Page:Scientific Memoirs, Vol. 2 (1841).djvu/423

$L\,K'$ are directly proportional to the lengths of the parts $A\,B$, $B\,C$ and $C\,D$, and inversely proportional to the products of the conductibility and section of the same part, consequently the relations of the lines $G\,F'$, $I\,H'$ and $L\,K'$ to each other are given. Further, that $G\,F'+I\,H'+L\,K'=G\,H-K\,I+(D\,L-A\,F=L\,M)$ is also known, as the tensions represented by $G\,H$, $K\,I$ and $D\,L-A\,F$ are given. From the given relations of the lines $G\,F',I\,H',L\,K'$ and their known sum, these lines may now be found individually; the figure $F\,G\,H\,I\,K\,L$ is evidently then entirely determined. But the position of this figure with respect to the line $A\,D$ remains from its very nature still undecided.

If we recollect, that proceeding in the same direction $A\,D$, the tensions represented by $G\,H$ and $D\,L-A\,F$ or $L\,M$ indicate a sudden sinking of the electric force at the respective places of excitation, that represented by $I\,K$ on the contrary a sudden rise of the force; and that tensions of the first kind are regarded and treated as positive quantities, while tensions of the latter kind are considered as negative quantities, we find the above example lead us to the following generally valid rule: If we divide the sum of all the tensions of the ring composed of several parts into the same number of portions which are directly proportional to the lengths of the parts and inversely proportional to the products of their conductibilities and their sections, these portions will give in succession the amount of gradation which must be assigned to the straight lines belonging to the single parts and representing the separation of the electricity; at the same time the positive sum of all the tensions indicates a general rise, on the contrary the negative sum of all the tensions a general depression of those lines.

I will now proceed to the determination of the electric force at any given position in every galvanic circuit, and here again I shall lay down as basis fig. 3. For this purpose let $a$, $a'$, $a''$ indicate the tensions existing at $B$, $C$, and between $A$ and $D$, so that in this case also $a$ and $a''$ represent additive, $a'$ on the contrary a subtractive line, and $\lambda$, $\lambda '$, $\lambda ''$ any lines which are directly as the lengths of the parts $A\,B$, $B\,C$, and $C\,D$, and inversely as the products of the conductibilities and sections of the same parts; further, let

$a+a'+a''=A$

2 e 2