TOOTHED GEARING 381 
Let APB and CPE (Fig. 582) be the pitch circles of two wheels with 
involute teeth in gear with one another. Let aTb and cTe be the out- 
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Fig. 582. 
lines of the surfaces of two teeth in contact at T, these outlines being 
involutes of the base circles 5, and §, respectively. Since a line drawn 
from any point on an involute to touch the base circle of that involute 
is a normal to the involute at that point, it follows that the common 
normal to the two involutes in contact at T must be a common tangent 
MN to the two base circles. Hence the point of contact is always on 
the line MN, and a portion of that line is the path of contact. 
Comparing the similar triangles O,PM and O,PN, it is clear that 
if the ratio of the radii of the base circles be the same as the ratio of the 
radii of the pitch circles, the common normal to the curves of the teeth in 
contact must pass through the pitch point. 
If the centres of the wheels be pushed closer together or further 
apart, the wheels will still work correctly, because this is equivalent to 
altering the radii of the pitch circles without altering their ratio. This 
is a special property of involute teeth, and is a valuable one in cases 
where the distance between the centres of the two wheels cannot be 
maintained constant. This property also makes it possible to regulate 
the amount of side clearance or back lash between the teeth. Altering 
the distance between the centres of the wheels obviously alters the 
inclination of the path of contact. The angle 6 which the path of 
contact makes with the common tangent to the pitch circles is usually 
from 14} degrees to 154 degrees. In designing involute teeth the 
direction of the path of contact is first fixed, and the base circles are then 
drawn to touch it. 
If on the tangent at P, PH be made equal to the pitch of the teeth, 
measured on the pitch circles, and if HK be drawn perpendicular to MN, 
then since PK: PH::0,M:0O,P, PK must be the pitch of the teeth 
measured on the base circles. The pitch PK is called the normal pitch. 
Tt is usual to make the parts of the path of contact on opposite sides of 
P equal to one another, then if two pairs of teeth are to be in contact, 
and PL be made equal to PK, KL will be the minimum length of the 
path of contact, and circles through K and L with centres at O, and O, 
respectively will be the minimum addendum circles. There should be a 
