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transformation. The cause of this is that they are in a definite 
relation to the plane at infinity and in particular to the spherical 
points in that plane. This obliges us to give a separate consideration 
of the cases of the first order of exception, where umbilies reach 
infinity. It was a priori not improbable that this would be accom- 
panied by the occurrence of multiplicity in all or in some of those 
cases, as really it proved to be for some. 
The method of investigation with respect to this was as follows : 
first the umbilies were exchanged for a more general kind of 
singular points which are capable of projective transformation. To 
this end it is sufficient to observe that an umbilie can be defined 
as such a point of a given surface which — when regarded as 
a node of its section of the tangent plane — has the property 
that both nodal tangents pass through the circular points of the 
tangent plane. 
After applying the general projective transformation the problem of 
the umbilics of the original surface is in this way reduced to the 
following : 
Given a surface w, a plane a, and in that plane a conic ce; to define 
on the surface w the points @ which have the property that the two 
nodal tangents of the section of the tangent plane 9 im 2 pass 
respectively through the points A, and A, where e is cut by g. 
For this more general problem the plane at infinity has been 
replaced by the plane a and we have but to study the points 2 
which as singularities of the first order may appear in the section d of 2 
and «@ which can be performed by choosing an appropriate system 
of axes with such a point for origin, by calculating for this system 
of axes the approximate equation of the surface, and by then applying 
a slight deformation. The results obtained in this way can be imme- 
diately applied to umbilies. 
In this manner it became evident that umbilies can appear in four 
different ways at infinity as singular points of the first order of 
exception, which we shall successively describe in short. 
e. The point of contact of a point-general surface with the 
plane at infinity as a fourfold umbilic. 
8. It is clear that whenever the surface touches the plane a, 
such a point of contact must be regarded as an @-point; for its 
tangents in the section of the tangent plane will certainly meet the 
conic c in the plane «. By regarding the surface as a quadrie we 
ean then by returning to the problem of the umbilics decide without 
calculation that the point under observation is a fourfold @-point. 
