278 Mr. Airy on the Forms of the Teeth of Wheels. 
the line perpendicular to the surfaces of the teeth, at the point 
of contact, intersect the line joining the centers at a fixed point, 
which divides that line into two parts, the ratio of which is the 
mechanical power. When this holds, the proportion of the angular 
velocities will be constant. For let 4 and B (Plate XV. Fig. 1.) 
be the centers of the wheels, C the point through which the line 
of action passes: D the point of contact: upon moving the wheels 
with the teeth still in contact through a very small angle, D in 
one tooth will be carried to F, and in the other to G, FG being 
ultimately parallel to the tangent at D, or perpendicular to CD, and 
DF, DG, perpendicular to AD, BD respectively. Then, 
BC AC. 
BD AD’ 
ee : FD GD 5 
therefore the angular velocities, which are as AD * BD’ will be 
FD : GD:: sn G : sn F :: sm BDC: sin ADC :: 
as BC : AC, a constant ratio. If then with centers 4 and B circles 
be described passing through C, and these circles revolve so as 
to make the velocities of their circumferences equal, the teeth of 
the wheels, if properly formed, will be in contact, and the normals 
to both will pass through C. These circles we shall call the 
principal circles of the wheels. 
If the normals from every point of the tooth should be equally 
inclined to the tangents of the circle at the points where they meet 
the circle, they evidently would if produced be tangents to a circle, 
whose radius : radius of circle described :: cosine of inclination 
of normal with tangent of circle described : 1. In this case both 
teeth would be involutes of circles. If the inclinations are not 
equal, we must make use of the following theorem. It is always 
possible to find a curve which by revolving upon. a given curve, 
shall by some describing point, in the manner of a trochoid, 
generate a second given curve: provided that the normals from 
all points of the second curve meet the first. 
