Mr. Atry on the Forms of the Teeth of Wheels. 279 
To prove this let 4B, (Fig. 2.) be the first curve, 4C the 
second ; from the points C and E, which are very near, draw the 
normals CD, EF; if a describing point P be taken, and PQ, PR, 
be made respectively equal to CD, EF, and QR equal to DF, and 
this process be continued, a curve will be formed, which by re- 
velving upon Bd, will, by the describing point P, generate the 
curve AC. For if Q coincide with D, then R will afterwards 
coincide with F, and so on for all succeeding points, since QR = DF. 
Also DC= QP, &c. And the angles made by these with the 
tangents are equal. For the cosines of these angles, drawing DG, 
: Su bez ; 
QS, perpendicular to EF, PR, are = and ee in which the 
numerators are the differences of equal lines, and the denominators 
are equal. Hence P will describe 4C. And the formation of 
the curve RQ is always possible, because RQ is greater than 
RS; for FD is necessarily greater than FG. As an example of 
this, suppose it were required to find the curve, which revolving 
on one straight line 4B, (Fig. 3.) would generate another straight 
line 4C. Since the angles made by the line PQ with the tangent, 
must be constant, it follows, that the curve weuld be the loga- 
rithmic spiral, P being its pole. 
The entire theory of the teeth of wheels, may now be included 
in this proposition. If the tooth HD, (Fig. 4.) be generated by 
the revolution of any curve on the outside of the circle HC, and 
if DK be generated by the revolution of the same curve in the 
same direction, in the inside of the circle KC, then the normal 
at the point of contact of the teeth, will pass through C. For 
let the generating curve be brought to the position LC, so as to 
touch the cirele HC at C; DC will be the normal of HD at D; 
and that the teeth may be in contact, the same generating curve 
in the other circle must touch AC at C; in which case it will 
coincide with this; D therefore will be in the surfaces of both 
