“This arrangement is not mathematically perfect, by reason of the angular vibration of the lever. This is, however, very small, owing to the length of the lever; it might have been compensated for by the introduction of another universal joint, which would practically have introduced an error greater than the one to be obviated, and it has, with good judgment, been omitted.
“The gear-cutter is turned nearly to the required form, the notches are cut in it, and the duty of the pantagraphic engine is merely to give the finishing touch to each cutting edge, and give it the correct outline. It is obvious that this machine is in no way connected with, or dependent upon, the epicycloidal engine; but by the use of proper templates it will make cutters for any desired form of tooth; and by its aid exact duplicates may be made in any numbers with the greatest facility.
“It forms no part of our plan to represent as perfect that which is not so, and there are one or two facts, which at first thought might seem serious objections to the adoption of the epicycloidal system. These are:
“1. It is physically impossible to mill out a concave cycloid, by any means whatever, because at the pitch line its radius of curvature is zero, and a milling cutter must have a sensible diameter.
“2. It is impossible to mill out even a convex cycloid or epicycloid, by the means and in the manner above described.
“This is on account of a hitherto unnoticed peculiarity of the curve at a constant normal distance from the cycloid. In order to show this clearly, we have, in [Fig. 110], enormously exaggerated the radius c d, of the milling cutter (m of [Figs. 105] and [106]). The outer curve h l, evidently, could be milled out by the cutter, whose centre travels in the cycloid c a; it resembles the cycloid somewhat in form, and presents no remarkable features. But the inner one is quite different; it starts at d, and at first goes down, inside the circle whose radius is c d, forms a cusp at e, then begins to rise, crossing this circle at g, and the base line at f. It will be seen, then, that if the centre of the cutter travel in the cycloid a c, its edge will cut away the part g e d, leaving the template of the form o g i. Now if a roller of the same radius c d, be rolled along this edge, its centre will travel in the cycloid from a, to the point p, where a normal from g, cuts it; then the roller will turn upon g as a fulcrum, and its centre will travel from p to c, in a circular arc whose radius g p = c d.
“That is to say even a roller of the same size as the original milling cutter, will not retrace completely the cycloidal path in which the cutter travelled.
“Now in making a rack template, the cutter, after reaching c, travels in the reversed cycloid c r, its left-hand edge, therefore, milling out a curve d k, similar to h l. This curve lies wholly outside the circle d i, and therefore cuts o g at a point between f and g, but very near to g. This point of intersection is marked s in [Fig. 110], where the actual form of the template o s k is shown. The roller which is run along this template is larger, as has been explained, than the milling cutter. When the point of contact reaches s (which so nearly corresponds to g that they practically coincide), this roller cannot now swing about s through an angle so great as p g c of [Fig. 110]; because at the root d, the radius of curvature of d k is only equal to that of the cutter, and g and s are so near the root that the curvature of s k, near the latter point, is greater than that of the roller. Consequently there must be some point u in the path of the centre of the roller, such, that when the centre reaches it, the circumference will pass through s, and be also tangent to s k. Let t be the point of tangency; draw s u and t u, cutting the cycloidal path a r in x and y. Then, u y being the radius of the new milling cutter (corresponding to n of [Fig. 109]), it is clear that in the outline of the gear cutter shaped by it, the circular arc x y will be substituted for the true cycloid.
The System Practically Perfect.
“The above defects undeniably exist; now, what do they amount to? The diagram is drawn purposely with these sources of error greatly exaggerated, in order to make their nature apparent and their existence sensible. The diameters used in practice, as previously stated, are: describing circle, 71⁄2 inches; cutter for shaping template, 1⁄8 of an inch; roller used against edge of template, 11⁄8 inches; cutter for shaping a 1-pitch gear cutter, 1 inch.