602 
P’ of Of is the centre of the sphere of curvature which has five 
points in common with the path of a point P of On. 
CONCLUDING REMARK. 
In § 7 we remarked: 
If a point P describes a twisted curve y and the point P’ the 
curve which is the locus of the centres of the spheres of curvature 
of y in P, the condition that such a sphere has with y a contact 
of the fourth order is expressed by putting the velocity of P’ equal 
to zero. 
We shall briefly indicate the proof of this proposition. 
If we represent y by the equation developed in § 5 (footnote), 
and the sphere of curvature by 
1 
d 
© + y? + 27 — 2 hy — 2 it ena 
we find for the condition for a contact of the fourth order 
A ed las eo Re 
ora aur rs tau 
and for the arc described by P’ (see e.g. Brancui-Lukat: Vorlesun- 
gen iiber Differential-Geometrie, I, zweite Auflage, p. 25) 
R d dR 
de = 0,3 bh = 7 dT ; 
ds ds 
From var = 0 follows d, = 0 as aT = o is excluded. Inversely 
: i | 
d, is only equal to zero, when vg,’ is equal to zero, as for 57 0 
4 
Vax becomes generally infinite and d, differs from zero. It would 
lead us too far if we entered further into this. 
