Fundamental Concepts of Electrical Theory. 413 



condition is satisfied two lines of force winch intersect twice 

 can separate if the points of intersection approach one 

 another, coincide fand then become imaginary. The con- 

 dition may sometimes imply that two lines of force touch 

 one another, i. e. have a common tangent. 



If (E, H), (E , H ) are the electric and magnetic intensities 

 in the two fields the condition implies that 



(EH o ) + (E H) =0. 



This equation may also be written in the form 



d(V,T,V ,T ) 



~d(x, y, z, t) 



= 0; 



consequently it also implies that there is incidence between 

 the lines of magnetic force. 



Our condition may also be obtained geometrically by 

 considering the lines L and L which are used in the 

 representation of the vectors belonging to the two fields 

 at a point (#, y, z, t). The condition implies that the two 

 lines intersect and that there is consequently a velocity v 

 which can be regarded as the velocity of the two lines 

 of force through the point. The condition also implies 

 that the polar lines L' and L ' intersect. It should be 

 noticed that if d is the shortest distance between the lines 

 L and L and the angle between them, we have 



(EH )-r-(EoH)= S S d sin0, 



where S and So denote the magnitudes of the electric 

 intensities. 



Now d can be regarded as the relative velocity of the two 

 lines of force which pass through the point x, y, z at time t. 

 Hence if (EH ) + (E H) is not zero one of these lines of force 

 cuts through the other. If, on the other hand, (EH ) + (E H) 

 vanishes and changes sign as t varies, the relative velocity 

 changes sign and so the two lines of force meet and re- 

 bound without cutting through one another. Finally, if 

 (EH ) + (E H) vanishes but does not change sign, the two 

 lines of force generally cut through one another. 



It is evident that if L cuts through L as t varies, then L' 

 will also cut through L '. Hence, if there is cutting of lines 

 of electric force there is also cutting of lines of magnetic 

 force ; if lines of electric force collide without cutting 

 through one another, some lines of magnetic force do 

 the same. If there is permanent incidence between two 

 lines of electric force along a curve C, there is generallv 



