462 ON PHYSICAL LINES OF FORCE. 



of the second term of our equation indicates the general law, which is quite 

 independent of the direction of the lines of force, and depends solely on the 

 manner in which the force vanes from one part of the field to another. 



We come now to the third term of the value of X, 



1 dp da\ 

 



Here /i/J is, as before, the quantity of magnetic induction through unit of area 

 perpendicular to the axis of y, and -/- -r- is a quantity which would disap- 



pear if adx + fidy + ydz were a complete differential, that is, if the force acting 

 on a unit north pole were subject to the condition that no work can be done 

 upon the pole in passing round any closed curve. The quantity represents the 

 work done on a north pole in travelling round unit of area in the direction 

 from +x to +y parallel to the plane of xy. Now if an electric current whose 

 strength is r is traversing the axis of z, which, we may suppose, points 

 vertically upwards, then, if the axis of a; is east and that of y north, a unit 

 north pole will be urged round the axis of z in the direction from x to y, so 



that in one revolution the work done will be = 47rr. Hence I -/- -7- ) repre- 



4?r \dx ay] 



sents the strength of an electric current parallel to z through unit of area ; and 

 if we write 



1 [dy d$\ 1 (da dy\ 1 /d/3 da\ 



_ I L _ _JLT I = /n I ' 1 n _ I v I Q i 



4\3y dzj P ' ^\dz dx)~ q> 4ir\dx dy)~ 



then p, q, r will be the quantity of electric current per unit of area perpen- 

 dicular to the axes of x, y, and z respectively. 



The physical interpretation of the third term of X, /u.y8r, is that if pfi is 

 the quantity of magnetic induction parallel to y, and r the quantity of electricity 

 flowing in the direction of z, the element will be urged in the direction of x, 

 transversely to the direction of the current and of the lines of force; that is, 

 an ascending current in a field of force magnetized towards the north would 

 tend to move west. 



To illustrate the action of the molecular vortices, let sn be the direction 

 of magnetic force in the field, and let C be the section of an ascending mag- 

 netic current perpendicular to the paper. The lines of force due to this current 



