Fundamental Concepts of Electrical Theory. 415 



of sets of values o£ x, y, z, t when r, t , V, aud Y are con- 

 nected by one or more relations, and they may perhaps be 

 inconsistent when the relations are not satisfied. 



The three equations (11) give go 1 sets of values of a?, y, z, 

 and t which generally represent a point moving along a 

 straight line with constant velocity, but if all the deter- 

 minants of the array 



f"_Yf', v'-Vv'y £"-V(T, t"-Vt', 0"-V0\ 



fo"-V fo', Vo"-y«Vo', &''-Vo&', t "-V o t o ', o "-V«A', 



% — fib ^?-^0, f— ?0» T ~ TO, #~ #0> 



vanish, the three equations are not independent and give 

 qo 2 sets or even co 3 sets of values of x, y, z, and t. In 

 the first case equation (10) is satisfied only if the velocity 

 is the velocity of light and f, 77, f, t is one position of* the 

 moving point. For a similar reason f , t) , f 0j t must also 

 be a position of the moving point. This case may be rejected 

 as trivial. 



In the second case the determinants of the array may all 

 vanish on account of either three, two, or only one relation 

 between V, V , t, and t . If three relations are required 

 for the vanishing of the determinants, it means that it 

 is possible to pick out a set of co 1 lines of magnetic force of 

 one field such that each of these lines of force is permanently 

 incident with a finite number of lines of force of the second 

 field. The condition (EH ) +(E H) =0 is then satisfied for 

 co 2 sets of values of a?, y, z, t ; i. e., at points of a moving 

 curve. It follows from § 6 'that there is also either permanent 

 incidence or contacts of a tangential nature between lines of 

 electric force of the two fields at points of this moving 

 curve. 



Two relations are sufficient for the vanishing of all the 

 determinants of the array when two equations of type 



Zf ■+■ mtf + ?if + pr + q6 +r = 0, 

 /f + mri + n £ + p^o + qd + r=Q, 



are satisfied, Z, m, n, p, q, r being constants. If we regard 

 (?> V> £> ww") (?o 5 Vo> So? ?CT o) as the rectangular coordinates 

 of two points in a space of four dimensions S 4 , the equations 

 signify that the paths of the two point charges are repre- 

 sented by two curves which lie on the same hypersphere or 

 hyperplane. In the latter case the two point charges lie at 

 any instant in a plane which moves with uniform velocity in 



