TOOTHED GEARING 
To understand the theory of the forms of the teeth of bevel wheels, it 
is desirable to refer again to the way in which the forms of the teeth 
tof wheels were derived. In Fig. 595, 
is the pitch circle of a spur wheel. 
aPR is the rolling circle which is used to 
describe the epicycloid ad, which is the 
_ of the face of a tooth on the wheel. 
pitch surface of the wheel is a cylinder, 
and if the rolling circle aPR be taken as the 
end of another cylinder, the two cylinders, 
being of the same length, and having their 
axes parallel, the face aa,b,b of a tooth on 
the wheel is formed by the straight line aa, 
on the surface of the rolling cylinder as the 
latter rolls on the pitch surface of the wheel. 
Fia. 595. 
385 
For a bevel wheel (Fig. 596), the pitch surface of which is the frustum 
ABB, A, of the cone OAB, the rolling cylinder of Fig. 595 becomes the 
frustum of a rolling cone, 
and the curve ab becomes a 
spherical epicycloid. The 
face of a tooth on the wheel 
is formed by a straight line 
on the surface of the 
rolling frustum as_ the 
latter rolls on the outside 
of the pitch surface of the 
wheel. The flank of a tooth 
is formed in like manner 
by a straight line on a roll- 
Fig. 596. 
ing frustum when the latter rolls on the inside of the pitch surface. 
As the cone OaPR (Fig. 596) rolls on the cone OAB, the point a 
which describes the curve ab is always 
at a distance from O equal to the 
length of the slant side of the cone 
OaPB; the point a therefore moves 
on the surface of a sphere whose 
radius is OA, and whose centre is at 
O. The surface of the outer ends of 
the teeth formed in this way on a 
bevel wheel is therefore a portion of 
the surface of a sphere, and cannot 
be developed. If, however, a cone 
be taken enveloping the sphere and 
having for its circle of contact the 
pitch circle AB, this cone will cut the 
true face of a tooth in a curve which, 
when developed, will for all practical 
purposes in ordinary cases be an epi- 
cycloid. Hence the practical method, 
due to Tredgold, of designing the forms 
of bevel wheel teeth, shown in Fig. 597. 
(°) 
Fiaq. 597. 
2B 
