The drawing-in of the carriage in a sense causes the other two operations to be performed. With respect to the second of these, viz., revolving the spindles and thus winding the threads upon them, it may be said this action causes what is termed the "Winding Chain" to pull off a small drum of six inches diameter, thus rotating the latter and thereby the spindles. Here, however, comes in now the action of the very beautiful and effective piece of mechanism, "Roberts' quadrant" (see Fig. 26). The winding chain just mentioned is attached to one extremity to the arm of the quadrant, and the peculiar manner in which the quadrant moves in relation to the winding drum gives the variable motion to the spindles that is required. When commencing a new set of cops it may take about eighty revolutions of the spindles to wind on the 64 inches of thread to each spindle, representing one stretch. The bare spindle may be about a quarter of an inch in diameter, but it may finally attain a diameter of an inch and a quarter (_i.e._, the cop upon the spindle). This cop will only require about twenty revolutions to wind on the 64 inches, which are only one-fourth of the revolutions necessary for the empty spindles. It is the action of the quadrant which gives this variation in speed to the spindles during winding-on. But as has been pointed out previously, the quadrant imparts a "differential winding" motion to the spindles in two distinct and different ways, and the second motion is even more important than the first. It is necessary for practical purposes that the cop of yarn should be built up of a conical shape in the upper part, as shown in the illustration. Now it must be obvious to the least technical of the readers of this story, that to wind a given portion of yarn upon the thin apex of a cone, will require a greater number of revolutions than would be necessary to wind the same length of yarn upon the base of the same cop. All the way between the apex and the base of the cone are also other varying diameters, and during each return movement of the mule carriage the thread is wound upon all the varying diameters of the cone in succession. This implies the necessity for the revolutions of the spindles to a varying quantity all the time of the return or inward movement of the spindle carriage. The quadrant gives this varying speed in a manner which is all but mathematically correct, any slight deviation from any such mathematical correctness being easily compensated for in other ways. For the specific manner in which this quadrant works, the reader is referred to any of the recent text-books on cotton spinning. The third primary and important operation, which takes place during each return movement of the carriage, is the guiding of the thread upon the spindles in a correct manner. This operation is closely associated, however, with the action of the quadrant.