the pitch or musical sound that generally accompanies it. When, however, a prolonged vibration is difficult, and the impulses brace each other up by frequent repetition, we have possibly in the first case a very short series, and in the other only one vibration. Such vibrations are as capable of rendering all complex acoustic combinations as vibrations of the push-and-pull kind.

It would be a matter of mere speculation to guess how the conditions of the vibrating helix are transferred to vibrating rods or discs of iron in an intermittent magnetic field. I would only say that the same double vibration is clearly traceable in them. To illustrate this in the case of rods, I took a small coil of No. 20 wire, 2 inches in diameter and about an inch high with a hollow axis of inch, and sent the interrupted current of a five Bunsen cell battery through it. Inside the axis I put a soft iron pin 2 inches long and inch square (fig. 10). To the upper end a fine copper wire was soldered to act as the thread of a telephone. When rightly placed the pin supported itself in the hollow, and kept dancing up and down symmetrically without much friction against the inside of the bobbin hollow. Here the mechanical motion was not so clearly eliminated as in the case of the helix; yet the ticking was heard, and it alone, when the motion of the pin was stopped by the hand. The impossibility of stopping the ticking of the pin was

Fig. 10.

shown by securing its ends between the jaws of a vice and making it as tight as possible, when the vice itself took up the tale of electric interruption, and made itself heard all round. A curious change was observed in long iron rods when this coil was placed round the middle of them and when shifted to the end. In the former position the sound was a stuccato rendering of the longitudinal note of the rod, and in the latter this sound was lost in a dull tapping. In

the latter position there was a pronounced tendency to push and pull.

I adopted a similar arrangement with the vibrating disc of the telephone. To the middle of one I cemented an india-rubber tube to act as a yielding handle, and the paper telephone wire was soldered near the middle. The disc was made to move in front of a telephone core excited by the coil just named. The movements of the disc were very violent, and made in all directions, making the connecting string jingle loudly, so that the isolation of the ticking sound was not so satisfactory as in the two previous cases. Still it was heard, and when the motions of the disc were kept in one direction its loudness did not grow with the extent of the motion. With a fine coil and a water cell the ticks were distinct enough; but the mechanical displacement was too small to yield the required comparison. The impossibility of stopping this ticking was illustrated in the following way :-A ferrotype plate was held tightly between two thin pieces of plate glass, the space between being filled up with sealing wax so that the whole was a solid mass of glass and wax. This was brought near a core excited by a water cell, when the sound was loudly rendered. Another illustration to the same effect was that of cementing by sealing wax the ferrotype plate of a telephone to a disc of thick microscopic glass, and putting this in the telephone with the glass side to the ear or mouth. Its articulate functions, though much impaired, still continued.

Lastly, to test whether the tick in a coil was due to electric conduction, I screwed a pin into the core of the telephone so as to act as a prolongation of the core; round this I placed a coil of fine wire, to which the string of a paper telephone was attached. There was no ticking heard so long as the circuit of the coil was broken; but the moment it was closed the ticking began. The coil was, of course, clear of the core. At the same time, however, there was mechanical action between the coil and core, illustrating the difficulty in such cases of determining by direct observation whether the single mechanical pull may not also make itself heard.

In conclusion, I would say, by way of summing up the evidence of this paper, that at the sending station the evidence of molecular action, though suggestive, is by no means conclusive; while at the receiving station the existence of molecular as well as mechanical

action amounts to demonstration. How the actual performance of the receiving instrument is to be apportioned between these, it is, of course, difficult to say. Taking into account Professor Tait's calculations as to the infinitesimal strength of a current that can make a telephone tick, and assuming that that tick is purely molecular, as we have done, molecular action must be there not the less considerable.

2. Sketch of the Arrangement of Tables of Ballistic Curves in a medium resisting as the Square of the Velocity, and of the Application of these Tables to Gunnery. By EDWARD SANG.

The motion of a body in a medium whose resistance is proportional to the square of the velocity, has been the subject of many inquiries. Its intimate connection with the theory and practice of gunnery has produced for it the attention of almost every cultivator of the higher analysis; but, like several other seemingly elementary problems in mechanics, it has hitherto received no complete solution.

If nothing else than the fluid's resistance influence the motion the investigation is comparatively easy; thus, taking the time as the primary variable, the

velocity at any future time; or backwards, what those had been at previous times. But since the velocities are inversely proportional to the times elapsed from some fixed epoch, it follows that, 4 P

VOL. IX.

at that epoch, the velocity must have been infinite, so that although the body may have come from an infinite distance, setting out therefrom with an infinite velocity, it must have begun that motion a finite time ago.

If we represent time by distances measured along OF, one of the asymptotes of a hyperbola, the ordinates, such as Pp, drawn parallel to the other asymptote, are proportional to the corresponding velocities. Thus the present velocity being Pp, that at the future time OA will be Aa, while that at the former date OB had been Bb; and the areas BbpP, PpaA represent the distances passed over during the intervals of time BP and PA. The distance corresponding to the finite previous time OP is thus infinite, and so also must have been the velocity of projection at the date O.

When the motion is affected by some influence other than the resistance, the investigation becomes more intricate. The case of a constant gravitation in a fixed direction is the simplest of these complications, and the simplest case of this is when the directions of the motion and of gravitation coincide.

If a stone be thrown straight upwards, its motion is impeded both by its weight and by the air's resistance; in the subsequent descent the motion is accelerated by gravity, but retarded by the air; so that, for the ascent, the soliciting influence takes the form g+cv2, and for the descent becomes g - cv2. Now the change in the sign of the velocity from +v in the ascent to v in the descent, is not accompanied by any change in the sign of v2, and therefore both parts of the motion cannot be represented by any one algebraic formula. Accordingly we find the upward motion to be represented by circular functions; the downward motion by the corresponding catenarian ones.

In fig. 2, the left hand row of dots represents the upward motion graduated to equal intervals of time. The stone is first shown at A as having come from some indefinite distance below; its speed, rapidly diminished, is altogether extinguished at N. In order to avoid confusion, the descent is shown on the adjoining right hand column of dots.

In descending, the acceleration due to gravity becomes less and less; it would cease altogether if the velocity could become so great as to cause a resistance equal to the weight; the tendency, there.

fore, is to reach a definite terminal velocity, and the stone ultimately If it had been

moves almost uniformly.

thrown downwards at a rate greater than this

terminal velocity, its motion would have been retarded, but less and less so as it approximated to the same limit of uniform motion.

We may study the ascent by tracing it backwards from the highest point, fancying the air to have then the quality of hastening the motion. In this case the velocity would increase to become infinite; but this infinite velocity would be acquired in a finite time. In fact, the time being represented by a circular arc, the velocity would be proportional to the tangent of that arc, so that in the time corresponding to a whole quadrant, the velocity would become infinite. Thus it seems that, however rapidly a stone may be thrown upwards, its motion is extinguished within a finite time determined by the coefficient of resistance.

Each particular body has its own terminal velocity depending on the weight and on the extent and peculiarities of the surface exposed to resistance; but the motions of all follow exactly the same law, so that one diagram may serve for all, the units of comparison alone needing to be changed.

5

1,0

Fig. 2.
N

.5

• 1,0

• 1,5

Also, one table of the positions and velocities may be made to do for all cases. In the arrangement of such a table we have to seek for the most convenient system of units; now, on contemplating the motion of a projectile independently of our measures of time and distance, we perceive that the terminal speed is the only standard with which we can compare the velocities at the various parts of the path, wherefore wo adopt this terminal velocity as the tabular unit of speed.

A

• 2.0

Ꮓ