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At the same time ensues from the behaviour of the quadries that 
when there is a real contact with the plane at infinity, the point 
of contact, if it appears in the section of the tangent plane as an 
isolated point, breaks up at the deformation into two real and two 
imaginary umbilics in whatever direction the deformation may take 
place. In the opposite case we have to do with four imaginary 
umbilies. So transition from real umbilies to imaginary ones never 
takes place in this way. 
d. The point of contact of a point-general surface with the curve 
of the spherical points at infinity as a double umbilic. 
9. It goes without saying that when w touches c the point of 
contact must be an @-point, for the points A, and A, coincide with 
this point of contact and so they are situated on the nodal tangents 
in this same point. 
By analysis it proves to be a double @-point. As the spherical 
points at infinity are all imaginary, these umbilics and the single 
ones into which they break up, are also always imaginary. 
e. The points of infinity of the spinodal line as single wmbilics, 
when the tangent of the spinode lies in the plane at infinity. 
10. If we consider a point in which the spinodal line of w cuts 
the plane a, it is easy to see that this point must be regarded as an 
2-point as often as the cuspidal tangent of the section of the tangent 
plane lies in plane «, which isa single condition. It appears, however, 
that this point cannot be driven asunder by deformation, so it must 
be regarded as a single @-point and the umbilic corresponding to it 
likewise as a single umbilic. This umbilic can be real or imaginary. 
The manner indicated here is the only one in which real umbilics 
can reach infinity without passing into a multiple umbilic, i. e. 
without meeting other umbilies there. 
f. The points of intersection of the surface with the curve of 
the spherical points at injinity as single umbilics, when 
one of the nodal tangents in the section of the 
tangent plane lies in the plane at infinity. 
11. It is immediately evident that the corresponding points on w are 
Q-points and after investigation they prove to be single ones. As 
umbilies they are of course always imaginary. 
Application to quadries. 
12. The equation of a quadric can be brought with an appropriate 
