Mr. Airy on the Forms of the Teeth of Wheels. 281 
the circle CH, and another on the inside, making the same curves 
revolve on the inside and outside of CA respectively, and thus 
an infinite variety of curves may be found. The construction 
last mentioned gives forms approximating most nearly to the 
usual forms of teeth. We may even give different forms to dif- 
ferent teeth ; but this probably would not be desirable. 
It may be desirable to know when the nature of the teeth 
will admit of an alteration in the distance of the centers of the 
wheels. Suppose then DL and FP, (Fig. 6.) to be the principal 
circles when the wheels are in the first position; KS and HR, 
the principal circles when the distance of the centers is increased. 
Suppose in the first position C was in contact with Z, and M 
with O; suppose in the second position, G and Q are in contact 
with E and O; draw normals to all these points as in the figure. 
Since the wheels in the first position work correctly, by sup- 
position, the angles at D and N will equal those at Fand P. And 
if they work correctly in the second position, HG will = KE, &c. 
HR will = KS, and the angles at H and R will equal those at K and 
S. By attending to this condition, when the tooth ZO is given, 
we can always form a tooth CQ, which will work with it in two 
positions of the wheels. Since the angles at H and R equal those 
at K and S, the angles at L and T will equal those at F and P; 
and therefore will equal those at D and N. It is evident that this 
condition will always be satisfied, if CQ be the involute, and 
therefore if the teeth be involutes, the distance of the centers 
may be altered to any degree, allowing the teeth to act on 
each other. 
In all, however, that has yet been stated, we have only con- 
sidered the mathematical conditions of the contact of two curves. 
That these forms may be applicable in practice, it is necessary 
that the curvature of the convexity of one tooth, should be greater 
than that of the concavity of the other, or else that both should 
