( 358 ) 



{c' — v') x' — 2 (r .V =F a c) r ^ a' — r' = (J . . . (20) 



the upper sign relating to the roots t^, the lower one to the roots 

 Tj. The product of the two roots r^ or the two roots t., is: 



.J _ „2 



As r ^ (2 in a point of the exterior, this product is negative, if 

 c y> V, it is positive, if c <^ v. We conclude : 



If tlie velocity is less than that of light, the two roots of our 

 quadratic are real and have opposite signs. Each of the two equa- 

 tions therefore has one available positive root. 



J f the velocity exceeds that of light, the two roots may be con- 

 jugate-imaginaries ; if they are real, they have equal signs, and 

 therefore they may both be either positive or negative. Each of the 

 two equations lias therefore either no roots or two available positive roots. 



We distinguish between real and imaginary roots by consulting 

 the discriminant of our equation (20). The roots are imaginary if 



[v X qc acf < (r" — a^) {v' — c'j, 



for Avhich we may also put 



(c..=pai;)^<(2/^ + ^')(i''-c') (21) 



We introduce the abbreviations : 



/? = - , § = .'6- rp a I? , ^' = 2/' + s' 

 c 



so that q means the distance of the point in question from the 

 direction of motion, § the distance of the same point, measured in 

 the direction of motion, from two points P^ P.^ (see fig. 3) of which 

 the coordinates are x =^ ± a /?. 



Replacing in (21) the sign <::^ by =, we get 



ë^ = ^M/i'-l) (22) 



This defines a cone of revolution about the direction of motion, 

 of which the apex lies in the point P^ or P^ according to the 

 meaning of § and of which the generatrices are inclined towards 



the direction of motion in the angle arctg (/i' — 1)~ . For points in 

 the interior of these cones, i. e. between the conical surface and the 

 conical axis, the roots of (20) are real, for points in the exterior 

 they are imaginary. 



In case of the reality of roots the distinction between positive 

 and negative values results from the sign of the coefficient of r in 

 equation (20). 



