( 390 ) 
All this being stated it is not diffieult to guess how in general the 
form of the lines of curvature must be, shortly after the breaking 
up of the double umbilic *). 
Fig. 4. 
Fig. 6. 
That form is represented in fig. 4, where QO, indicates the umbilie 
of the first kind, O, that of the second. At O, the angle of the two 
other lines of curvature, starting from the umbilic, which contains 
O, O, is a little larger than a right angle, at O, on the contrary it 
is a little smaller. 
If after that we allow the umbilies to coincide again, they meet 
at about half the distance and the figure now formed where the lines 
of curvature situated at some distance to the right and left of O, 
and QV, must have retained in general the same direction, can hardly 
be otherwise but such as has been indicated in fig. 5°), apart from 
the symmetry which in general does not exist of course, no more 
than in any of the other figures. 
1) After the publication of the Dutch version of this paper we found that Mr. 
A. Guuistranp already in 1900, in his memoir “Allgemeine Theorie der mono- 
chromatischen Aberrationen und ihre nächsten Ergebnisse für die Ophtalmologie” 
(see Nova Acta Regiae Societatis Scientiarum Upsaliensis, ser. 3, vol. 20, pp. 90 
and 114) arrived also, starting from other considerations, at the investigation of 
the double umbilie and its breaking up and that we obtained the same results. 
2) However, a closer investigation of this subject by another method would not be 
unwished for. It would have to be a systematic study of the lines, if possible in 
their entire length, satisfying the differential equation : 
dy»? 
pets) jee ++ Be, + He, — 2c,*)ay Hey] + 
de 
di 
+o [6d,y + 2(e,—Ge, + 4e,*)e? + 6(e,—e,)uy + 2(6e,—e,—40,)y?] =0. 
For this is the form which the differential equation of the lines of curvature 
assumes in the neighbourhood of a double umbilic at second approximation, 
when we place the X-axis in the direction in which the two single umbilics diverge 
by a slight deformation of the surface. We then have d;=0O and d; = 3d); the 
Jatter on account of (5). 
