an exact revolution. This hand shaft, d, instead of having the change wheels g g' applied to it and to the end of the feed screw, operated an auxiliary shaft, d'. The hand shaft having the locking device ef was provided with a gear of 22 teeth, and the intermediate shaft d', upon which the change wheels were applied, had a gear of 21 teeth. It is obvious that the intermediate shaft d' would revolve faster than the other, and the result was that this screw 8, which was inch pitch, would feed the blank to cut a rack tooth to correspond with a wheel of 6 to the inch pitch. That 6 to the inch pitch is equal to this decimal of an inch, .5236. The change wheels g g' were connected by an intermediate gear a, as in screwcutting lathes. Now, the reason why that ratio of 21 to 22 is correct arises from this fact, that what we call the "diametral" pitch of "6 to the inch" corresponds to the circumference of a circle 1 inch in diameter. A circle 1 inch in diameter has 3.1416 circumference; of that is .5236 of an inch. The pitch, therefore, of a tooth "6 to an inch" in pitch is a little over an inch, and the reason why the gears come 22 to 21 in proportion is because, if you assume the diameter of a circle to be inch, which is the pitch of the screw s, the circumference of this half-inch circle would have to be divided into three parts, to get the same figure, .5236. Now of a circle inch in diameter is the same as of a circle 1 inch in diameter, that is, .5236, and the ratio of these gears is in the ratio of the diameter of the circle and of its circumference. Now, the diameter of the circle we will assume is 7, and the circumference being 22, of the circumference would be 73. The ratio between 7 and 7 is 21 to 22. By using change wheels of equal diameter upon this intermediate shaft, which revolves in a suitable proportion, the rack carriage would be moved suitably to cut a pitch of six to one inch, and by altering these change wheels, as you do in a screw-cutting lathe, you could move this screw faster or slower, but always in this peculiar ratio. You would always get the diametral pitch as the feed of the screw. By putting on other change wheels, you could get teeth of larger diametral pitch, the change of rotation between the hand shaft h and the screw 8 being in the proportion of 22 to 21 all the time.

}

One consideration has occurred to me in respect to the use of this system for cutting worm wheels, and that is, that in my experience a worm wheel never operates on a worm the same as a spur wheel operates on another spur wheel, but the divergence of the lines of

the teeth from the center of the wheel, which is of no consequence in spur wheels, can be compensated for in the case of a worm, where a straight line is opposite to those two radial lines, by diminishing the pitch of the spur wheel in some measure. It has been a very successful practice in shops I have been connected with, where the pitch is under inch, to cut the teeth by the Manchester rule of diametral pitch by regarding the exterior of the wheel as the pitch circle, and it has proved very convenient. The worm teeth match the worin quite accurately where they are obtained by that rule instead of assuming the true pitch circle as the proper line.

I would like to mention that the rack-cutter I have referred to is manufactured in Newark, N. J., by Gould and Eberhardt, and is their own invention.

Mr. S. W. Powel.*-As to the ratio of transforming gears for rack-cutting machines, it is immaterial whether you use a ratio of 7 to 22 or 21 to 22, provided you have the change gears necessary to arrange the machine for the work in hand.

Since the meeting at Scranton, one of the writers has seen a change-gear table for rack-cutter, made about ten years ago at the Pratt & Whitney Co.'s works in Hartford, Conn., in which a set of lathe change gears was used with the addition of one other gear, and which gives all the diametral pitches in common use.

Now, as to taking the outside diameter of a worm gear as the pitch diameter, we are convinced that this is not correct, and can give an instance in which we are supported by the practice of one of our best tool shops. A worm and gear were designed to run together, having bearings for both shafts bored in the same casting, and therefore not adjustable. By some mistake the gear was cut with one tooth less than the correct number, and it was a failure. Another gear was made, and, by order of the foreman who was building the machine, was cut with one tooth more than the correct number, and was likewise a failure. A third gear made and cut correctly is no doubt in the machine yet.

Where a small worm drives a gear with a large number of teeth under light duty, we are aware that one tooth more or less makes very little difference in the running of the mechanism. But where a large worm of quick pitch is used to drive a gear with few teeth, it is our opinion that the correct number of teeth will be found more satisfactory than an incorrect number.

* Author's closure, under the Rules.

CCCXXV.

THE MECHANICS OF THE INJECTOR.

BY J. BURKITT WEBB, HOBOKEN, N. J.

(Member of the Society.)

THE fact that the injector wastes no heat except a small amount by radiation, is usually accepted as proving that the instrument has a very high efficiency; when, however, we make a careful comparison of it with a good steam pump, which forces its water through a heater heated by exhaust steam, we may be surprised to have the latter come out the best. I desire to call your attention to a mechanical principle upon which the injector works, and to show that it is an unfavorable one, and one which accounts largely for the difference in favor of the steam pump.

If a mass of clay or putty be projected against an equal mass at rest it will set it in motion and the two united masses will move on with half the velocity given to the first mass; if, however, the projected mass contains but one-tenth, instead of one-half of the whole amount, the final velocity will be but a tenth of that of projection.

The principle governing such cases is called in mechanics the "conservation of the motion of the center of gravity," which means that the velocity of the center of gravity of the united masses is the same as the velocity of their center of gravity before they united.

In the first instance, the two masses being equal, their center of gravity lies always midway between them, and therefore moves along with half the velocity of the projected mass; after impact the center of gravity is in the center of the united mass, and as the impact does not alter its* velocity, we know at once what velocity the united mass must have. In the second case one-tenth of the mass being in the striking and nine-tenths in the struck mass, the center of gravity will lie nearest the latter and at a distance from it equal to one-tenth of the distance separating the two masses. The velocity of the center of gravity will therefore be one-tenth of the

* The velocity of the center of gravity.

velocity of projection, and consequently the masses after uniting will have a velocity of one-tenth that of projection.

=

In both these cases, supposing the first mass to be m, and the second to be ma, and representing the velocity of the first mass by v and that of the center of gravity by V, we find that before impact the energy ism1v2, while after impact it is only (m + m2) V. In the first case m1 = m, and V v, so that half the energy disappears at impact, being converted into heat by the blow and lost. In the second case, m1 + ma ten times m1, and Vis only one-tenth of v, consequently the energy after impact is but one-tenth of what it was before, or nine-tenths is lost by the blow.

Looking more closely into the condition before impact, we see that the energy consists then of two parts, viz.: the energy of the whole system of two masses, moving with the velocity V, and the energy with which the two masses approach each other, that is to say, we may calculate the energy on the principle that the pair of masses is moving forward with the velocity V of their center of gravity, and then that mass one has an additional forward velocity = Vin the first case, and 9 Vin the second, while mass two has an additional backward velocity Vin both cases, thus causing the latter mass to stand still and making the velocity of the first

mass V.

=

Having made this division of the energy, we find, as might be expected, that only the first part of the energy is preserved while the energy of approach is lost by the blow; and this holds for all bodies which are not sufficiently elastic to separate again after the blow is struck.

Now, in the injector, the water is almost at rest when it is struck by steam, moving with a high velocity, and thus set in motion. If the steam is, say, one-fifteenth of the water, the velocity of the mixture will be but one-sixteenth of that of the steam, and fifteensixteenths of the mechanical energy of the moving steam will be lost by the blow. This mechanical energy has been developed by allowing the steam to flow from the boiler into the vacuum chamber and thus to get up a high velocity, but, however economical such a method of generating mechanical power from steam may be, it is neutralized by the wasteful way of using the power, for impact is, as has been shown, a wasteful method. In this respect the injector is like a slowly moving impact water-wheel, where almost all of the kinetic energy acquired by the water in running down to

the wheel may be lost in heat when the water strikes and dashes into foam and yet in such a wheel, were it desirable to warm the water, it might be claimed that no energy was lost.

In the injector a greater part of the energy even than calculated is lost by the blow, from the fact that it is not struck exactly in the direction in which the water.is to move.

In reasoning upon the efficiency of the injector it is not enough to state that no heat is wasted, because there would be none wasted if the steam were condensed into a tank of water for the purpose of heating it, while if our object were to get mechanical power it would all be wasted, whereas in a proper engine we might get out of it the legitimate amount of power. The steam used by the injector is at boiler temperature, whereas the heat when returned is at feed-water temperature, and we should therefore charge against the injector the amount of power which a good engine working between these temperatures would develop from the amount of steam used by the injector, and not credit it with heating the feed-water, except so far as we might not be able to do so with exhaust steam.

DISCUSSION.

Mr. Wm. Kent.-Prof. Webb is no doubt correct, if we consider the injector as a means of raising water from one level to a higher one, but if it is used for feeding water into a boiler, say from a tank on or above the boiler level, then the injector has a perfect efficiency, less the heat lost by radiation, which, if the injector and pipes connected to it are felted, is almost nothing. In this case the efficiency is the same as that of a steam trap, which feeds a boiler without any expenditure of energy other than that necessary to open and close the valves, and loses no heat except that due to radiation from its external surface. As a pump for lifting water, the injector is very inefficient, but as a boiler feeder, its efficiency is almost perfect.

Prof. Jas. E. Denton. In this paper Prof. Webb gives a physical interpretation of the limits imposed upon the mechanical execution of an injector, which, I believe, is original with him, and is certainly a very welcome addition to any previous mathematical treatment of this subject. The discussion of the distribution of energy in an injector by equations of momentum is nothing new. All mathematical discussions of the instrument have presented