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relationship between any point A .of a, as point common to two 
tangents a,,a,, and the point B common to the corresponding 
tangents 6,, 6. | 
If A describes a line /4 its polar line with respect to «° will rotate 
about a fixed point, whilst the pair a,,a, generates an involution. 
3ut then 5,,6, must also generate an involution, so that B describes 
a line /p. So the point fields (A), (B) are in projective correspond- 
ence (collinear, homographic). 
By projecting the field (A) out of any point P unto 6 we obtain 
in 8 two projective collocal fields, admitting three coincidencies. So 
the congruence of the lines 4B is of sheaf degree (order) three. Its 
field degree (class) however is one; for if A describes the section 
of «a with any plane M, B will arrive once in J, i.e. II contains 
only one line AB. 
The congruence (3,1) found here is generated, as we know, by 
the axes (= biplanar lines) of a twisted cubic y’, i.e. any line AB 
lies in two osculating planes of y’. 
Evidently any line AB is double edge of the complex cone of any 
of its points P. However the complex rays through A form the 
pencil A (a) counted twice and the pencils determined by the lines 
b,,6,; for B the analogous property holds. 
2% 
§ 8. Evidently the three double edges of the complex cone of P 
are the mutual intersections of the three osculating planes of y° 
passing through P. 
Likewise the complex curve in II has for double tangent the axis of 
y® lying in that plane, the other two double tangents coinciding with 
the intersections of WZ with « and 3. For, each of the lines 0’, 6" 
which concur in the point cH determines a complex ray lying in 
II, which lines coincide both with « 7. 
An osculating plane 2 of 7’ contains «* axes, enveloping a conic 
w’. Any plane 2 is singular for the congruence (AB). So the com- 
plex curve in @ is the conic w’ counted twice. 
As the congruence (3,1) cannot admit singular points, no point 
bearing more than three planes 2, no complex cone can degenerate 
but those corresponding to the principal points and the points of 
the principal plane. We already remarked this for @ and #; for any 
point of a single principal plane the complex cone breaks up into 
a pencil and a rational cubic cone. 
The complex cone of any point of the developable with y’* as cus- 
pidal edge admits an edge along which two sheets touch each other 
(the plane section has two branches touching each other). For any 
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