592 
or 
zat Pet (Oe (ee 
Um,x— — Pmzs Un,y == — Um, y. Um,z— TT Omz ee © © (5) 
§ 4. We now assume 7 to be connected with a fixed system. 
It is required in the first place: 
to find “the locus of the points describing a point of inflexion in 
their paths — i.e. their paths relative to 7,” 
The projection of the acceleration of a movable point on the 
binormal of its path is always equal to zero; if also the projection 
on the principal normal is to be zero, it is necessary and sufficient 
that the radius of curvature of the path be infinite’), in other words 
that the point deseribe a point of inflexion in its path. A point 
describes therefore a point of inflexion when the velocity and the 
acceleration are equally directed, hence when À can be determined, 
so that 
Jar — Ag, 2 = JanS ay = dae hogs =e 
or, because we consider only points fixed to 7, so that 
Jn, a — Auma = Jin, y naz Avm, y= Jm,z A Omz 0. 
From this we find — see (16) and (3) — 
AN a Se _ 4, (a) 
— , = ee 
A(2) A(a) A(a) | 
where A, (a), A,(2), A, (2) and A(2) are functions of the third 
degree in A. 
The locus is therefore a twisted cubic. 
If we make OZ coincide with the instantaneous screw-axis, 
we have 
Eg 
Ome == Omy 10, Ome =F, 
f Ing = Eta — yt ee! 
Jmy = "IH = 
A= (5) + (5) | 
1) Here it is assumed that the point moves; the cases where at the moment in 
consideration either the whole fixed system or a line of points is at rest, might 
be treated separately (with little difficulty). 
