Page:The Kinematics of Machinery.djvu/180

 curvature of that element, and from the centre of curvature of the corresponding element of bP a circle with a radius smaller by the same amount, we obtain two circular arcs touching each other on the normal at P v and having their normal passing through the instantaneous centre 0, in common with the elements at P of the original profiles. This procedure, carried out for every point in the two profiles, furnishes two new profiles, a^ P l and\ P v which are equidistantsor parallels to the first curves, and may equally

. ill.

. 112.

serve as profiles for elements.* This gives us a further immense variety of profile forms, which are only limited by the conditions as to usefulness mentioned in section 31. These equidistant curves often possess advantages, as in the case when they are employed to represent the point-profiles already described. They give us then a circle or circular arc for a profile instead of a point. We have this applied practically in the "pin" teeth of "lantern" pinions, which once were frequently used, and even now are occasionally seen. Fig. 112 is an example. Instead of the point a and the epicycloid a b, the circle of radius a a^ and the line a^\ equidistant from the epicycloid are employed.

As a further illustration we may employ the forms already treated in another way, the curve-triangle and square. In Fig. 113 02' 3' is the centroid of the one and 0234 that of the other element of a higher pair, whose profiles we wish to determine. Using the method of 34 we place in 2' 3' an auxiliary centroid of the same figure as itself. We choose a describing point at B, a point

profile with radii equal to the difference between the original and the intended radii, and not by drawing arcs from the centres of curvature of each element with the actual increased or decreased radii in the- way described. The method of the figure gives the same result as the latter, and is obviously much more simple.
 * In the figure the equidistants are found by drawing arcs about points in the