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 BhpP, 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+c2, 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 2, 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 5 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.

Fig. 2.
N

1,0.

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

This terminal velocity has to be considered along with the intensity of gravitation, which intensity, if acting alone, would generate velocities proportional to the times; wherefore the time in which the heavy body, falling freely, would acquire the terminal velocity, presents itself as the proper tabular unit of time, and, as a necessary accompaniment, the tabular unit of distance must be that over which the body would pass with its uniform terminal velocity in the time during which that terminal velocity was acquired; or, in other words, must be double the height of the free fall needed for the acquisition of the final velocity.

The accompanying table has been constructed for these assumed units; the details of the rise are given on the left hand, those of the fall on the right hand side, the times being reckoned before and after the instant of culmination::

Opposite the time 13 in the first column, we find the velocity 3.60210, this means that if the stone be thrown upwards with the

velocity 3.60210, as at A in fig. 2, it will reach its highest point in the time 13, the height to which it will rise being 1.31864, as shown in the third column. The subsequent fall is shown in the fourth column, and there we find that the same level is reached just before the time 20; the whole time of flight being somewhat less than 3.3.

To translate this into ordinary measures, let us suppose a body whose terminal velocity is 320 feet per second; the time in which this velocity is acquired in free fall is 10 seconds, and therefore all the tabular times must be multiplied by 10, the tabular velocities by 320, and the linear distances by 3200. Hence if such a body were thrown upwards with a velocity of 1152 feet per second, it would rise in 13 seconds to a height of 4230 feet, and would thence descend in 1993 seconds, striking the ground with the velocity of 308 feet per second.

This table enables us to interpret easily the results of experiments made on falling bodies. Thus, if the height and the time of descent be accurately observed, we may thence deduce the terminal velocity; from which, again, knowing the weight and the extent of surface, we may discover the constant of the coefficient of resistance. In order to facilitate the computations for this class of experiments, we may annex another column to our table, namely, one containing the ratio of the time of free fall to the actual time of fall. Thus the fall through the distance 132500, which is done in the time 2.0 with resistance, would be accomplished in 162788 if there were no resistance due to velocity, and the ratio, as shown in the annexed column, is 81394.

Suppose now that a body has been dropped from a height of 400 feet, and that the observed time of the fall is 6 seconds; the time of falling freely from this height is known to be 5 seconds, and therefore the ratio is 83333. The table shows that this ratio belongs to the tabular time 1.8271, and that, consequently, 3.284 seconds is the time in which the terminal velocity is acquired in falling freely; that terminal velocity, therefore, must be 105 feet per second.

In experiments of this kind, the disengagement at the beginning. and the stroke at the end of the fall may be made, by help of the