In the Quarterly Mathematical Journal for 1860, I gave a quaternion process for obtaining in a very simple manner, from Ampère's experimental data, his well-known expression for the mutual action between two elements of currents. As one of the data the assumption was made, after Ampère, that the action is a force whose direction is that of the line joining the middle points of the elements, i.e., it was assumed that the necessary equality of action and reaction holds, not merely for two closed circuits but, for each pair of elements of these circuits.