282 Mr. Arry on the Forms of the Teeth of Wheels. 
be convex. For this purpose we must investigate the curvature 
at any point. 
Take then two points on the circle near each other, and 
the two points of the generating curve which will touch them; 
join these with the center of curvature of the generating curve, 
and with the describing point; let ¢, 0, ¥, (Fig. 7.) be the small 
angles at the center of curvature, the describing point, and the 
center of the circle; suppose the lines from the describing point, 
when in contact with the circle, to be produced respectively, and 
let the angle at their point of intersection = x. Also let a and 
B be the angles which those lines make with the radii of the 
circle. Then we shall have 
OP =a BY Xe ig ee Nt 
But calling R the radius of the circle, r the radius of curvature, 
s the distance of the describing point, z the distance of the point 
of intersection, 
are are are . cos a are. cosa 
= —: —————— SS = ——_——_ 5 
ie R?’ . rT s > Xx x 
cosa _1 1 cos a. COs a 
ee fee ed ; bie COS a 
Rt s 
Lo eail 
Rt? 
r+s=s = rad. of curvature of tooth; 
1,1 _ cosa 
Ro rT s 
Lae COs a COS a 
Lee” or s 1 ° 
“. curvature = = .———_—__ = - — é 
$s aye $s ies ie 
Ri ir Roo? 
From an examination of this expression, it appears, that 
when «a is < 90°, r may be positive or negative, but must be 
less than the radius of the circle in the same direction; when 
