How did electrical conductivity provide evidence for the dissociation of acids in water?

Wilhelm Ostwald utilized electrical conductivity measurements to provide empirical support for the theory of electrolytic dissociation proposed by Svante Arrhenius.1

Arrhenius’s theory posited that acids dissociate into electrically charged ions when dissolved in water, and that the varying chemical activities of different acids were directly related to the concentration of these active ions. Ostwald recognized that if this theory were correct, the degree of dissociation could be mathematically linked to the acid's concentration and an equilibrium constant.12

To test this, Ostwald measured the electrical conductivities of more than 200 organic acids. Because conductivity in solutions is dependent on the presence and concentration of ions, these measurements allowed him to calculate the degree of dissociation ($\alpha$) for each acid. By applying the law of mass action to these results, he derived his famous dilution law:13
$$\frac{{\alpha }^2}{(1-\alpha )c}=k$$
This mathematical consistency between his conductivity data and the dissociation theory provided the evidence needed to substantiate the work of Arrhenius and establish a foundational principle of modern physical chemistry.1