Fig. 35.
recognisable in any way from their constructive form. Even the
ratio between the pitch radii R and R l does not give us the
relative diameters of the throats
of the axoids. The wheels which
I have elsewhere described* as
hyperboloidal face gear, form
a case in which the twisting
motion becomes very visible.
Here one axoid is a circular cone,
the other a plane hyperboloid,
or one in which the generator moves
always in a plane normal to the
axis of rotation (Fig. 36). The slid- ing of the instantaneous axes as
they pass the line of contact is
here very distinct, especially near the vertex of the cone,
We have seen that all axoids belong necessarily to one particular
class of geometric forms, viz. ruled surfaces, that pairs of
such surfaces can, therefore, express all possible motions. We
are therefore justified in considering these, in preference to any of
the other geometric forms above mentioned (p. 80), as the general represen- tatives of the motions occurring in machines. The most general charac- teristic of the relative motions of the axoids is their rolling. This exists even in the special case where of the two motions constituting the twist the
turning becomes infinitely small and the sliding only remains, for
the latter may itself be considered as a turning about an infinitely
distant axis,t and is therefore only a particular case of cylindric
rolling.
- Der Constructeur; 3rd Edition, p. 451.
t This is the special case of the proposition given on p. 61, where the two positions P Q and P l Q l are parallel, and where therefore the normals to P P l and Q Q-^ intersect only at infinity.
G 2
Fig. 36.
