TBANSACTIONS OF SECTION A. 569 



-where, as in Article xxvii. (November 1846) of my Collected Matliematical and 

 Physical Papers (vol. i.)— 



Vo d"- , d" , d- /p\i 



d.v^ dy' dz^ 



This (5) is the promised proof of § 3. 



6. Let now u, v, w denote the components of electric current at (.r, y, s) in 

 the electric system of § 4 ; so that — 



47rM = M3 = - V'Wi ; 4n-u = i'3 = - V'^i ; 47rW = W3 = - V"«^i • (7) 

 -which, in virtue of (4), give — ■ 



^Sl%^"' = («) 



rf.r dy dz 



Hence the components of electro-motive force due to change of current lieing 



rtKj a "3 _ ««'3 



~df' 7lt' dt' 

 are equal to — 



A „ ndu , „'> dv A —..odw (n\ 



4.V--^, 4.v-^^. 4.V ^^- ... (J) 



and therefore if ^ denote electrostatic potential, we have, for the equations of the 

 electric motion (§ 5) — 



If odu d<^\ 1 / „_„d» d^\, ,„ l/^-i w d^\ na\ 



"=;;(^"^-rf^);''%-(^ Vr.!,;- "=-;^v^ d-r^)- ^^^^ 



where k denotes ^tt of the specific resistance. 



7. As ^ is independent of t, according to § 4, we may, conveniently for a 

 moment, put — 



d"^ d^ r, d'ir (^^\ 



Kd.v Kdy ' Kctz 



J-v'W; f = vM.«. | = vM.v) ... (12 



and so find, as equivalents to (10)- 

 da 0/ \ d^ 

 ^=^'(''°)' di 



The interpretation of this elimination of SE' may be illu.strated by considering, for 

 example, a finite portion of liomogeneous solid conductor, of any shape (a long thin 

 wire with two ends, or a short thick wire, or a solid globe, or a lump, of any shape, 

 of copper or other metal homogeneous throughout), with a constant flow of elec- 

 tricity maintained through it by electrodes from a voltaic battery or other source 

 of electric energy, and with proper appliances over its whole boundary, so regulated 

 as to keep any given constant potential at every point of the boundary ; while 

 currents are caused to circulate through the interior by varying currents in circuits 

 exterior to it. There being no chnrujing electrification by our supposition of § 4, 

 ^ can have no contribution from electrification within our conductor ; and therefore 

 throughout our field — 



V^'I' = (13) 



which, with (8) and (11), gives — 



^ + f + ^V = (14) 



dx dy dz 



Between (12) and (14) we have four equations for three unknown quantities. 

 These in the case of homogeneousness (k constant) are equivalent to only three, 

 because in this case (14) follows from (12) provided (14) is satisfied initially, 

 and the proper surface condition is maintained to prevent any violation of it from 

 supervening. 



' Maxwell, for quaternionic reasons, takes V' the negative of mine. 



