( 1085 ) 
P He. for P. A ray of the complex to be found is represented by 
the equations | 
k 
(p,tAp,) sin? —(p,+hkp,) cos D EP ly Ps 
1 
and 
E k 
(p,tp,) sin (P+ a) — (p,+p,) cos (P+ a) = — py — ds Ps 
a 
2 
Elimination of ® furnishes as equation of the complex : 
(pst kp)? sin? a + (p, + Ap,)* sin? a = 
k k k k P 
= Pe % v.) on? af Ps v)( Pe var os erf Pe %s ps) (3) 
a, a, a, a, 
Equation (8) represents the complex £ when the following five 
relations have been satisfied: 
sin” a À A, 
k —+ ee) 
a,” — 2a, a, cos a + a,” = — AB ® « (6) 
k? . 
=== (¢," — dS Ed SAD an ne AT) 
derden 
| a; ly | Y nl a) 
k iin 2+) —+ — ] cos “ mn AA Ne oe tice (CG 
Te a 
Equation (5) furnishes two values for & differing only in sign; 
the absolute value is V —OB,? =WV —OB,?. From (6) and (7) ensues: 
=|/\ Ri (2) 
u, a, = DE 5 . . . . . . . 7 
out of (8) in connection with (4), (5), and (6): 
.(C—kE 
GOS == Tidal ———— ae ee tae roek Set (19) 
kB 
Kor each / we find one value of cosa. As however all conditions 
remain satisfied when @ changes signs, we may add to P, instead 
of P, also the image P,’ of P, with respect to OP,. Out of (4) 
follows for every & one 2; finally we find a, and a, out of (6) 
and (7); they are the radi of the circles according to which #0 
is cut by O,? and O,?. By division of (4) and (7) is evident, that 
the distance from O to P, P, (or PP’) must be the same for both 
values of 4. This was to be foreseen; P,P, must touch, being a complex 
ray, the complex conic of «Oy, being a circle with O as centre. 
Its radius is: 
; 71 
Proceedings Royal Acad. Amsterdam. Vol. XIII. 
