( 394 ) 
choice of axes when the origin is placed in one of its umbilies, into 
the finite form: 
z=, (# + y°) + hive + hye + ke? Ee (OU) 
Bringing the value of z into the second member this furnishes 
the development in series 
ze, (#7? + y’) + khee? + hem y + hoeey* + hoe,y? + . (12) 
Comparing this to (1) it is immediately evident that for the 
umbilies on a quadrie we always find d, =d,, d, =d,, so K, < 0. 
Furthermore the cubic (7) passes into (d‚n — d,) (n° + 1) == 0; so 
K, <0. From this it is evident, as indeed is known, that on a 
quadrie never other umbilics than those of the third kind can appear. 
From this ensues again immediately that on a quadrie no common 
double umbilics can appear. Indeed beside the nodes the only 
possible multiple umbilies at finite distance on a quadric are the 
vertices of a surface of revolution; but these are fourfold umbilies 
whose occurrence on surfaces of higher order would demand more 
than one relation between the coefficients of the equation. So it is 
not astonishing that for such vertices the lines of curvature bear 
themselves in an entirely deviating way. 
13. Passing now to the umbilies of quadrics at infinity we observe 
that the case given sub ec appears for paraboloids. If, however, we 
regard more closely the section with the plane at infinity, then this 
is evidently degenerated into two right lines. Each of these right 
lines meets the curve of the spherical points in two points. If we 
make tangent planes to appear in those points, then also there the 
section of the tangent plane degenerates, namely, into one of the 
recently considered right lines and into another. These two must at 
the same time be regarded as the tangents of the section of the 
tangent plane. One of these tangents therefore always happens to lie 
in the plane at infinity and we are in case //. 
To the fourfold umbilie at infinity four single umbilies are in this 
way added for the paraboloid. For finite distances four such points 
only are thus left, which furnishes here the proof to the sum. 
Inversely case d requires as is easy to see, at least for quadrics 
with real equation, that these should pass into surfaces of revolution. 
There is then double contact of the surface and the curve of the 
spherical points. Indeed in this case four umbilies pass into infinity; 
the eight remaining ones coinciding four by four in both vertices. 
The remaining case e cannot make its appearance for quadrics. 
The case f has just been discussed. It can as is easy to see make 
its appearance for quadries only in the manner indicated there. 
