It seems to me that we have here a new principle for making integrators which might be called linkage integrators; viz., if we take a linkage and move one point T along a given curve, then any other point T' will describe another closed curve whose area is dependent on that of the given

curve.

As a simple example, take a Peaucillier cell with fixed pole O (fig. 18). Give QT a wheel W and QT' a wheel W'. We have now two planimeters, OQT and OQT'. If OT=r, OT'=r', then we have always rr'=k2, where k2=OQ2-QT2.

If T describes a closed curve, T' describes another, and

(T)=aw, (T')=aw', where QT=a

2do if 0 is the angle which OT makes with a fixed line, and

Also (T)

(T) = $fr2dt

Hence, as T describes a closed curve, T' describes another whose area is proportional to

If ds denotes an element of the area (T), the last integral becomes by Stokes' Theorem about the conversion of a line- into a surface-integral

The origin must be outside the area (T) to avoid r=

=0.

On Methods that have been adopted for Measuring Pressures in the Bores of Guns. By Captain Sir A. NOBLE, K.C.B., F.R.S., M.Inst.C.E.

[Ordered by the General Committee to be printed in extenso.]

THE importance of ascertaining, with some approach to accuracy, the pressures which are developed at various points along the bores of guns by gunpowder or other propelling agent is so great that a variety of means have been proposed for their determination, and I purpose, in this paper, to give a very brief account of some of these means, pointing out at the same time certain difficulties which have been experienced in their employment, and the errors to which these methods have been in many cases subject.

The earliest attempt, by direct experiment, to ascertain pressures developed by fired gunpowder was that made by Count Rumford in his endeavour to determine the pressures due to different densities of charge. He assumed, the principles of thermodynamics being then unknown, that charges fired in a small closed gun-barrel would give pressures identical with those given by charges doing work both on the projectile and on the

products of combustion themselves; but even this error was a small one compared with that which led him to adopt, as correct, his extravagant estimate of the pressures developed.

For a density of unity—or, in other words, for a charge approximately filling a chamber in which it was fired-he estimated the pressure at over 101,000 atmospheres, or at 662 tons per square inch.

He adopted this pressure notwithstanding the great discrepancy which he found to exist between the two series of experiments which he made, and he meets the objection that, were the pressure anything approaching that which he gives, no gun that ever was made would have a chance of standing by assuming that the combustion of powder is exceedingly slow, and lasts the whole time occupied by the projectile in passing through the bore.

It is sufficiently curious that a man so eminent for his scientific attainments as was Rumford should have fallen into so great an error, both because any attempt at calculation would have shown him his mistake, and because Robins, sixty years earlier, had conclusively proved that with the small grain powders then used-and it must be remembered that Rumford's powder was sporting of very fine grain-the whole of the powder was fired before the bullet was very greatly removed from its seat. Robins's argument--and it is incontrovertible-was, that were it otherwise a much greater energy would be realised from the powder when the weight of the projectile was doubled, trebled, quadrupled, &c. ; but his experiments showed that under these circumstances the work done by the powder was nearly the same.

For other objects, on a much larger scale, and with appliances far superior to those which the great man I have named had at his disposal, I have had occasion to repeat Robins's experiment, and the results are interesting. With a charge of 10 lb. of the powder known as R.L.G2 and a shot weighing 30 lb. a velocity of 2,126 f.s., representing an energy of 971.6 foot-tons, was attained. The same charge being used, but the weight of the projectile being doubled, the velocity was reduced to 1,641 f.s., while the energy was increased to 1,125 foot-tons. With a shot weighing 120 lb. the velocity was 1,209 f.s. and the energy 1,196 foot-tons. With a shot of 150 lb. the velocity was 1,080 f.s. and the energy 1191-5 foot-tons; while with a shot of 360 lb. the velocity was reduced to 691 f.s., representing a muzzle energy of 1,191.9 foot-tons. These energies were obtained with maximum chamber pressures respectively of 13.5 tons, of 17.25 tons, of 19 tons, of 20 tons, and of 22 tons per square inch. It will be noted that the maximum energy obtained was realised with the shot of 120 lb. weight, the energy given by a shot of 360 lb.-i.e. three times that weight, or twelve times the weight of the original shot-being nearly exactly the same.

Very different, however, were the results when one of the modern powders, introduced with the special object of insuring slow combustion, was compared with the R.L.G2 experiments which I have just quoted.

With brown prismatic or cocoa powder an exactly similar series was fired. The 30-lb. shot gave a velocity of 1,515 f.s. and an energy of 493-4 foot-tons; the 60-lb. shot gave 1,291 f.s., and an energy of 693-4 foot-tons; the 120-lb. shot, 1,040 f.s., and 877.5 foot-tons; the 150-lb. shot, 948 f.s. and 920-7 foot-tons; while with the heaviest shot, the 360 lb., the velocity attained was 654 f.s., equivalent to an energy of 1,064.7 foot-tons. The maximum chamber pressures in this series varied from 4.8 tons per square

inch, with the lightest projectile, to 96 with the heaviest ; and with this powder it will be observed that the energy developed increased steadily and considerably with each increment in the weight of the shot, while the low chamber pressure shows that, even with the heaviest shot, the projectile must have moved a considerable distance from its seat before the charge can be considered to have been entirely consumed.

I have mentioned the discrepancy between Rumford's two series of experiments. This discrepancy was very great, the one series giving, for a density of unity, a tension of about 190 tons per square inch, or 29,000 atmospheres, the other series giving a tension of over 101,000 atmospheres. It is remarkable that Rumford makes no attempt to explain this discrepancy, but, as he deliberately adopts the higher tension, it is not improbable that he was led to this conclusion by an erroneous estimate of the elastic force of the aqueous vapour contained in the powder or formed by its explosion. He considered, relying on M. de Betancourt's experiments, that the elasticity of steam is doubled by every addition of temperature equal to 30° F., and his only difficulty appears to have been-he expressly leaves to posterity the solution of the problem-why the tension of tired gunpowder is not much higher than even the enormous pressure which his experiments appeared to indicate.

It will be remembered that Rumford's apparatus consisted of a small but strong wrought-iron barrel, terminated at one end by a small closed vent, so arranged that the charge could be fired by the application of a red-hot ball. At the other end it was closed by a hemisphere upon which any required weight could be placed. His method was as follows:--A given charge being placed in the bore, a weight judged to be equivalent to the expected gaseous pressure was applied. If the weight were lifted, it was increased until it was just sufficient to confine the gases, and the pressure was then assumed to be that represented by the weight.

It seems probable that Rumford's erroneous determinations were mainly due to two causes :

1st. To the weight closing the barrel being lifted, not by the mere gaseous pressure, but by the products of explosion (produced, it will be remembered, from a very 'brisante' powder and considerably heated by the red-hot ball) being projected at a high velocity against it. In such a case the energy acquired in traversing the barrel would add notably to the pressure due to the density of the charge; and it is again remarkable that the augmentation of pressure, due to this cause, was clearly indicated by an experiment designed for the purpose by Robins.

2nd. To the gases acting on a much larger area than was allowed for in his calculations; and this view appears to be confirmed by the résumé he gives of his experiments.

No attempt was made for very many years either to corroborate or amend Count Rumford's determinations; but, in 1845, General Cavalli endeavoured indirectly to arrive at the pressure developed by different kinds of powder in a gun of 16 cm. calibre. His method consisted in drilling holes in the gun at right angles to the axis, at different distances from the base of the bore, in which holes were screwed small barrels of wrought iron, so arranged as to throw a bullet which would be acted on by the charge of the gun while giving motion to the projectile. By ascertaining the velocities of these bullets he considered that the theoretical thickness of the metal at various points along the bore could be deduced. His experiments led him to some singular results.

He believed that with some very brisante Belgian powder with which he experimented a chamber pressure of 24,022 atmospheres (157·6 tons per square inch) had actually been reached, while with an ordinary powder and a realised energy of nearly the same amount the maximum chamber pressure was only 3,734 atmospheres (24.5 tons per square inch). With the brisante powder this erroneous conclusion was doubtless due to two principal causes, viz.—

1st. To the seat of the small bullet being at a considerable distance from the charge. Under these circumstances, as later on I shall have occasion to describe experiments to prove, a far higher pressure induces motion in the bullet than is due to the tension of the gases in a state of rest.

2nd. To the brisante nature of the powder. With such powders, especially in large charges, it has been proved that great variations of pressure exist in the powder chamber itself, in some cases the pressure indicated at one point of the chamber being more than double that at others.

It has further been proved that with brisante powders waves of pressure of great violence sweep from one end of the chamber to the other, and if Cavalli's small bullet were acted on by one of these waves an exceedingly high pressure would, without doubt, be indicated.

3rd. A third cause of error, but much slighter, is due to the muzzle pressure, when the small bullet quits its barrel, being both abnormally high and also abnormally sustained; hence there will be a considerable increment of velocity after the bullet quits the gun.

It is but fair to add that the results obtained by Cavalli with the powders which he terms 'inoffensive' are, if some correction be made for the third cause of error alluded to above, not far removed from the truth.

A Prussian Artillery Committee, under the presidency of General Neumann, made, in 1854, a great improvement on the plan proposed and employed by Cavalli.

Their mode of procedure consisted in drilling a hole in the powder chamber of the gun to be experimented with, in which hole was placed a small barrel of about six inches in length. Now, when the gun was loaded, if in the small barrel were placed a cylinder of a length equal to that of the projectile, it is clear that, on the assumption that the pressure in the powder chamber is uniform, the cylinder and the projectile will describe equal spaces in equal times; hence, if we determine the velocity of the cylinder when it quits the small barrel, we know the velocity of the projectile when it has moved six inches from its seat. By altering the length of the column of the cylinder placed in the small barrel, and ascertaining the resultant velocity, the velocity of the projectile at any desired point of the bore can be determined.

General Neumann's Committee carried out their experiments only in very small guns and with the grained powder used in those days. Their results were probably not far from the truth, although subject to one of the defects to which I alluded in reviewing General Cavalli's experiments. Indeed, these results were examined and entirely confirmed by the distinguished Russian artillerist General Mayevski, in a very elaborate memoir; but the experiments of the Prussian Committee were chiefly remarkable for being, so far as I know, the first to recognise the variations of pressure which may exist in the powder chamber itself, variations which may, under certain circumstances, attain great magnitude, and to which I have already drawn attention.

The results of the Prussian experiments showed, with every charge fired, two distinct maxima of tension. Other relative maxima no doubt existed, but the mode of experimenting was not sufficiently delicate to render them perceptible.

Before passing to the more modern methods adopted for determining the tensions in guns, I must advert to one which has been repeatedly resorted to during the last 150 years. I mean the method of firing the same weight of charge and projectile from guns of the same calibre but of different lengths, or, as has sometimes been done, by successively reducing the length of the same gun by cutting off a determinate number of calibres from the muzzle.

It is obvious that if, under the circumstances supposed, we know the muzzle velocities of a projectile from a gun of, say, twenty-five calibres in length and from a gun of thirty calibres in length, we are able from the increased obtained to deduce the mean pressure acting upon the energy projectile over the additional five calibres.

The earliest experiments with different lengths of guns appear to have been made in England as far back as 1736. These experiments, however, have but little value, as the velocities were not directly determined, and could only be deduced from the observed ranges. The same objection applies to the long series of experiments carried on in Hanover in 1785, and those cited by Piobert in 1801; but the interesting observation that the ranges obtained from guns of twelve, fifteen, nineteen, and twenty-three calibres in length were relative maxima cannot be relied on in any way as showing abnormal variations in the muzzle pressure accompanying variations in length.

In Hutton's experiments, made with guns varying in length from fifteen to forty calibres, the muzzle velocities were obtained by means of the ballistic pendulum; and, between these limits of length, the mean powder pressure he realised can with sufficient certainty be deduced.

This remark applies also to the numerous similar experiments where the muzzle velocities have been obtained by the more accurate chronoscopes that have been for many years in common use; but this mode of determining the pressure has many inconveniences, and ceases to be reliable when the bore is of a very reduced length and the pressures approach their maximum value.

To the important and extensive series of experiments carried on by Major Rodman for the United States Government in 1857 to 1859, the main object of the experiments being to ascertain the effect which the size of grain of the powder used has upon the pressure, we are indebted for that officer's most ingenious pressure gauge; and the crusher gauge, which is now so extensively used, can only be considered a modification of Major Rodman's instrument designed to remove certain difficulties attending the use of the original instrument.

Major Rodman's gauge is well known, but its construction is shown in the accompanying drawing (fig. 1). Major Rodman applied his gauge in the following manner :

Desiring to ascertain the pressure at various points along the bore of a gun, he bored at these points channels to the interior surface of the bore, and in these channels cylinders with small holes drilled down the centre were inserted; to this cylinder is fitted the indicating apparatus, carried by Major Rodman on the outside of the gun, and consisting of an indenting tool & with its knife (shown in elevation and section). Against the knife