1258 
quadruple points in F,, F,, triple points in F’,, F,. 4° further contains 
the straight lines g,,y, and the nodal lines s,,s,; the latter arises 
from the observation that / intersects two curves g° meeting >, ors, 
in a point S, or S, lying on them. 
The intersection of the surfaces A belonging to two straight lines 
ll’ consists of: 6 curves 9%, resting on / and /’, the nodal lines 
Si Ss dj, and ‘the straight: lines! dip. d, ‚does deer Gas Oe 
The cubic transformation, which, in tetrahedrical coordinates is 
determined by 
Oi, es — 4 — Cf 
transforms this congruence into the bilinear congruence of rays, 
which has the images s,*,s,*, of s,,s, as directrices’). The surface 
A passes in consequence of this into the ruled surface formed by 
the straight lines r, which rest on _ s,*,s,* and on the curve 2° 
passing through the four points # into which / is transformed. The 
image of A is apparently a ruled surface of order four with nodal 
lines s,*,s,*. As this, apart from the points / has six points in 
common with an arbitrary curve 9° laid through those points, it is 
found onee more that A must be of order six. 
Now that the surface A is completely known, the characteristic 
numbers of the congruence may be found in the usual way’). 
In an arbritary plane ® this congruence determines a cubic 
involution possessing three singular points of order two (the inter- 
sections of d,,, s,, s,) and six singular points of order one (the 
intersections of d,,, dy, dass dass Js, g.)- It has been more fully 
described in my paper on “Cubic involutions in the plane”. *) 
3. Let us now consider the congruence [g°|, which possesses 
three fundamental pots F,, F,, F, and four singular bisecants 
Si Sis Sos 52, Of which the first two pass through F, the other two 
through /. Here too two pencils of quadratic cones that can pro- 
duce it are easily pointed out, while the four straight lines s are 
again singular straight lines of order two. 
The degenerate figures (d, J?) now form the following groups: 
A. The conic d* passes through #, and rests on the five straight 
lines d,,= FF, FE, s,, 8, 53, 8, ; the locus of d? is the ruled cubie 
surface L*, of which d,, is the nodal line, the second transversal ¢ 
') This transformation may effectively be used in the investigation of Rreys’s 
congruence (see my paper in volume XI of these Proceedings, p. 84), 
*) Cf. my paper in vol. XIV (p. 255) of these Proceedings, 
8) These Proc, XVI, p 974 (8 6), 
