378 APPLIED MECHANICS 
motion of two teeth in gear will not be altered if one of the pitch circles 
is considered to be at rest and the other pitch circle is supposed to roll 
on the first. Let the pitch circle PQ (Fig. 577) be at rest, and let the 
_ pitch circle PR roll on PQ. The direction of the motion of the point 0 is 
perpendicular to Pb, because, in the position considered, J is rotating 
about P, and in order that the pure rolling of PR on PQ may not be 
interfered with, and in order that the two teeth in gear may remain in 
contact, the direction of the motion of b must be tangential to the curves 
of the teeth at aor 6. Therefore the common normal to the curves of 
the teeth passes through P. 
322. Cycloidal Teeth.—Let APB and CPD (Fig. 578) be the pitch 
circles of two wheels. Let the outline of the flank of a tooth on APB be 
a portion of the hypocycloid aQé, described by 
the rolling of the circle PQR on the inside of 
the pitch circle APB. Let the outline of the 
face of a tooth on CPD be a portion of the 
epicycloid:cQd, described by the rolling of the 
circle PQR on the outside of the pitch circle 
CPD. Next let the face cQ be brought round 
so as to touch the flank aQ, and let Q be the 
point of contact. The point Q must be on 
the rolling circle PQR when the latter touches 
both pitch circles, because the normal to the 
hypocycloid at Q must pass through the point of contact of the rolling 
Fria. 578. 
circle and the circle APB when the former is describing that part of the — 
hypocycloid at Q, also the normal to the epicycloid at Q must pass 
through the point of contact of the rolling circle and the circle CPD when 
the former is describing that part of the epicycloid at Q, therefore, since 
the two normals coincide, the rolling circle when it passes through Q must 
touch both pitch circles, 
Since the common normal to the curves aQb and cQd, at their point of 
contact Q, passes through the pitch point P, the wheels will work correctly 
if the faces of the teeth on one are epicycloids and the flanks of the teeth 
on the other are hypocycloids, described by the same rolling circle. 
It is evidently not necessary that the flanks of the teeth of two 
wheels which gear together be described by the same rolling circle, but 
_ the rolling circle which describes the flanks of the teeth on one wheel 
must be used to describe the faces of the teeth on the other. 
Since the hypocycloid becomes a straight line passing through the 
centre of the pitch circle when the diameter of the rolling circle is equal 
to the radius of the pitch circle, it follows that the flanks of wheel teeth 
may be made radial. 
If a number of wheels are to be interchangeable, that is, if any one 
of them is to be capable of working correctly with any of the others, 
it is obvious that the faces and flanks of the teeth on each must be 
described by the same rolling circle. 
323. Path of Contact.—In the preceding Article it has been shown 
that the point of contact of two cycloidal teeth must be on one or other 
of the rolling circles when the latter are at the pitch point; it follows, 
therefore, that the path of contact of two teeth must be made up of ares 
of these rolling circles. 
