Fig. 27.
In [Fig. 27] let a a and b b represent the pitch circles of two wheels of equal diameters, and therefore having the same number of teeth. On the left, the wheels are shown with the teeth in, while on the right-hand side of the line of centres a b, the wheels are shown blank; a a is the pitch line of one wheel, and b b that for the other. Now suppose that both wheels are capable of being rotated on their shafts, whose centres will of course be on the line a b, and suppose a third disk, q, be also capable of rotation upon its centre, c, which is also on the line a b. Let these three wheels have sufficient contact at their perimeters at the point n, that if one be rotated it will rotate both the others (by friction) without any slip or lost motion, and of course all three will rotate at an equal velocity. Suppose that there is fixed to wheel q a pencil whose point is at n. If then rotation be given to a a in the direction of the arrow s, all three wheels will rotate in that direction as denoted by their respective arrows s.
Assume, then, that rotation of the three has occurred until the pencil point at n has arrived at the point m, and during this period of rotation the point n will recede from the line of centres a b, and will also recede from the arcs or lines of the two pitch circles a a, b b. The pencil point being capable of marking its path, it will be found on reaching m to have marked inside the pitch circle b b the curve denoted by the full line m x, and simultaneously with this curve it has marked another curve outside of a a, as denoted by the dotted line y m. These two curves being marked by the pencil point at the same time and extending from y to m, and x also to m. They are prolonged respectively to p and to k for clearness of illustration only.
The rotation of the three wheels being continued, when the pencil point has arrived at o it will have continued the same curves as shown at o f, and o g, curve o f being the same as m x placed in a new position, and o g being the same as m y, but placed in a new position. Now since both these curves (o f and o g) were marked by the one pencil point, and at the same time, it follows that at every point in its course that point must have touched both curves at once. Now the pencil point having moved around the arc of the circle q from n to m, it is obvious that the two curves must always be in contact, or coincide with each other, at some point in the path of the pencil or describing point, or, in other words, the curves will always touch each other at some point on the curve of q, and between n and o. Thus when the pencil has arrived at m, curve m y touches curve k x at the point m, while when the pencil had arrived at point o, the curves o f and o g will touch at o. Now the pitch circles a a and b b, and the describing circle q, having had constant and uniform velocity while the traced curves had constant contact at some point in their lengths, it is evident that if instead of being mere lines, m y was the face of a tooth on a a, and m x was the flank of a tooth on b b, the same uniform motion may be transmitted from a a, to b b, by pressing the tooth face m y against the tooth flank m x. Let it now be noted that the curve y m corresponds to the face of a tooth, as say the face e of a tooth on a a, and that curve x m corresponds to the flank of a tooth on b b, as say to the flank f, short portions only of the curves being used for those flanks. If the direction of rotation of the three wheels was reversed, the same shape of curves would be produced, but they would lie in an opposite direction, and would, therefore, be suitable for the other sides of the teeth. In this case, the contact of tooth upon tooth will be on the other side of the line of centres, as at some point between n and q.
Fig. 28.