(326) The assumption that the point =A= is stationary was only made to illustrate the point at issue, and is not founded upon the actual facts of the case. If now it be assumed that not only the barrel but the point at which the chain is held makes a forward movement, a new set of conditions arises. In this case the unwinding of the chain during a given time will be diminished by the amount of the advance of the point =A= in the same period. Assuming the latter to be made at a regular rate it would be easy to calculate the extent of the unwinding. If the effect of the horizontal movement of the barrel from =B= to =C= be to unwind half of one coil of chain—say a length of 7 inches—and that in the same space of time the point =A= moved 3 inches, the amount unwound would be reduced to 4 inches. But this is not the actual condition of things in practice. The point moves at a variable velocity, its forward motion gradually diminishing, so that the acceleration of the rotary velocity of the barrel is greater at the end of its horizontal traverse than at the beginning. In other words, its terminal velocity is highest.
(327) The point of the attachment of the chain at =A= is made in an oscillating arm which, during the inward run of the carriage, receives a forward movement at a speed which is controlled by the velocity of the back shaft. As the latter is, in turn, commanded by the scroll shaft during this period—see paragraph 315—it follows that the variation in the forward movement of the arm is coincident with that of the carriage. Thus the advance of the point =A= will always be in strict correspondence with the velocity of the carriage traverse.
(328) Referring now to Fig. 167, and, assuming =A B= to be the arm to which the chain is fastened, and =O J= and =H C= to represent the arcs through which the point of attachment of the chain travels at different times, it will be seen that the periods of movement are well marked. In each case the arcs are of the same number of degrees, although the chord of one is shorter than that of the other. Dealing first with the inner arc, which represents the position of the point of attachment when nearer the centre, the whole period of movement is divided into equal parts. These are represented by the letters =J K L M N O=. Now, if vertical lines are drawn from these, until they terminate in a straight line drawn parallel to a horizontal line through the point =B=, a clear idea can be formed of the effect of the oscillation of the vertical arm =A B=. The lines terminate at =J^{ 1} K^{ 1} L^{ 1} M^{ 1} N^{ 1} O^{ 1}=. It can be easily seen that the horizontal movement of the point of attachment of the chain gradually becomes less as the arm is oscillated from its most backward position =B C= to its most forward one =B H=, this diminution occurring most after the point =L= is reached. In the movement from =J= to =K= and =K= to =L= the horizontal traverse is about equal. It shows a decrease from =L= to =M=, a greater one from =M= to =N=, and a still greater one from =N= to =O=. The same thing happens if the chain be supposed to be attached at the point =D=. In this case also the decrease in the horizontal forward traverse is variable, but occurs in the same way. The periods here are marked by the letters =C= to =H=, and the extent of the forward motion by those =C^{ 1}= to =H^{ 1}=. It will be noticed that the amount of the traverse is greater than that previously noted, the total space covered being respectively =J^{ 1}= to =O^{ 1}= and =C^{ 1}= to =H^{ 1}=. That is to say, the point at which the chain is fastened moves forward in the same direction as the barrel, but at a different speed. In other words, when the chain is held at =K=, the total forward movement is comparatively small, and if it were held at a point shown by the small inner circle, it would be still less. On the other hand, its attachment at =B= implies a greater total forward movement. It therefore happens that the retardation of the chain by the arm is less in the early part of the oscillation of =A B=—or, to put it differently, the delivery of the winding chain by the arm is greater when it is fixed at =D= than when it is fixed at =K=. Therefore the barrel is more slowly rotated during the same period in the former than in the latter case, but as it completes its lateral movement it is rapidly and considerably accelerated.