariable circumstances is carried over from one element to the other, and $$\chi$$ the conductibility between them, We will now endeavour to determine more precisely the quantity of electricity denoted by $$\varsigma$$. According to § 4 the quantity of electricity, which is transferred in an exceedingly short time from one element to the other, is, the distance being invariable, in general proportional to the difference between the electroscopic forces, the duration, and the size of each of the two elements. If therefore we designate the electroscopic forces of the two elements $$E$$ and $$E'$$ by $$u$$ and $u'$, and the space they occupy by $$m$$ and $$m'$$, we obtain for the quantity of electricity carried over from $$E'$$ to $$E$$ in the element of time $$d\, t$$ the following expression: where $$\alpha$$ represents a coefficient depending in some way on the distance $$s$$. This quantity changes every moment if $$u' - u$$ is variable; but if we suppose that the forces $$u'$$ and $$u$$ remain