WILSON AND LEWIS. — RELATIVITY. 461 



2-voetor O^P i* ^ simple plane vector in the i)l;me of the point Q 

 and of w. The 1-vector p lias everywhere the direction of the funda- 

 mental vector w; if 1 be the singular vector from the vertex of the 

 cone to the point Q, the scalar product l«p is constant. In fact 



are the expressions for the fields in terms of 1 and w. 



LfCt us now choose arbitrarily a time-axis k4, and then the perpen- 

 dicular planoid is our three dimensional space. We may resolve our 

 1-vector and 2-vector fields as follows. 



w v+ k4 



P = 



l-w a + /,k4).(v+k4) 



II V , k4 



where Is and p^ are the space components of 1 and p. As 1 is a singular 

 vector, ^4 is equal to the magnitude of 1^. 



^ IXW ^ _ /, _ .,N (l8 + /4k4)x(V+ k4) 

 (1 — V^) l^V (1 — V^) (h — ^4V)xk4 



(72) 



P 



(/4-L-v>^ (/4-l.-v)^ 



Of these two planes into which P is now resolved, the first lies in 

 ^' space" and the second passes through the time axis and is perpen- 

 dicular to "space." 



We shall attempt to show with the aid of a diagram (Figure 22) the 

 geometrical significance of the various terms which we have employed 

 in the above formulas. The origin, that is, the vertex of the hyper- 

 cone, is any chosen point on the given (5)-line w. A point upon the 

 forward hypercone is Q, and 1 is the element OQ. The unit vector n 

 is drawn along QJ from Q towards and perpendicular to the vector 

 W. The intervals OJ and QJ are equal, and equal to /{ = — I'W. 

 The vector p drawn at Q parallel to w and of magnitude 1/R is the 

 extended vector potential at (^ due to w. The 2-vector P lies in the 

 pl-due OJQ, and is equal in magnitude to 1/R-. The arbitrarily cho- 

 sen time-axis is Isu, and on the planoid perpendicular to k4 (that is, 

 on " space ") the vector 1 projects into 1, = O'Q. The intersection 

 of the line of w with the planoid is G (the point of the line W which is 

 simultaneous with Q). Similarly 0' is the intersection of k4 with the 



