To find the line of contact upon a given frustrum of the tangent-cone; let the [Fig. 166] be the plane of the frustrum; a the centre. Set off a e equal to the shortest distance between the axes (called the eccentricity), and divide it in c, so that a c is to e c as the mean radius of the frustrum to the mean radius of that with which it is to work; draw c p perpendicular to a e, and meeting the circumference of the conical surface at m; perform a similar operation on the base of the frustrum by drawing a line parallel to c m and at the same distance a c from the centre, meeting the circumference in p.
The line p c is then plainly the line of direction of the teeth. We are also at liberty to employ the equally inclined line c q in the opposite direction, observing only that, in laying out the two wheels, the pair of directions be taken, of which the inclinations correspond.
Fig. 167.
[Fig. 167] renders this mode of laying off the outlines of the wheels at once obvious. In this figure the line a e corresponds to the line marked by the same letters in [Fig. 166]; and the division of it at c is determined in the manner directed. The line c m being thus found in direction, it is drawn indefinitely to d. Parallel to this line and from the point c draw e to e, and in this line take the centre of the second wheel. The line c m d gives the direction of the teeth; and if from the centre a with radius at c a circle be described, the direction of any tooth of the wheel will be a tangent to it, as at c, and similarly if a centre e be taken in the line e d, and with radius e d, c e a circle be drawn, the direction of the teeth of the second wheel will be tangents to this last, as at d.
Having thus found the direction of the teeth, these outlines may be formed as in the case of ordinary bevel-wheels and with equal exactness and facility, all that is necessary being to find the curves for the teeth as described for bevel-wheels, and follow precisely the same construction, except that the square, [Fig. 162], marking the lines across the cones, requires to be set to the angle for the tooth instead of at a right angle, and this angle may be found by the construction shown in [Fig. 167], it being there represented by line d c. It is obvious, however, that the bottoms of the blocks to form the teeth must be curved to bed on the cone along the line d c, [Fig. 167], and this may best be done by bedding two teeth, testing them by trial of the actual surfaces.
Then two teeth may be set in as No. 1 and No. 6 in the box shown in [Fig. 148], the intermediate ones being dressed down to them.
