Page:Scientific Memoirs, Vol. 2 (1841).djvu/476

 464 (L) and (L′), may be comprised in the following general statement, which is true when the circuit consists of any number of parts whatever.

The electroscopic force of any place of a galvanic circuit, composed of any number of parts, is found by dividing the sum of all its tensions by its reduced length, multiplying this quotient by the reduced length of the part of the circuit comprised by the abscissa, and subtracting from this product the sum of all the tensions abruptly passed over by the abscissa; lastly, by varying the value thus obtained by a constant magnitude to be determined elsewhere.

If, therefore, we designate by $$A$$ the sum of all the tensions of the circuit, by $$L$$ its entire reduced length, by $$y$$ the reduced length of the part which the abscissa passes through, and by $$O$$ the sum of all the tensions to the points to which the abscissa corresponds, lastly, by $$u$$, the electroscopic force of any place in any part of the circuit, then where $$c$$ represents a constant, but yet undetermined magnitude.

Thus transformed, this exceedingly simple expression for the electroscopic force of any circuit will allow us hereafter to combine generality with conciseness, for which purpose we will, moreover, indicate by $$y$$ the reduced abscissa. This form of the equation has besides the peculiar advantage that, without anything further, it is even applicable when in any part of the circuit the tensions and conductibilities constantly vary; for in this case we should merely have to take, instead of the sums, the corresponding integrals, and to define their limits according as the nature of the expression required. Since $$O$$ does not change its value within the entire extent of the same homogeneous part of the circuit, and $$y$$ constantly varies to the same amount on like portions of this extent, the following properties, already demonstrated less generally with respect to the simple circuit, evidently apply to every galvanic circuit, and in these is expressed the main character of galvanic circuits:—

a. The electric force of each homogeneous portion of the circuit varies throughout its entire length constantly, and on like extents always to the same amount; but where it ceases and another commences, it suddenly