595 
the same plane. For the locus in question we find therefore the 
surface represented by: 
Um,x Umy Um,z 
Im,z Tiny Jm, = 0. 
(2) (2) (2) 
mx my M,z 
The locus of the points the paths of which have at the considered 
moment a stationary plane of osculation, is a surface of the third 
order. 
In the same way we show that the locus of the points the paths 
of which have at the considered moment a contact of the fourth 
order with the plane of osculation, is represented by 
Um,ax Um, y Um, z 
Jina Jn, y Jm, 2 
(2) (2) (2) 
Ji Fin: y aa Zz 
Tat) Gms Sh 2 
i.e. to this locus belong the common points of the four surfaces of 
the third order of which the equations appear by successively omit- 
ting a ray out of the above-mentioned matrix. The required locus 
is therefore according to a well known theorem of the determinants 
the curve of intersection of 
Um, x Vin, y Un. z Um, x Um, y Un, z 
Jm, x Jm, y Jm, z == 0 and Jm, x Tmyy Im, z == 0 
(2) (2) (2) (3) (3) (3) 
m, X Jm, y Jm, 2 Jm, x Jm, y Jm, z 
provided we do not count the points defined by 
Um, x Un, y Vin, z 
Pee Jae diep 
Le. the cubic we found before as the locus of the points of inflexion. 
The common points of the four cubical surfaces form therefore a 
twisted curve of the sixth order. 
The locus of the points of which the paths at the considered moment 
have a contact of the fourth order with the plane of osculation, is 
a twisted curve of the sixth order. 
i=. 
§ 6. Let P(w,y,z) be a point of the fixed system connected to 
T, and P’ (w', y',z') the centre of the sphere of curvature of the 
path of P; in this case P’ has the characteristic property that the 
