In performing this operation by actual measurement of the lines, I have had in view to trace a path for those of my readers who may not have the tables, or may be unaccustomed to use them. The process, generally, is to take the diameter of any bevil wheel O [fig. 4], in the middle of it’s face; and supposing it a spur wheel, to find it’s plate by the method above given: and then to multiply the length of that plate by the line A r and divide the product by the line A w, both measured on the same scale of equal parts.
It may be well to observe, likewise, that the same method of finding the plates, applies to bevil wheels of every description or angle: but that it does not give equal plates for every pair, except in the above case of wheels placed at right angles to each other.
I would just remark that by the [figure] near B, is shewn a section of the Machine on which I centre the wheels to be cut on this Engine. It is an inverted cup s t, into which the arbor is screwed in a true position; and this cup is fixed on the top of the shaft A B, by the three pressure screws near s t, which enter a triangular neck made round the shaft, against the upper slope of which, the screws press so as to draw the cup downward in the act of centering it. This I say is my present method; but it is in a measure accidental, the shaft not having been perforated to receive arbors of the usual kind. Mine, however, have their utility in the ease with which they are varied in size, and changed on the Machine: but on their comparative usefulness I give no opinion. The other is the most solid method.
In the description of my differential Steel-yard, (see [page 163]) I stated that the load P was wholly collected in the point o; and that dividing the line A C by the line A o, the power of the Machine was known. But I should have shewn that this line (A o) is equal to one half the difference between the arms A D and A E. To do this, here, (see [Plate 23], [fig. 4]) I take the Machine in the state of infinite power, before mentioned; and observe, that in moving the point of suspension from o towards A, I at once lengthen the arm A E, and shorten the arm A D: by which process, (supposing each arm to have been called a) that which I lengthen by any quantity d becomes a + d, and that which I shorten by the same quantity becomes a - d, and the difference of these quantities, is 2d: so that the line A o is in reality one half the difference between the two arms A D and A E as was required to be shewn.
But we may go a step further: The two arms of the equibrachial lever x y may likewise be made unequal: and the line s a be subdivided in any ratio: which division will augment still more the power of this Machine. If for example, we hang the load on the point v, halfway between a and s, that power will be doubled; for the line c v (representing the space moved through by the load in this case) is only one half of that w s, or o q, and might be still less at pleasure. Thus the whole power of the Machine is now found by dividing the length of the long arm, beyond D, by the line a v, instead of the former line A o, or dividing the motion of it’s extremity upward, by the line c v, the motion downward, of the load P.
It has been further suggested, that the description of my excentric Bar Press was not sufficiently explicit. I have therefore added the [figure 2] of [Plate 22], to assist in elucidating that description. I had, perhaps made an undue use of the principle of virtual velocities by saying, too concisely, ([page 174]) that “as the whole approaches toward B C, the relative motion (of the cheeks s and B) becomes insensible, the circles parallel, and consequently, the power infinite.” It is however vulgarly said that power cannot be gained without losing time—which implies that if time is lost, power will be gained: and the principle of virtual velocities says the same thing, though in more appropriate terms—that if a small movement be given to a system of bodies actually counterpoising each other, the quantity of motion with which one body ascends, and the other descends perpendicularly, will be equal: so that, as remarked in [page 50], by “whatever means a slow motion is obtained, dependent on that of a moving force, the power is great in the same proportion.” Now, in the eccentric Bar Press, (see [fig. 2]) this is so in an eminent degree: for when the bars are in the position A B, the distance of the cheeks is equal to B s; and they must move, circularly, as far as A f, to bring them closer to each other by the quantity s a: dividing therefore, the distance B g by the line s a, we find (near enough for practice) the power of the Machine within the limits A g B. It is nearly as 10 to 1. In like manner this power at A e g, is equal to the arc e g divided by the line f b; and at A l n to the arc l n divided by the line d k, namely by the difference of the lines k l and m n. From the above it appears that the nearing motion of the cheeks of the press, becomes slower and slower as the bars A and C come nearer to the point C: insomuch that the difference between the lines m n and o p is nearly imperceptible, and that between the lines o p and C q entirely so. But according to the above process, the distance p C should be divided by this imperceptible line, to find the power of the press at the point C; which therefore is immense. Another proof of this may be drawn from the supposition (see [fig. 3]) that the small lever a d is turned round the centre o by a bar o C fixed to it, and of equal length with the line A C [fig. 2]. [Fig. 3] shews that the lines or bars C d, and a C are moved endwise by the circular action of the points a and d; and therefore (by statics) their motion is the same as though caused by the perpendiculars b o and o c let down from the centre o, on each of them. Hence the power of this Machine is found by dividing the distance o C by the sum of the lines b o and o c; which sum (when these lines vanish by the union of the bars over the centre) becomes infinitely small: the quotient of which division therefore is infinitely great—as was to be shewn.