4 ∫√1 − a2(x − a) dy0 2a √1 − a2 = arc cos a√1 − a2 = 1,
which is the formula used to calculate our table above.
As to the weight balls can safely carry in any bearing, below will be found results of experiments and calculations made by Professor Robinson, of the Ohio State University. This article is the result of careful, exhaustive work, and I am under great obligations for the privilege of introducing it here, as it has never before been in print.
“To find the load which a single hardened steel ball will safely carry in any ball-bearing, either when running between two flat surfaces or between two equally grooved surfaces of hardened steel, in each case the following formula may be applied,—viz.: Load in pounds = 190 d2 √1 + dd′ − d, where d equals the diameter of the ball in inches and d′ equals the diameter of the groove in which the ball runs, either top or bottom. For flat surfaces, for top and bottom bearing of ball d’ = ∞ and dd′ − d = 0, so that, for a ball between hardened flat plates, Load = 190 d2. For n balls in a nest, all in equally fair bearings, the load equals n 190 d2 √1 + dd′ − d; for example, a one-inch ball between flat surfaces will carry one hundred and ninety pounds safely.
“Again a one-half-inch ball will carry 1904 = 47.5 pounds; and again a one-inch ball in a groove of one and one-eighth-inch diameter top and bottom will carry 190 √1 + 11′ (8 − 1) = 570 pounds. So that there is great advantage in supplying grooves for the balls to run in. Again, suppose the ball be one inch and the grooves one and one-eightieth inches in diameter; then the load equals seventeen hundred and ten pounds. Again, if the ball is one-half-inch diameter, and the groove nine-sixteenths-inch diameter, the load equals 142.5 pounds, etc.
“Hundreds of experiments in all were made on this subject, and the above formula deduced by theory was found to agree almost exactly with the experimental results for hardened steel for balls and track for same. When a much greater load than the above is attempted to be carried, the balls will indent a groove of their own until the necessary bearing surface is obtained.
“I am not aware that the coefficient of friction for ball-bearings is definitely known. Experiments made with the Lick telescope, in which the weights of some parts had to be guessed at, gives .00175 for the value of friction coefficient for one-inch balls; but this, though the best I have, is not a reliable figure. It is for hardened steel on hardened steel.”
Mr. Robinson here shows an advantage in the groove so far as capacity for resisting strain is concerned, but he would hardly construct a ball-bearing with grooves fitting the balls after a careful perusal of our section on grooves and friction.
As to ball-heads to bicycles, they have been highly recommended by a few makers and much admired by some riders. As before said, the balancing of the bicycle is accomplished by means of the steering apparatus, and the easier the head swivels the less work the rider has to do to effect his object. If simple steering—that is, changing the general course of the rider’s progress—happened to be all for which the head is swivelled, it would make little difference whether it moved very easily or not; nay, it would be better to have it move a little stiff, since it would then stay in place. But when it comes to balancing, the head is constantly moving, and every resistance is work to be overcome by the rider’s muscular exertion. To say that a head cannot swivel too easily, would be a valid axiom in the art of balancing; hence a ball-head could do no harm, and might do some good. In the Rover or Safety pattern, ball-heads are quite common and are rather a valuable acquisition, especially in the telescope. In the Stanley head, however, it is very questionable whether the advantage gained is sufficient to justify the extra complication and weight of the parts. Conical heads can be, and are, made to work so smoothly and the amount of motion is so small that the same question in regard to friction does not apply as in the case of other bearings about the machine. It is the opinion of the writer that every other part about a wheel should be about perfect, and of the very highest grade, before the question of ball-heads should be considered at all.
In regard to the patents on, and general use of, ball-bearings in cycles, I think the necessity of using the prominent lateral adjusting bearings is really not so absolute as many suppose; of course this is the most artistic form and the most easily-made pattern of all, and is in every way adapted to cycle use; but it would not be policy to throw aside any other advantage in a wheel to gain the lateral adjustment in the bearings. There are some other styles of ball-bearing boxes which answer the purpose very well, the chief difficulty being that a greater amount of work is necessary for their adjustment. If the boxes are split in a plane through the geometric axis of the axle, they will be slightly out of round after adjusting, but when it is taken into consideration that the weight is all on one side or, as in a bicycle, on the top, the fault will not be noticeable; it is more serious when the boxes revolve than when they are stationary.
The patents now existing on lateral adjusting bearings have caused many attempts at other methods of taking up the wear. The validity of these patents is questioned by many, and considerable litigation has been the result, though in many cases makers prefer to use other devices to running the chance of a law suit. The happy medium adopted by others is to pay the royalty demanded; this is, perhaps, the best course to pursue if the said royalty is not made burdensome. Every maker, however, should assure himself, by special examination, if his particular bearing really infringes any patent before paying; the fact of it being a ball-bearing with a lateral adjustment is not an incontrovertible reason that it should infringe, since both of these elements are, in themselves, old. It is only a special ball-bearing with a special adjustment that is patented. Unhappily, however, the special adjustment is a screw. How the patent will stand, time alone can tell; its validity is certainly questionable.
A word here in regard to paying royalties in general. Makers are too scrupulously averse to such payments, even when small, and buyers have the idea that any one who pays a royalty is naturally working at a disadvantage. This is not necessarily the case. Some would save more by the use of an ingenious machine for making the parts than several times the royalty often amounts to. In the manufacturing business there are so many ways of saving and losing money, that unless a careful watch is kept all round the little matter of royalty on some one part will fall into insignificance as compared with other leaks.
The advertisement of a maker that he pays no royalty gives us but little assurance that he can make a better machine for less money. When a patent is evaded by slight changes, such, for instance, as the increase or decrease of an inch in the diameter of a wheel, it shows not so much a great shrewdness on the part of the pirate as a frailty in the patent; this sort of evasion of royalties is considered to be perfectly legitimate, however, and means that either the attorney who took out the patent was incompetent, or that there was but little invention to be claimed.