Page:Scientific Memoirs, Vol. 2 (1841).djvu/420

 408 Consequently if that which has been stated with respect to the circuit of two members is to acquire a sense no longer subject to any arbitrary explanation, this uncertainty must be removed. The first fundamental law effects this in the following way:—For since the state of the ring alone, independent of the time, is regarded, each section must, as has already been stated, receive in every moment the same quantity of electricity from one side as it gives off to the other. This condition occasions upon such portions of the ring as have perfectly the same constitution at their various points, the constant and uniform change in the separation which is represented in the first figure by the straight line $$F\, G$$, and in the second by the straight lines $$F\, G$$ and $$H\, I$$. But when the geometrical or the physical nature of the ring changes in passing from one of its component parts to another, the reason of this constancy and uniformity no longer obtains; consequently the manner in which the several straight lines are combined into a complete figure must first be deduced from other considerations. To facilitate the object, I will separately consider the geometrical and physical difference of the single parts, each independently.

Let us first suppose that every section of the part $$B\, C$$ is $$m$$ times smaller than in the part $$A\, B$$, while both parts are composed of the same substance; the electric state of the ring, which is independent of time, and which requires that everywhere throughout the entire ring just as much electricity be received on one side as is given off from the other, can evidently only exist under the condition that the electric transition from one particle to the other in the same time within the portion $$B\, C$$ is $$m$$ times greater than in the portion $$A\, B$$; because it is only in this manner that the action in both parts can maintain equilibrium. But in order to produce this $$m$$ times greater transition of the electricity from element to element, the electrical difference of element to element within the portion $$B\, C$$ must, according to the first fundamental position, be $$m$$ times greater in the portion $$A\, B$$; or when this determination is transferred to the figure, the line $$H\, I$$ must sink $$m$$ times more on equal portions, or have an $$m$$ times greater "dip" than the line $$F\, G$$. By the expression "dip" (Gefälle), is to be understood the difference of such ordinates which belong to two places distant one unit of length from each other. From this consideration results the following rule: The dips of