( 388 ) 
is broken up into two at the deformation of this surface, which two 
umbilies diverge in general, real at a variation of the parameter in 
one sense, and imaginary in the other. So we have to do with 
a double umbilic, namely with such a one at whose effective *) 
occurrence a transition takes place from the real to the imaginary. 
3. Before considering the further properties of this double umbilie 
we wish to observe that the condition A, =O was already known 
as an important characteristic. It characterises namely the case 
of transition between two of the three general kinds of umbilies 
distinguished for the first time by Darsoux*) according to the manner 
in which the lines of curvature bear themselves in their neighbourhood. 
For the first kind, see fig. 1, lines of curvature are starting from 
the umbilic in three different directions — namely in each direction a 
Fig. 1. Fig. 2. Fig. 3. 
single one, which we have represented by a right line because 
its curvature depends on the terms of higher order of the equation 
(1), to begin with those of the fourth. Those three directions have 
the property that they cannot be represented in one quadrant, 1. e. 
each of them lies inside the obtuse angle formed by the two others. 
For this kind K, > 0°). 
For the second kind, see fig. 2, also lines of curvature start from 
the umbilie in three different directions; these directions are however 
such that one of them falls inside the acute angle formed by the 
two others, so that the three can now be contained in one quadrant. 
Moreover an infinite number of lines of curvature — five of which, the 
right line included, are indicated in fig. 2 — start in the firstmentioned 
direction which might be called the :iddle one. For this kind A, < 0. 
For the third kind, see fig. $, only one line of curvature starts 
from the umbilic, the right line of that figure. The two other directions 
1) See for the meaning of this term page 289 of the paper quoted in the first note. 
2) G. Darroux. Legons sur la théorie générale des surfaces. Quatrième partie. 
Gauthier-Villars, 1896, p. 448—465, 
8) This characteristic K,>O means moreover as is proved in the dissertation 
in a simple way, that the lines of curvature turn in the neighbourhood of O every- 
where their convex side to the umbilic, but for K, <0 on the contrary their con- ~ 
cave side. 
