Page:Scientific Memoirs, Vol. 2 (1841).djvu/469

$fx_{2}+c.$

If we now call the given tension or difference of the electroscopic force $a$, we have

$a=\pm (x_{1}-x_{2}).$

But $x_{1}-x_{2}$ evidently represents the entire, positive or negative, length of the prismatic conductor; if we designate this by $l$, we obtain accordingly

$a=\pm fl,$

whence the constant $f$ may be determined. If we now introduce the value of the constant thus found into the equation (c), it is converted into

$u=\pm {\frac {a}{l}}x+c,$

so that only the constant $c$ remains to be determined. We may consider the ambiguity of the sign $\pm$ to be owing to the tension $a$, by ascribing to it a positive value when the extremity of the conductor, belonging to the greater abscissa, possesses the greatest electroscopic force, and when the contrary a negative. Under this supposition is then generally

$u={\frac {a}{l}}x+c.$

(d)

The constant $c$ remains in general wholly undetermined, which admits of our allowing the diffusion of the electricity in the conductor to vary arbitrarily, by external influences, in such manner that it occupies the entire conductor everywhere uniformly.

Among the various considerations respecting this constant, there is one of especial importance to the galvanic circuit, I mean that which supposes the circuit to be connected at some one place with a perfect conductor, so that the electroscopic force has to be regarded as constantly destroyed at this place. If we call the abscissa belonging to this place $\lambda$, then according to the equation (d)

$0={\frac {a}{l}}\lambda +c.$

By determining from this the constant $c$, and placing its value in the same equation (d), we obtain

$u={\frac {a}{l}}(x-\lambda ),$

from which the electroscopic force of a galvanic circuit of the