If, however, the axes of the shafts are at any other angle we may find the distance the lathe tool must travel per lathe revolution to give teeth of the required angle (or in other words the pitch of the spiral) by direct proportion, thus: Let it be required to find the angle or pitch for wheels to connect shafts at an angle of 25°, the wheels to have 20 teeth, and to be of 10 diametral pitch.
Here, 20 ÷ 10 = 2 = diameter of wheel at the pitch circle. The circumference of 2 inches being 6.28 inches we have, as the degrees of angle of the axes of the shafts are to 45°, so is 6.28 inches (the circumference of the wheels, to the pitch sought).
Here, 6.28 inches × 45° ÷ 25° = 11.3 inches, which is the required pitch for the spiral.
When the axes of the shafts are neither parallel nor meeting, motion from one shaft to another may be transmitted by means of a double gear. Thus (taking rolling cones of the diameters of the respective pitch circles as representing the wheels) in Fig. 190, let A be the shaft of gear _h_, and B _b_ that of wheel _e_. Then a double gear-wheel having teeth on _f_, _g_ may be placed as shown, and the face _f_ will gear with _e_, while face _g_ will gear with _h_, the cone surfaces meeting in a point as at C and D respectively, hence the velocity will be equal.
When the axial line of the shafts for two gear-wheels are nearly in line one with the other, motion may be transmitted by gearing the wheels as in Fig. 191. This is a very strong method of gearing, because there are a large number of teeth in contact, hence the strain is distributed by a larger number of teeth and the wear is diminished.
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.
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.