of departure have become imaginary. For this kind too A, <0. To 
distinguish it analytically from the preceding one we can notice the 
sign of the diseriminant of the cubic 
de n° + (2d, — 3d) n° + (Bd, — 2d,)n—d,=0. . . ( 
which proves to serve for the determination of the three directions 
of departure. If we call this discriminant A, chosen in such a way 
that for A, > 0 the three roots are real, we have for the first kind 
Kee ike OR for the second Ke <0, K, > 0; for. the third 
ere he Ons Ay fourth kind Ke > 0; KC Or does mot: exist; 
because as is demonstrated also algebraically A, > 0 includes A, > 0. 
4. As is apparent from this explanation the double umbilic forms 
the ease of transition between the first and the second kind, for which 
case of transition A, must of necessity be equal to nought, and A, > 0. 
The form of the lines of curvature now becomes very simple as long 
as one confines oneself to the approximation which has led to the 
figures 1, 2 and 3. Out of the differential equation 
dy \°* 
{d,«+d,y] |: — B |+ [(d, — 3d,)a + (8d, —d) 4] 
C 
ly x 
=0 , (8) 
dx 
C 
which serves to determine the lines of curvature, a factor separates 
itself namely d‚v + d‚y, which made equal to zero represents a 
right line, whilst the remaining furnishes two mutually perpendi- 
cular pencils of parallel lines. In this manner, however, from each 
point of the first mentioned right line three lines of curvature 
would start, so that there would be an entire line of umbilics. This 
is of course in general not the case, so that this representation of the 
lines of curvature must undergo a considerable modification as soon 
as the terms of higher order are taken into consideration. We shall 
soon refer to this again. 
5. We shall first mention the results of a closer investigation of 
the deformation of the double umbilic. From this we were able to 
prove, 1st. that for a variation of parameter in the sense in which the 
two single umbilics diverge in a real manner, this diverging shall always 
take place in the direction of the just discussed right line d,a—+-d,y—0, 
which after that represents in first approximation for each of the 
two separated umbilics one of the directions of departure of lines 
of curvature, 2"¢. that these separated umbilics are always of a dif- 
ferent kind, namely one of the first kind, the other of the second. 
Moreover dr + d,y =O indicates for that of the second kind the 
middle direction of departure, whilst also the remaining directions 
of departure of the diverged umbilics nearly correspond to the direc- 
tions of departure of the original double umbilic discussed in § 4. 
