600 
We can therefore also write (7): 
i) zel me oe 
(—2') vr aha) vp he ea vee =0.. 15.8 
that means, in the inverse motion P lies in the normal plane of the 
path of the point P’ fixed to 7’. 
Lf at any moment in the direct motion P’ lies in the normal plane 
of the path of the point P fixed to T,, then in the inverse motion 
P les in the normal plane of the path of the point P’ fixed to Ty. 
We have already seen that the condition for P’ to be the centre 
of the sphere of curvature of the path of P in the direct motion, 
is expressed by the equations (7) or (8); what are then the conditions 
for P to be the centre of the sphere of curvature of the path of 
P’ in the inverse motion? 
The equation of the normal plane of the path of P’ in the inverse 
motion is 
(Ke) omer + (Ly) omy + (Z—2) ome = 0; 2. (10) 
the centre of the sphere of curvature of the path of P’ is therefore 
defined by this equation and two more derived from it by differ- 
de’ dy’ “dz” 
dt’ dt’ dt 
entiation with respect to ¢; for the values must be taken 
which follow from 
Var! = Vay' = Var = 0 
for instance 
qa’ 
dt 
In order to express that P(w,y, z) is the centre in question, we 
must substitute 
= — 6 + gz! — ry). 
Kz an Say eee A 
But then (10) is transformed into (8a), and we have already seen 
that in the way indicated before the equations (86) and (8c) appear 
from (8a) ($ 7). Hence: 
If P’ is the centre of the sphere of curvature of the path of P 
in the direct motion, P is the centre of the sphere of curvature of 
the path of P’ in the inverse motion, in other words the cubical 
transformation is reversed together with the motion. 
We can go one step further. 
The locus of the points P fixed to 7’, the paths of which relative 
to 7’; — hence in the direct motion — have a contact of the fourth 
order ($ 7) with the spheres of curvature in P, is at any moment 
