Page:The Kinematics of Machinery.djvu/105

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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

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