CONTENTS.

Art.		Page
503.	His method of experimenting	146
504.	Ampère’s balance A sensitive instrument used by Ampère to measure the force between electric currents.	147
505.	Ampère’s first experiment. Equal and opposite currents neutralize each other	147
506.	Second experiment. A crooked conductor is equivalent to a straight one carrying the same current	148
507.	Third experiment. The action of a closed current on a segment original: "element" of another current is perpendicular to that segment	148
508.	Fourth experiment. Equal currents in systems that are geometrically similar produce equal forces	149
509.	In all of these experiments the acting current is a closed circuit	151
510.	Both circuits may, however, for mathematical purposes be conceived as consisting of elementary portions, and the action of the circuits as the total result original: "resultant" of the action of these elements	151
511.	Necessary form of the relations between two elementary portions of lines	151
512.	The geometrical quantities which determine their relative position	152
513.	Form of the components of their mutual action	153
514.	Resolution of these in three directions: parallel to the line joining them, and parallel to the elements themselves	154
515.	General expression for the action of a finite current on the element of another	154
516.	Condition provided by Ampère’s third case of equilibrium	155
517.	Theory of the directrix A line used to determine the direction of force. and the determinants of electrodynamic action	156
518.	Expression of the determinants in terms of the components of the vector-potential of the current	157
519.	The part of the force which is indeterminate can be expressed as the space-variation of a potential	157
520.	Complete expression for the action between two finite currents	158
521.	Mutual potential of two closed currents	158
522.	Appropriateness of quaternions A complex number system used in 19th-century physics to represent rotations and vectors. in this investigation	158
523.	Determination of the form of the functions by Ampère’s fourth case of equilibrium	159
524.	The electrodynamic and electromagnetic units of currents	159
525.	Final expressions for electromagnetic force between two elements	160
526.	Four different admissible forms of the theory	160
527.	Of these, Ampère’s theory is to be preferred	161