393 KINEMATICS OF MACHINERY.
formed as it were a tooth common to both cones, the slot in the plate was the hollow in the double-faced wheel I in which it geared,* Davies soon noticed, however, that it was impossible thus to obtain a satisfactory joint between the slot and the diaphragm, because of the special spheric-cycloidol form of the teeth, and he seems then to have thrown the whole arrangement at once aside. The question as to what form the piston and the surface of the chamber would have if the motion between them were to be pure rolling is an interesting one. The forms are simply those of the axoids between the links b and d of the chain (C^C 1 -).
These axoids can be determined without difficulty from the well- known formula for the relative motions in the universal joint. In it the link a is fixed, and the links b and d have in turning such a relative velocity that if the corresponding angles of turning be in- dicated by &> and w v
tan G),
-- = cosct, tanw
where a is the angle between the axes of b and d, that is the angle of the link a.f From this we obtain the ratio of the angular velocities w and w l of the two axes of the expression
u \ _ cos a
10 ~~ 1 - siii 2 <o sin 2 a *
This formula expresses at the same time the relative distances of the instantaneous axis, that is the line of contact of the axoids, from the axes of the two shafts.^ If we now suppose that the latter, instead of being convergent, are parallel, the axoids become cylinders instead of cones, and the equation gives us the radii of these cylinders, and we have for the centroids, or normal sections of the axoids, curves of the form represented in Fig. 276. In the
w position shown the ratio is a maximum, when contact occurs
between B and B l it is a minimum, after the next turning of 90 there is again a maximum, and after a third similar motion another minimum.
- There is a wooden model of this machine in the Patent Museum at Kensington.
The patent is in the names of Taylor and Davies, 1836-8.
f See Rankine, Machinery and Millwork, p. 203, &c.
$ It will be remembered that the instantaneous axis must be in the same plane as the axes of the revolving shafts.
