382 APPLIED MECHANICS 
small clearance between the root circle of one wheel and the addendum 
circle of the other. 
If the parts of the path of contact on opposite sides of the pitch point 
_ are equal, and if there are two pairs of teeth always in contact, then PM 
or PN, whichever is least, will be the maximum value of the normal 
pitch of the teeth. Let r=radius of the smaller of the two base circles, 
p=maximum normal pitch, and »=the minimum number of teeth, then 
2rr=np, but p=r tan O, therefore nr tan 0=27r, and n=2z7/tan 0. 
When 6=15°, m=24. With only one pair of teeth in contact at a time, 
n=/tan 0, or n=12 when 0=15°. 
When the pitch circle becomes of infinite diameter, as in a rack, the 
base circle will also become of infinite diameter, and the involute will 
become a straight line. Hence in a rack 
which gears with a wheel having involute 
teeth, the teeth are straight on face and 
flank, as shown in Fig. 583. The faces 
and flanks are perpendicular to the path 
of contact, and therefore make an angle : 
of 90° — 6° with the pitch line. TGs, FE: 
The essential condition that two wheels, or a wheel and rack, 
having involute teeth, may gear correctly together, is that the teeth shall 
have the same normal pitch. Two or more wheels having different 
numbers of involute teeth of the same normal pitch can be arranged to 
rotate about the same axis and gear correctly with one wheel or one 
rack. The base circles of the wheels on the same axis will of course be 
of different diameters, and the paths of contact will be inclined at | 
different angles.* 
326. Internal Teeth.—The theory of the forms and the methods of 
drawing the outlines of the teeth for internal or annular wheels in which 
the teeth are on the inside of the rim, as shown in Figs. 584 and 585, 
are the same as for external teeth. 
In the case of cycloidal teeth (Fig. 584) the face ab is a hypocycloid 
of the pitch circle ABC described by the rolling circle which describes 
Ui 
NW 
A’ ; 07 
Fig. 584. Fig. 585. 
the hypocycloidal flanks of the teeth on the wheel which is to gear with 
ABC, and the flank dc is an epicycloid of the pitch circle ABC described 
by the rolling circle which describes the epicycloidal faces of the teeth of — 
the other wheel. 
In the case of involute teeth (Fig. 585) the curve adc is the involute 
of a base circle which must be concentric with the pitch circle ABC, and 
which must touch the straight line, which is the path of contact. 
327. Pin Wheels. — When the rolling circle which describes the 
hypocycloidal flank of a tooth on a wheel A has a diameter equal to that 
* Except when the single wheel becomes a rack, in which case the paths of 
contact are inclined at the same angle. 
