598 
fixed to T,, and the centre P’ of the sphere of curvature which has 
a contact of the third order with the path of P relative to Ty, are 
conjugated to each other. 
§ 7. We directly find again the locus of the points the paths of 
which have at the considered moment a stationary plane of oscula- 
tion; they are those points P of which the conjugated points P’ 
lie at infinity, hence the points defined by 
Um, x Um, y Um, z 
Jm, x Jm, y Jm, z == 
Juz Jay Jm 
Now we require 
the locus of the points P the paths of which have with the circles 
of curvature a contact of the third order (i. e. four points in common). 
To such a point P not one single point P’, but a line of points 
P’ is conjugated; they form the singular points of the transfor- 
mation through which the points of the space (P) are transformed 
into those of the space (P’). This locus is therefore defined by: 
Um, x Un,y Vin, z = Er 
| 
(i | 
Im, Im,y Jin, z SI Pe Ei v | 
2 7 ds (3) 
Fim Jab Fee 328 67 En > & © 
In the same way as in $ 6 it appears that: 
The points of the system fixed to T,, the paths of which have with 
the circles of curvature a contact of the third order (four points in 
common), form a twisted curve of the sixth order. 
Let it also be required to find those points P of which the paths 
have at P a contact of the fourth order (five points in common) 
with the spheres of curvature. 
If a point P is to belong to this locus and if P’ is to be the 
centre of the sphere of curvature, we have’) 
Vq, x! = Va,y' = Va, = 0. 
The coördinates of P must therefore satisfy besides (8) also the 
equations resulting from them by differentiation and by the substi- 
tution of 
de, dy' 
OTN DERK 
es ee ee), ae (y + rv — pe), 
' See the concluding remark. 
