( 220 ) 



determines a complex £1 whose rajs form with the axis a constant 

 angle, so they cut a circle lying at infinity. 

 The equation 



<^W=p:+P^ (16) 



furnishes a complex Si, whose rays cut the circle .r' -}-?/' = a'. 

 For XOY cuts each cone of the complex according to this circle. 

 If I represents the distance from a ray then 



I, _ P,' +P,' +Pe' .^rj. 



~ P^" + P2' -^ Pz' 



If XOY is displaced along a distance c in its normal direction, 

 p, and ps pass into (p^ — ci\) and (ps -\- cp^). So for the distance 

 /, from a ray to the point (0, 0, c) we have 



, , ip^ + Ps' + Pe') -i- 2c {p,p, - p,p,) + 0^ (Pi" + Pu') ,, «, 

 l^ — _ . (ly) 



Pr -hP, +Pz 



If in this equation we substitute — c for c we shall find a relation 

 for the distance 4 from the ray to point (0,0, — c). 



The equation 



a, I,' + «, I,' = /? 

 furnishes a complex £2 with the equation 



-}-^{a,-a,)ö{p,p,-p,p,) = (19) 



This symmetrical complex is very extensively and elementarily 

 treated by J. Neuberg {Wiskundige Opgaven, IX, p. 334 — 341, and 

 Annaes da Academia Polytechnica do Porto, I, p. 137 — 150). The 

 special case «1 ^1 + «2 ^2 = ^ "^^^^ treated by F. Corin {Mathesis, IV, 

 p. i77_i79, 241—243). 

 For ^1 = 4 we find simply 



P.P. - P.P. = ^ (20) 



This complex contains the rays at equal distances from two fixed 

 points. As c does not occur in the equation the fixed points may 

 be replaced by any couple of points on the axis having as centre ^). 



§ 5. When there is a displacement in the direction of OZ the 

 coordinates of rays p^, p„ p^ and p„ do not change whilst we obtain 



P, = /" + hp, and p, = p, — hp„ 

 so 



Pi Pa + Pa Ps = Pi Pi + P. Ps' 



The forms {p,^ + Psl and (Pi Pi — P. P.) are now not invariant. 



1) This complex is tetraedral. See Sturm, Liniengeometrie, I, p. 364. 



