901 
and seh, are thus each = 6 x 13 = 78, and ope was 80, so the class 
of the focal surface = 78 + 78 — 80 = 76. 
I may be permitted to point out in passing a slight inaccuracy 
committed by ScHuBeRT on page 64 of his “Kalkiil” where he gives 
formulae for order and class of the focal. surface of a congruence 
taking the number ope, called by him ce, only onee into account; in 
PASCAL-SCHEPP’s well known “Repertorium” vol. II, page 407 we 
find indicated the exact formulae, with the rank number 7 counted 
twice. | 
In a congruence of rays appear in general o' rays whose two 
foci coincide; these too are easy to trace in our congruence. For, 
according to $20 in order to find the foci of an arbitrary ray s, 
we must apply in the focus P, the complex cone and the tangential 
plane to ° and intersect these by each other; the foci of the lines 
of intersection are the foci of s, and the tangential planes through s, 
to the complex cones of the foci the focal planes. So as soon as the 
complex cone of P, touches the tangential plane 2" along a line 
f, the two foci of s, will coincide in the focus of ¢ and the focal 
planes will coincide in the tangential plane through s, to the complex 
cone of the only focus. 
The points P, whose complex cones touch 2° are to be found 
again with the aid of Scuupmrt’s “Kalkiil’. We conjugate the two 
rays s, along which the complex cone of a point P?, of 2° cuts the 
tangential plane in that point, to each other; so we obtain in that 
manner a set of oo? pairs of rays and we apply to it the formula: 
EOP = Ode + Ohe + Gp” — Ope’); 
The left member namely indicates the number of coincidences whose 
points of intersection lie in a given plane, that is thus evidently the 
order of the curve which is the locus of the points ?, to be found. 
dye indicates the number of pairs of rays whose component g lies 
in a given plane; this plane cuts out of 2° a plane curve £° which 
possesses no other singularities than three nodes and which is so 
of class 6.5 —2.3—= 24, and all the complex rays in this plane 
envelop a conic; so there lie 48 complex rays g in this plane 
touching 2°, If we apply in one of the points of contact the tangential 
plane to 2", then there lies in it one ray /; so og, is 48 and likewise 
of course dhe. 
With op? we must trace the number of pairs of rays whose points 
of intersection lie in two given planes at the same time, thus on a 
given line; this line intersects $2° in six points and in the tangential 
1) ScHUBERT 1. c. page 62. 
59 
Proceedings Royal Acad. Amsterdam, Vol, XV, 
