594 
We remind *) that the distance from the point (w + Az, y+ Ay, 
A 3 
zt Az) to the plane of osculation at P is equal to + (Geet e). 
1 1 ard 
where — and — represent the curvature and the tortuosity in Pand 
R iM 
e 
As? 
approaches to zero at the same time with As; a stationary plane 
of osculation appears therefore only when is equal to zero (i.e. 
1 
RT 
besides at the points of inflexion there, where the tortuosity is equal 
to zero). 
We calculate the projections see jen JP of the acceleration of 
the second order on the tangent, the principal normal and the binormal. 
If a, a,,@, are the cosines of the angles which these make with the 
fixed X,-axis, we find from: 
v? dv 
Jz == 0, R + Lara 
by the application of the formulae of FRENET—SERRET 
2 d dv v Sov ww ah v® 
De =al— —— Fa Se Te 
aU leet dt? __R? INR dt BR dt aOR 
Hence 
d'v v® Sv dv v' dR v? 
B, SOG Cart) EN VEREEN 
herma za 
Nd 
BL 
In the motion relative to /y of a fixed system connected with 
i bs is is therefore equal to zero in those points where the plane 
of osculation is stationary and inversely, because — at least in the 
general case — no points appear where v,, is equal to zero. 
The velocity being directed along the tangent and the acceleration 
(of the first order) lying in the plane of osculation, this plane is 
stationary at those points and at those points only where the velo- 
city, the acceleration and the acceleration of the second order lie in 
1) See e.g. L. P. EisENHART : Differential Geometry (Ginn and Co, Boston) p. 21, 
Ex. 10. If we lay the axes OX, OY, OZ along the tangent, the principal normal 
and the binormal at an ordinary point of the curve, this can be represented for 
sufficiently small values of s by: 
s* s4 Zn Meik s dR slr da? a 1 1 5 
BR ERG ITR GR de ZldeR RET IR 
s? stal dal Te dl 
AS he A burrie 
GRT „24M de BT T ds R i 
LZS 
