(391 ) 
If we then continue the deformation in the same manner so that 
now the two umbilies diverge imaginarily, a figure seems to be 
formed as is represented in fig. 6. 
In no case there occurs a transition proper from the first kind 
to the second on a point-general surface continuously deforming itself. 
When the relation A,=O meets its fulfilment then we find that 
two umbilies of different kinds approach each other to disappear from 
the surface after the coincidence. 
b. The nodes of a point-general surface as 
twelvefold umbilics. 
6. When there is a node, the equation of the surface in its neigh- 
bourhood cannot be given in the form indicated in equation (1). After 
a fit choice of the axes we can however start from: 
ani +t byt dez 4+ A, fA, +....=20.... (9) 
or after a slight deformation, from: 
a+ Bx + By + Bz + aa? + by? Heet 4....=0. . (10) 
It is soon evident that to determine at first approximation the 
umbilies which appear in the neighbourhood of the place where 
formerly the node existed, the terms of order two are sufficient. 
So the surface may be treated there as a quadric, which immedia- 
tely makes the behaviour of the umbilies clear. If namely we have 
to do with an isolated point, made to appear after the gradual 
disappearance of a sheet, then at the very instant four real *) umbilies 
disappear, which were situated on that sheet, whilst eight others were 
imaginary and become so again after the disappearance of the sheet. 
If the node is a conical point then, when the two sheets are disunited, 
four real *) umbilics make their appearance, becoming imaginary at the 
union, whilst eight others again meet likewise for a moment in the 
node, but are previously and afterwards imaginary. For an imaginary 
node of course all the twelve umbilies coinciding there for a moment 
remain imaginary. 
The umbilics at injinity. General considerations. 
7. The umbilies are distinguished from the plaitpoints and many 
other singular points by the fact, that they cannot stand a projective 
1) These are at first of the third kind. They can, however, gradually pass during 
a continued deformation into those of the second kind without giving rise to the 
appearance of a double umbilic. 
2) Also for those holds good what was remarked in the preceding note, 
