394 M. G. Kirchhoff on the Motion of Electricity in Wires. 



upon an equal quantity at the unit of distance, produces the unit 

 of force. In general, all quantities which appear in this investi- 

 gation — strength of current, resistance, &c. — shall be regarded 

 as measured by a mechanical unit, in the manner often described 

 by W. Weber in his ' Electrodynamic Determinations/ We 



•&V 

 have there ^ as the force with which the free electricity 



strives to move the unit of positive electricity at the point under 

 consideration in the direction in which s increases. An equal 

 force tends to move the negative electricity in the opposite direc- 



tion. Therefore we have — 2-r— a8 the electromotive force de- 



^s 



rived from the free electricity, and acting at the point in question. 

 In developing the value of V, I will assume that no other free 

 electricity acts upon the wire than that which is in the wire itself. 

 The quantity of free electricity which, at the time t, is contained 

 in the element of the wire which corresponds to the element ds 

 of the axis I will denote by eds ; let ds' be a second element of 

 the axis, and ^ds' the quantity of electricity contained in the 

 corresponding element of the wire. I picture to myself a por- 

 tion of the wire, whose centre lies in ds and the length of which 

 is 2e, where e denotes a quantity which is to be regarded as infi- 

 nitely small in comparison with the length of the whole wii'e, 

 but as infinitely great in comparison with the radius of its cross 

 section. When the element of the wire in which the quantity of 

 electricity e'ds' is contained lies outside the above portion, we 

 can imagine, in the calculation of V, its electricity to be concen- 

 trated in the line ds', and the point to which V refers situated 

 in the line ds. Hence the portion of V derived from the whole 

 wire, with the exception of the portion alluded to, is 



'-r- 



where r denotes the distance of the elements ds and ds', the in- 

 tegration being extended over the whole of the central line, with 

 the exception of the length 2e. With regard to the portion of 

 V derived from the part separated, this can only be calculated 

 when the distribution of the electi'icity within a cross section is 

 known. I will assume that here, as in the case of a constant 

 current, and of electricity in equilibrium, free electricity is to be 

 found upon the surface only, and besides that its density is the 

 same at all points of the periphery of a cross section. Denoting 

 by a the radius of the cross section, we have, according to this, 

 the density of the free electricity at any point of the surface of 



e 

 the portion of wire under consideraticsi = ^ — ; hence, as on 



