376 APPLIED MECHANICS 
acting surface of the addendum is called the face, and the acting surface 
of the root is called the jlank. Circles con- 
centric with the pitch circles, and passing 
through the tops and bottoms of the teeth, 
are called the addendum circle and root circle 
respectively. In the case of an internal wheel 
the addendum is inside and the root is outside 
the pitch surface. 
In a mortice wheel (Fig. 576) the teeth are Fig. 576. 
made of wood, and have tenons formed on them 
which fit into mortices in the rim of the wheel. The teeth in this case 
are called cogs. The wood used is generally hornbeam or beech. 
319. Ordinary Proportions of Teeth.—The following proportions 
represent average practice for cast-iron teeth. Pitch =p=are ABC (Fig. 
575). Thickness = AB = 0-47p. Width of space=BC=0°53p. Total 
height=e=0°7p. Height beyond pitch line=a=0°3p, Depth within 
pitch line=c=0°4p. Width=2p to 3p. For heavy mill-gearing the 
width is sometimes as great as 5p. 
The cogs of mortice wheels have a thickness = 0°6p, and the iron teeth 
which gear with them have a thickness=0-4p, so that there is no side 
clearance when the teeth are new. 
320. Frequency of Contact of a Pair of Teeth.—If N, and N, be 
the numbers of teeth on two wheels A and B which gear with ne 
another, then the ratio of their angular velocities is as N, is to N,. Let 
m, and n, be the quotients got by dividing N, and N, respectively by their 
greatest common divisor, then if a particular tooth on A gears with a 
particular tooth on B, the same pair will again come in contact after My 
revolutions of A and m, revolutions of B. Also one tooth on A will in 
turn gear with n, teeth on B, and one tooth on B will in turn gear with m, 
teeth on A. For example, if 'A has 60 teeth and B has 20, the same pair of 
teeth will come in contact after every revolution of A or ‘after every three 
revolutions of B. Also a particular tooth on A will come in contact with 
only one particular tooth on B, and a particular tooth on B will come in 
contact in turn with three particular teeth on A. If the number of teeth 
on A be increased to 61, the velocity ratio will be altered to a small 
extent only, but the same pair of teeth will now only come in contact 
after 20 revolutions of A or 61 revolutions of B: Also each tooth on 
one wheel will now come in contact in turn with every tooth of the other 
wheel. The extra tooth added in this case is called a hunting tooth or 
hunting cog. The effect of the hunting cog is to cause the teeth to wear 
more uniformly. 
321. Condition to be Fulfilled by the Curves of the Teeth of Wheels 
in order that they may Work correctly.—Two toothed wheels, in gear 
with one another, are said to work correctly when the ratio of their 
angular velocities is exactly the same at every instant as that of their 
pitch surfaces working in rolling contact without slipping. 
In Fig. 577, O, and O, are the centres of two toothed wheels whose 
pitch lines PQ and. PR are in contact at P. The shaded curves represent 
portions of two teeth, one on each wheel, which are in contact at the 
point ab, a being the point on the tooth of the one wheel which is in 
contact with the point 2 on the tooth of the other. In geometry it is 
