Fig. 192.
[Fig. 192] (from Willis’s “Principles of Mechanism”) is another method of constructing the same combination, which admits of a steady support for the shafts at their point of intersection, a being a spherical bearing, and b, c being cupped to fit to a.
Rotary motion variable at different parts of a rotation may be obtained by means of gear-wheels varied in form from the true circle.
Fig. 193.
The commonest form of gearing for this purpose is elliptical gearing, the principles governing the construction of which are thus given by Professor McCord. “It is as well to begin at the foundation by defining the ellipse as a closed plane-curve, generated by the motion of a point subject to the condition that the sum of its distances from two fixed points within shall be constant: Thus, in [Fig. 193], a and b are the two fixed points, called the foci; l, e, f, g, p are points in the curve; and a f + f b = a e + e b. Also, a l + l b = a p + p b = a g + g b. From this it follows that a g = l o, o being the centre of the curve, and g the extremity of the minor axis, whence the foci may be found if the axes be assumed, or, if the foci and one axis be given, the other axis may be determined. It is also apparent that if about either focus, as b, we describe an arc with a radius greater than b p and less than b l, for instance b e, and about a another arc with radius a e = l p-b e, the intersection, e, of these arcs will be on the ellipse; and in this manner any desired number of points may be found, and the curve drawn by the aid of sweeps.
“Having completed this ellipse, prolong its major axis, and draw a similar and equal one, with its foci, c, d, upon that prolongation, and tangent to the first one at p; then b d = l p. About b describe an arc with any radius, cutting the first ellipse at y and the line l at z; about d describe an arc with radius d z, cutting the second ellipse in x; draw a y, b y, c x, and d x. Then a y = d x, and b y = c x, and because the ellipses are alike, the arcs p y and p x are equal. If then b and d are taken as fixed centres, and the ellipses turn about them as shown by the arrows, x and y will come together at z on the line of centres; and the same is true of any points equally distant from p on the two curves. But this is the condition of rolling contact. We see, then, that in order that two ellipses may roll together, and serve as the pitch-lines of wheels, they must be equal and similar, the fixed centres must be at corresponding foci, and the distance between these centres must be equal to the major axis. Were they to be toothless wheels, if would evidently be essential that the outlines should be truly elliptical; but the changes of curvature in the ellipse are gradual, and circular arcs may be drawn so nearly coinciding with it, that when teeth are employed, the errors resulting from the substitution are quite inappreciable. Nevertheless, the rapidity of these changes varies so much in ellipses of different proportions, that we believe it to be practically better to draw the curve accurately first, and to find the radii of the approximating arcs by trial and error, than to trust to any definite rule for determining them; and for this reason we give a second and more convenient method of finding points, in connection with the ellipse whose centre is r, [Fig. 193]. About the centre describe two circles, as shown, whose diameters are the major and minor axes; draw any radius, as r t, cutting the first circle in t, and the second in s; through t draw a parallel to one axis, through s a parallel to the other, and the intersection, v, will lie on the curve. In the left hand ellipse, the line bisecting the angle a f b is normal to the curve at f, and the perpendicular to it is tangent at the same point, and bisects the angles adjacent to a f b, formed by prolonging a f, b f.