583 
With a view to this we remark, that the four barmonical points 
P,A,, P', A, of a curve g° from each of its chords s are projected 
by four harmonical planes. This must remain the case, if we let the 
e° degenerate into a conic &? and a straight line d. 
In the degeneration considered in $ 2, the points 4, and A, lie 
both on the conic £°. The following two cases are now possible: 
1. The point P lies also on the conic 4°. If we take as chord s 
a common secant of the conic £? and the line d, we see that also 
the point P' lies on #? and is harmonically separated from P by A, 
and A,. 
2. The point P lies on the line d. If we take the chord s in the 
same way, we see that the point P’ lies on &? and by 4, and A, 
is harmonically separated from the point of intersection D of the 
two components £* and d. 
To the point D', which is harmonically separated from D, all the 
points of the line d are therefore associated; for the rest there belongs 
to every point P of the degenerate 0° one definite other point P’. 
In the degeneration considered in § 3, the point A, lies on the 
line d, the point A, on the conic £° (or inversely). Two cases are 
again possible: 
1. The point P lies on the conic £°. If we again take as chord s 
a secant of &? and d, we see that the point P’ lies also on the conic 
k* and is harmonically separated from P by the points A, and D. 
2. The point P lies on the line d. If we take as chord s a straight 
line in the plane of the conie k°, we see that the point P lies also 
on the line d and is harmonically separated from P by the points 
A, and D. 
To each point of this degenerate 0’ belongs consequently a definite 
other point. If P coincides with D, the point P' does the same. 
If the g° is degenerated into three straight lines, considerations of 
the same kind hold good. 
§ 6. Singular points of the involution (P, P’). On every non dege- 
nerate g*® the points A, and A, are associated to themselves; it 
appears from the preceding § that this is also the case for the dege- 
nerate y*. These points are therefore not singular points of the invo- 
lution. On the contrary the points of the lines a,,a, and a, are 
singular points. Let us consider e.g. a point A, of the line a,. In 
order to find the point A,’ associated to A, on a curve @° passing 
through A,, we must bring through the bisecant a, of this curve 9° 
a plane which by the planes a,, and a,, is harmonically separated 
from the plane whieh touches the curve gy’ in the point A, and 
