256 Prof. R. Clausius on the Determination of the 



closed, the thing is simplified in this way — that, besides the 

 intensity of the current, we have only to consider the position 

 and direction of the current-elements, without taking sepa- 

 rately into consideration the two electricities present in them. 

 Thereby we then arrive at formulas of extraordinary s implicity, 

 which afford great facilities for the determination of the pon- 

 deromotive and electromotive forces, and bring into clear view 

 the entire field of mathematical developments having reference 

 thereto. 



These formulas I will take leave to develop in the sequel — 

 and not merely from the electrodynamic fundamental law ad- 

 vanced by me, but also from those of Riemann and Weber. 

 It will be seen that the results of this formulation correspond- 

 ing to the three laws differ from each other only by solitary, 

 easily determinable terms, and hence can very conveniently 

 be compared with one another. 



§2. 



Let a moved quantity of electricity, the amount of which 



does not affect the question, and which we will therefore take 



as a unit of electricity, be at the time t in the point x, y, z 



dx dn d z 

 and have the velocity-components -j-> ~j -=-. Given, further, 



a galvanic current /, which may likewise be in motion. Fo r 

 the sake of simplicity we will provisionally assume that the 

 current is linear, since this involves no important limitation, 

 as a current which is not linear can always be supposed to be 

 divided into an infinite number of infinitely thin current- 

 threads which can be regarded as linear currents. 



Let us now contemplate in the conductor of the current, 

 first, a single particle of the flowing electricity: this has a 

 double motion — first, the motion of the current in the con- 

 ductor, and, secondly, the motion of the conductor itself. In 

 order to distinguish from one another the variations of the 

 quantities which come into consideration occasioned by these 

 two motions, we will, in like manner as I have already done 

 in a previous investigation*, introduce the following method 

 of designation. The coordinates of a fixed point in the con- 

 ductor we consider simply as functions of the time t ; while 

 for the determination of the coordinates of the particle of 

 electricity flowing in the conductor we take as an auxiliary a 



* " Ueber die Behandlung der zwischen linearen Stromen und Leitern 

 stattfindenden ponderomotorischen und electromotorischen Krafte nach 

 dem electrodynamischen Grundgesetze," Verhandl. des naturhist. Vereins 

 der preu88. Rheinl. u. West/, vol. xxxiii. 1876; Wied. Ann. vol. i. , and 

 Clausius, Median. Wiirmetheorie, Bd. ii. Abschn. x. 



