TOOTHED GEARING 377 
shown that when two curves touch one another they have a common 
normal, Let M,M, be the common normal of the curves of the teeth at 
ab, and let O,M, and O,M, be the perpendiculars from the centres of the 
wheels on to this common normal. The point a moves in a circle aA 
Fia. 577. 
whose centre is O,, and the point / moves in a circle bB whose centre is 
O,. At the instant that the points a and 0 are in contact the point a is 
moving in the direction ac, the tangent to the circle aA at a, and the 
point b is moving in the direction ad, the tangent to the circle bB at b. 
Let v, and v, be the linear velocities of @ and U respectively in the 
directions in which they are moving at the instant when they are in 
contact. Make ac=v,andad=v,. Now, although the points a and dare 
moving in different lines with different velocities, the components of 
these velocities in the direction M,M, must be the same, otherwise the 
points a and } would move relatively to each other along the line M,M,, 
but for a small movement of the wheels so long as the teeth remain in 
contact the only possible relative motion of @ and 4 is in a direction © 
perpendicular to M,M,. Therefore if ce be drawn at right angles to 
M,M, it will pass through d, and ae=v will be the component velocity 
of a and also of } in the direction M,M,. Hegcee the ratio of the angular 
velocities of the two wheels must be 
go ~ 6 - OMe OS 
O,M, O,M, O,M, 0,8’ 
where § is the point of intersection of the lines O,O, and M,M,. But 
with rolling contact between the pitch lines PQ and PR the ratio of the 
angular velocities of the two wheels would be equal to os: Therefore if 
1 
os = * the point S must coincide with the point P, 
1 1 
The condition to be fulfilled by the curves of the teeth is therefore as 
follows. The common normal to the curves of the teeth in contact must 
pass through the pitch point, the pitch point being the point of contact of 
the pitch lines. 
Another way of proving that the common normal to the curves of the 
teeth should pass through the pitch point is as follows. The relative 
