Page:Scientific Memoirs, Vol. 2 (1841).djvu/494

But how must, in each separate case, a given galvanic apparatus be arranged so as to produce the greatest effect? Let us suppose, in solving this problem, that we possess a certain magnitude of surface; for instance, of copper and zinc, from which we can form, according to pleasure, a single large pair of plates, or any number of smaller pairs, but in the same proportion, and, moreover, that the liquid between the two metals is constantly the same, and of the same length, which latter supposition means nothing more than that the two metals between which the liquid is confined retain, under all circumstances, the same distance from each other.

Let $\Lambda$ be the reduced length of the body upon which the electric current is to act, $L$ the reduced length of the apparatus when formed into a simple circuit, and $A$ its tension; then, when it is altered into a voltaic combination of $x$ elements, its present tension will be $xA$, and the reduced length of each of its present elements $xL$, accordingly the reduced length of all the $x$ elements $x^{2}L$, consequently the magnitude of the action of the voltaic combination of $x$ elements is

${\frac {xA}{x^{2}L+\Lambda }}.$

This expression acquires its greatest value ${\frac {A}{2{\sqrt {\Lambda \cdot L}}}}$ when $x={\sqrt {\frac {\Lambda }{L}}}$. We hence see that the apparatus in form of a simple circuit is most advantageous, so long as $\Lambda$ is not greater than $L$; on the contrary, the voltaic combination is most useful when $\Lambda$ is greater than $L$, and indeed it is best constructed of two elements when $\Lambda$ is four times greater than $L$, of three elements when $\Lambda$ is nine times greater than $L$, and so forth.

27. The circumstance that the current always remains the same at all places, affords us the means of multiplying its external action, as in the case when the current influences the magnetic needle. We will, for perspicuity, suppose that, in order to test the action of the current on the magnetic needle, each time a part of the circuit be formed into a circle of a given radius, and so placed in the magnetic meridian that its centre coincides with the point of rotation of the needle. Several such distinct coils, formed of the circuit in exactly the same manner, will, taken singly, produce, on account of the equality of the current in each, equally powerful effects on the magnetic