386 APPLIED MECHANICS 
OO, is the axis of the wheel. ACDA, is one half of the pitch surface, 
O being the vertex of the pitch cone. OAOQ, is a right angle. O,AC is 
one half of the cone, already referred to as enveloping the sphere whose 
centre is at O, and whose radius is OA.. AE is an are of a circle struck 
from O, as centre. This arc is the development of part of the base of 
the cone O,AC. Then AE is considered as part of the pitch circle of a 
spur wheel “of radius O ,A, and the teeth are constructed on this as for a 
spur wheel. <A thin templet, made to the shape of the teeth on AE, 
may be used to mark off the shape of the teeth on ihe edge of the bevel 
wheel blank. 
The theory of involute teeth for bevel wheels may be developed in a 
similar manner to that of cycloidal teeth. In a spur wheel with involute 
teeth a plane is taken touching a base cylinder, and a line in this plane 
parallel to the axis of the cylinder describes the surface of a tooth as the 
plane rolls on the cylinder. In a bevel wheel the base cylinder of the 
spur wheel becomes a base cone whose vertex is at the vertex of the pitch 
cone of the wheel, and as a plane rolls on the base cone a line in the 
plane, and passing through the vertex of the cone, describes the surface . 
of a tooth on the wheel. Tredgold’s method is also applicable to ingolite 
teeth. 
When the diameter of a bevel wheel is mentioned without quulifin 
tion, the larger diameter of the pitch surface is understood. 
329. Stepped and Helical Teeth. —The smaller the pitch of the teeth 
of two wheels in gear the smoother is the motion, but the teeth are 
weaker the smaller the pitch. 'To combine the smoothness of the motion 
due to fine pitched teeth with the strength 
due to coarse pitched teeth, Dr. Hooke 
invented stepped teeth. These teeth are 
shown in Fig. 598. Imagine a toothed 
wheel having teeth of a pitch p to be 
divided into m dises of equal thickness 
by planes at right angles to the axis of 
the wheel, and let each dise be placed so Fra. 598. 
that the teeth on it are 1-nth of the pitch 
p in advance of the teeth on the disc in front of it. These discs would 
now form a wheel with stepped teeth, which would have the strength of 
teeth of pitch p, and which would work as smoothly as teeth of pitch p/n. 
If the number of steps on a stepped tooth be made infinite, its surface 
becomes a screw or helical surface, and the teeth formed in this way are 
called helical teeth. Simple helical teeth on a spur wheel have the 
appearance shown in Fig. 599. The out- 
line of the section of helical teeth by a 
plane at right angles to the axis of the 
wheel is designed as for ordinary teeth, 
and their outline in the direction of the 
width of the wheel is drawn by the rule 
for drawing helices or screw curves. It ; 
is obvious that two wheels gearing together Fig. 599. 
and having helical teeth must have their teeth of “ opposite hand,” that 
is, one must be right-handed and the other left-handed. It is also evident 
that the inclinations of the helices must be the same. . 
