or also by 
For its orthogonal coordinates of rays, i.e. the quantities 
' ’ = 
AE, PY Ds 
0, Ve p= ea ae), p= ay ya! ; 
we find 
u— u (b°—ce?) (u'—u) 
p= a a, etc. P,= IN Unen etc, 
From this ensues that the co? normals of the system of ellipsoids 
form a quadratic complex with the equation 
nn nne a eee (3) 
2. For the traces of the normal with YOZ and YOZ, we have 
successively Te a 5? and u = — das so 
c?—b? ca 
zi zy ande Zn 
c a 
Now follows from 
2": z' = (c?—8?) : (?—a?) 
that the complex can be built up out of oe* linear congruences, of 
which the directrices form two projective pencils of parallel rays 
situated in NOZ and YOZ having the direction OX and OY. So 
the complex is fetraedral and has as principal points O and the 
points X,, Y,,Z, lying at infinity on the axes. 
The trace of the ray of the complex with XOY is determined 
by wu" =—c’. If we notice that the parameter w is proportional to 
the distance of the point P indicated by it to the point P,, we see 
that out of 
! 
u —ú arc. 
mn DTE 
uu b-—c? 
the characteristic anharmonic relation of the complex is obtained, 
namely 
(PP peu Pp" 8) = (a°—c’) H (b?—c’). 
3. The footpoints P, of the normals let down out of P, lie 
evidently on the cubic curve 
wu by c?z 
t=, ee = STA ars eka hey | U (4) 
ate b° Hv ctv 
