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points coinciding in one of the coincidences of this involution. The 
straight line / is then principal tangent of one of the surfaces a’. 
The lines bearing the coincidences formed in this way, being the 
principal tangents of the surfaces a’, form a line complex of order 
9; for the rays of this complex, which lie ina plane, are inflectional 
tangents of a pencil of eubies along which this plane intersects 
the pencil (a); and these inflectional tangents envelop a curve of 
class 9. 
It appeared in $ 2 that an arbitrary point P belongs to o' groups 
of S*. The remaining points of these groups lie on a plane curve 
of order five, which has a node in P. These groups are formed 
by the intersections of the c* with the lines passing through P. If 
we now consider the tangents at the branches of c* passing through 
P, each of these two tangents has in P three coinciding points in 
common with c*. Therefore P is a coincidence of the two groups of 
S* lying on these lines. An arbitrary point P belongs therefore to 
2 coincidences of |S‘. 
At the c? mentioned 5 Xx 4-— 2—4= 14 tangents may be drawn 
out of P. To them belong the lines connecting P with the 9 inter- 
sections of the plane of c° with the base-curve 9° '). 
If Q is the point of contact of one of the remaining 5 tangents, 
two of the intersections of the line PQ with the curve c° coincide 
in Q, so that Q is a coincidence. An arbitrary point P belongs 
therefore to jive groups, which have a coincidence Q lying out- 
side P. 
Between the points P and Q exists evidently a correspondence 
(5, 4); for to each coincidence Q belong two points P, and each 
point Q belongs to 2 coincidences. 
§ 5. If the point P describes a plane JV, the points Q will 
describe a surface w, if the points Q describes a plane V, P de- 
scribes a surface ®. 
In order to find the orders of these surfaces we inquire their 
intersections with the plane WV. If the point P describes the plane 
V and Q also lies in it, the line PQ lies in this plane. As this 
line bears the coincidence lying in Q, it is an inflectional tangent 
of one of the curves of the pencil, along which the plane J” inter- 
sects the pencil (a*), while Q is the associated point of inflection. 
The locus of these points of inflection @ is a curve e* of order 
twelve. 
1) As will afterwards appear these lines also bear coincidences which, however 
arise in a different way from those considered in this §. 
