13 
by OO, and OO,; it is therefore a nodal curve wt. Through O 
there pass six of its tangents; according to a theorem found by 
Bertini the six points of contact, coincidences of the (2,2), lie on a 
conic’). The bearers of the coincidences of the (2,2) envelop conse- 
quently a curve of the swath class. 
§ 2. We arrive in the following way at a (2,2) for which n= 2. 
Let the conic a? and the pencil of conics (6%) be given. To the point 
P we associate the points ?, and P, in which the conic 6? through 
P is cut by the polar line p of P relative to a’. On a straight line 
r, (6?) defines an involution; as a rule this has one pair of points 
in common with the involution on 7 of the pairs of points that are 
harmonically separated by a’. This (2,2) belongs accordingly to the 
first class. 
The points of «* are evidently the coincidences of this (2,2). The 
straight line 7 is transformed into a nodal e*, which has the pole 
Rk of + as double point. For when P moves along r, its polar line 
p revolves round A and bears the two points P,,P, associated to P. 
The base points B; (4 = 1, 2, 3,4) of (6%) are singular points. On 
the polar line 6, of By, (6?) defines oo’ pairs of points P,, P, which 
are associated to B. If P gets into the intersection of 6, with 7, 
one of-the points associated to P coincides with Bj; hence o* passes 
through the four points By. 
The conic 6° through R cuts 7» in two points A, R,, which are 
associated to A; hence e* has a double point in A. 
The six tangents of of meeting in R bear double points ?, — P,; 
from this it follows again that the branch curve is a (V)°. It has 
double points in the base points of (b°); for the involution of the 
pairs of points on Od, associated to Bz contains two double points 
for which By, is a branch point. 
With a 6°(V)° has four points in common besides the double 
points Bp; they are the branch points of the correspondence (2,2) on 
b°. The curve (V)' touches a? in the six coincidences of the involu- 
tion /* in which (67) cuts a’. - 
§ 3. Any point A of a’ is a coincidence of the (2,2), but it is also 
associated to the point A’ which the tangent a at A has further in 
1) Relative to this conic w? as an invariant curve, wí is transformed into itself 
by a central quadratic involution (inversion) with centre O of which the other 
two fundamental points lie on the polar line of O relative to w?; this straight 
line contains the points of contact O,, O, of O. (See J. DE Vries, La quartique 
nodale, Archives Teyler, série Il, tome IX, § 12). 
