721 
sively as point p to the five others (which are then called g), and 
the whole number is o'. The quantity p in the formula points to 
the number of pairs, where the point p lies in a given plane; 
now this plane intersects 4’? in twelve points, which we can all 
regard as points p; through each of these passes one line of the 
regulus containing still five other points of £'*, which we shall call 
q; it is then clear that there are 60 pairs pg whose component p 
lies in a given plane. The symbol q has the same meaning as p, in 
this case for the points g; however, as in our case each point of £'* 
can be a p as well as a gq, the quantity g is also = 60. Finally 
the letter g indicates the number of pairs whose connecting line 
intersects a given line; now that given line intersects only two lines 
of the regulus, on each of which 30 pairs pq are situated; g is 
therefore 60, and in this way we find for ¢, the number of coincidences, 
e= 60 + 60 — 60 — 60. 
So there are cixty lines of the regulus containing two coinciding 
points of £'*; 6 of them correspond to the double generatrices of 
2°°, but a closer investigation shows us that these must be counted 
double; the remaining forty-eight are tangents of £'* and correspond 
to torsal lines: so °° contains forty-eight torsal lines of the first kind. 
The formula e=p + q— g, or written as: p + g=g Je, is namely 
deduced by assuming a system of oo! pairs of points p,g and by 
projecting {hese out of a line /. If a plane 2 through / contains 
p points p, we can connect the points q conjugated to these 
by planes with /, so that p planes are conjugated to À; if inversely 
a plane 2 contains q points g, then to this plane g others are con- 
jugated, and thus is generated a correspondence (p,g) with p + q 
coincidences, which are evidently furnished by means of the coinci- 
dences of the pairs of points themselves (e) and by the pairs of points 
whose connecting line intersects /. 
Let us now apply this to our case. A plane À intersects £7? in twelve 
points p; to each of these the five points g are conjugated lying 
with p on a generatrix of the regulus, so that to 4 sixty other planes 
are conjugated. A plane 2 through a nodal point D of 4? however 
contains of £'? besides D only ten more points, which give rise to 
fifty planes; so the ten remaining ones must be furnished by D 
itself. Now the generatrix of the regulus through D intersects 2° 
besides in JD only in four points more, the planes through these 
and / count double in the correspondence, because D itself counts 
double in the plane 7D, but this furnishes only four planes counting 
double, or eight single ones; so the two missing ones must coincide 
with the plane /D, i.e. /D is a double plane counting double (and 
