On the Resistance of the Air to the Motion of Elongated Projectiles Having Variously Formed Heads

XIV. On the Resistance of the Air to the Motion of Elongated Projectiles having variously formed Heads. By F. Bashforth, B.D., Professor of Applied Mathematics to the Advanced Class of Artillery Officers, Woolwich, and late Fellow of St. John's College, Cambridge. Communicated by Professor Stokes, Sec. R.S. Received January 30,—Read February 20, 1868. The famous theory of the parabolic motion of projectiles was at an early period found to give results not in accordance with practice. Manifestly, then, the air must offer a very sensible resistance to a body which is moving through it with a high velocity. This resistance will depend upon the form of the moving body, and upon the velocity with which it is moving. Hence, before the path of a projectile can be calculated, it will be necessary to determine experimentally the resistance opposed by the air to the motion of the projectile, corresponding to various velocities. According to Newton's law, the resistance of the air varies as the square of the velocity. But the velocities were low in the experiments made under his direction. In 1719 John Bernoulli gave equations for finding by the method of Quadratures the path &c. of a projectile, when the resistance of the air was supposed to vary according to any power of the velocity. But in spite of grave doubts respecting the accuracy of Newton's law, it has been adopted by most of the eminent mathematicians who have written on the subject, such as Euler (1753), Lambert (1765), Borda (1769), Bezout (1789), Tempelhof (1788–9), d'Ehrenmalm (1788), Lombard (1796), and Poisson. The first good experiments made with a view to determine the resistance of the air to the motion of projectiles were those of Robins in 1742. The projectiles used were leaden bullets of small size. When we consider the great density of the material used, its liability to change its form in the barrel of the gun, and the smallness of the solid projectiles, it is truly wonderful that Robins was able to accomplish so much with his ballistic pendulum. Afterwards Hutton carried on Robins' system of experimenting both with the whirling machine and ballistic pendulum, introducing additional precautions, and using iron projectiles of greater size. In recent times MM. Didion, Morin, and Piobert have carried on experiments in France with heavier spherical projectiles, by the help of an improved ballistic pendulum; but they have done little more than confirm the results of Robins and Hutton, and extend them to spherical projectiles of larger diameter. Robins came to the conclusions:—"First, That, till the velocity of the projectile surpasses that of 1100 feet in a second, the resistance may be esteemed to be in the duplicate proportion of the velocity; and its mean quantity may be taken to be nearly the same with that I have assigned in the former paper. Second, That, if the velocity be greater than that of 1100 or 1200 feet in a second, then the absolute quantity of that resistance in these greater velocities will be near three times as great as it should be by a comparison with the smaller velocities"*. Hutton remarks in a note on these conclusions:—"These suppositions are not nearly correct. In fact, by more accurate experiments with cannon-balls, it appears that the law of the resistance begins to increase above the ratio of the square of the velocity, from the very slowest motions, and thence goes on increasing gradually more and more above what is assigned by that ratio, till we arrive at the velocity of 1600 or 1700 feet per second, where it is at the greatest, amounting in that maximum state to only $2\frac{1}{10}$ times the quantity resulting from the ratio of the square of the velocity. And at the velocity of 1100 feet, instead of answering to that law, it amounts to 1·86 times the same." Euler, in the remarks which accompany his translation of Robins' 'Gunnery,' states that, the greater the velocity of the shot, so much the more does theory deviate from the truth†. Hutton's formula of resistance consisted of two terms, one varying as the velocity, and the other as the square of the velocity. In the year 1836 M. Piobert reexamined Hutton's experiments, and found that the resistance of the air for various velocities was sufficiently well represented by a formula of two terms, one of which varied as the square, and the other as the cube of the velocity. In 1839 and 1840 numerous experiments were made at Metz, under the direction of a commission, by means of an improved ballistic pendulum. The projectiles used were spherical solid shot of 24, 12, and 8, or 26·47 lbs., 13·38 lbs., and 8·86 lbs. in weight, and 5·85 inches, 4·66 inches, and 4·06 inches in diameter, and a shell 50·71 lbs. in weight and 8·67 inches in diameter. The distances from the gun at which the pendulum was placed were 49 feet, 82 feet, 164 feet, 246 feet, and 328 feet. The resistance of the air to these projectiles was found to be represented by the formula $$\pi R^2 v^2 \times 0·027(1+0·0023v)‡;$$ and the new calculation of Hutton's experiments gave $$\pi R^2 v^2 \times 0·02786(1+0·0023v)‡.$$ When spherical balls and smooth-bored guns were used, it was only possible to strike the receiver properly when at a moderate distance from the gun; and thus the variation of velocity to be measured was confined within very narrow limits. There was also the disadvantage that, as the velocity of the ball had to be reduced to that of the receiver in order to determine the striking velocity of the ball, only one velocity could be measured for each round fired. It would therefore be quite impossible to employ Robins' ballistic pendulum to find the velocities of the heavy elongated projectiles in use at the present day. * Robins' Tracts on Gunnery, by Hutton, 1805, p. 181. † Neue Grundsätze der Artillerie, 1745, p. 508. ‡ Didion, Traité de Balistique, 1860, pp. 61 & 64. Various attempts have been made to measure the velocities of cannon-balls by the aid of electricity. The machines with revolving cylinders were in general failures, because their inventors made their success depend upon the known uniform angular velocity of the cylinder. These failures opened the way for the introduction of Major Navez' electro-ballistic pendulum, and others of the same class, which worked with two screens, and therefore furnished no means for testing the probable accuracy of the velocity determined. The apparent convenience and portability of these instruments led to their general use both in Europe and America. Major Navez' instrument, in its original complicated form, is now out of fashion; whilst Colonel Benton's two-pendulum instrument and Colonel Leurs' modification of it are in common use, as they are simpler than Major Navez' instrument, and give results quite as much to be relied upon. Even if the electro-ballistic pendulum were perfect in itself as a measurer of time, considerable errors might be expected to arise from the imperfections of the indications of two screens placed at moderate distances apart. A reference to a paper by Major Navez, "Considérations sur les expériences de balistique en ce qui concerne la mesure du temps"*, will show how little had been accomplished when that was written (1865). And Colonel Benet has well remarked:—"Electro-ballistic machines heretofore used have been powerless to solve one of the most important problems in ballistics—the law of the movement of a projectile through the air,—and this because of the limited number of points of the trajectory that could be determined"†. From preliminary experiments already made, I feel certain that a simpler, cheaper, and better instrument might be substituted with advantage for electro-ballistic pendulums where such instruments can be used. The time occupied by a body in falling from rest through a given space, or the time occupied by the sound of the explosion in travelling over a given space, might be made the foundation of the measurement of a velocity; or the velocity of the shot might be directly compared with the velocity of the sound of the explosion. In the spring of 1864, when I was appointed Professor of Applied Mathematics to the Advanced Class of Artillery Officers at Woolwich, and Referee of the Ordnance Select Committee, I strongly recommended the construction of a chronograph capable of recording the time occupied by a projectile in passing over nine or more successive equal spaces. The principle of the chronograph used at the Greenwich Observatory was plainly the one to be adopted. The chief difficulties to be overcome were found (1) in the arrangement of a proper system of screens, so that the ball in passing might merely cause a momentary interruption (not a rupture) of the galvanic current, and that the resistance of the circuit might be kept perfectly constant during the experiment, (2) in the arrangement of a system of marking, which should give definite records on the surface of the cylinder when moving with a velocity of about 10 inches per second, and (3) in the compensation for the want of uniformity in the angular velocity of the cylinder; * Revue de Technologie Militaire, t. iv. † Electro-ballistic Machines, 1866, p. 39. The chronograph was completed in the summer of 1865, and in November and December of the same year it was tried with ten screens, placed at intervals of 120 feet. Satisfactory records were obtained for eleven out of eighteen rounds of elongated shot fired from a 12-pounder B.L. gun, which very plainly indicated that the resistance of the air varied as the cube of the velocity. A full account of the chronograph and of the method of using it, accompanied by a detailed statement of the eleven successful rounds, was printed in the Proceedings of the Royal Artillery Institution, Woolwich, for August 1866, and was also published in a separate form. At present it will be sufficient to state that the axis of the cylinder is vertical, and in a line with the axis of the horizontal fly-wheel, to which it is attached. The fly-wheel is spun by hand. When the gun is ready to be fired, the markers are brought into contact with the paper; and after the clock has recorded three or four seconds the gun is fired. The clock is allowed to record three or four seconds more, and then the markers are raised and the experiment is completed. The friction on the axis of the fly-wheel, the resistance of the air to the motion of the wheel, and the friction of the markers on the paper tend to reduce the angular velocity of the cylinder. But as the pendulum of a half-seconds' clock raises a lever once each double swing, it interrupts the clock galvanic current once a second, and thus the clock-records show what spaces have passed under the markers each second. The changing angular velocity of the cylinder thus becomes accurately known. For if we commence measuring from some arbitrary point taken two or three seconds before the screen-records, and measure along the spiral traced by the clock marker, noting each record of the clock, and continuing our measurements two or three seconds beyond the screen-records, and if we difference these quantities, we shall find whether the angular velocity has been sufficiently regular in its change during the experiment. If so, we can by interpolation find what would have been the records if one had been made every tenth of a second. It has always been found to be sufficient to suppose the angular velocity constant during each tenth of a second, and to calculate smaller intervals of time by proportional parts. As the clock goes on all day breaking the current once a second, every record of the clock is made under precisely the same circumstances. If there be a loss of time between the breaking of the current and the making of the corresponding record, the loss of time may always be expected to remain the same for any single experiment, and therefore there can be no error; for in experiments on gunnery the exact length of a second, only, is required, and not the exact time of the day. The galvanic current which works the screen-marker is kept constantly circulating through all the screens, excepting during the momentary interruption caused by the breaking or repairing of a screen, or some accidental rupture of the conducting wire. Although there is no necessity for this arrangement, it is found to be practically convenient. The ordinary screens used for other instruments are formed of fine copper wire stretched across a frame repeatedly, and through which the galvanic current circulates. When, then, a shot passes through the screen the current is permanently broken, unless some of the broken wires happen to rest in contact. But for my chronograph it was necessary to make provision for the breaking and immediate restoration of the current, and that without changing the resistance of the circuit. The galvanic current passes along the top of each of the ten screens simultaneously. Equal weights are attached by long pieces of sewing-cotton to certain wire springs which project through holes in sheet copper. When the shot cuts one or more threads, the corresponding springs are released and fly from the bottom to the top of their holes. So long as any single spring is not in contact with the side of the hole through which it projects, the current is interrupted. This kind of screen secures a perfectly constant resistance to the screen galvanic current. But it has been objected that if the galvanic current circulated about the screen electro-magnet for several minutes, which elapse between successive rounds, the loss of time between the breaking of the galvanic current and its corresponding registration would not be equal for all the ten screens. This error would be the more pernicious because it might be expected to follow some law, and therefore could not be eliminated. The following arrangements have been made partly to meet this difficulty, and partly for the sake of the convenience of keeping up a constant communication between the instrument and the range. A self-acting contact-breaker and ringer (figs. 1 & 2) is placed by the side of the gun. Ordinarily the lever \( ab \) is down, as in fig. 1, in which case the galvanic current takes the direct course, \( abcd \). When the lever \( ab \) is raised, the current is permanently interrupted, but the insertion of a metallic pin \( f \) (fig. 2) opens a passage through the contact-breaker, \( e \), when all the bells in the circuit are continuously rung. After the screens have been mended the lever \( ab \) is raised to try whether the current is good. If so, it is lowered and the gun is loaded. When the range is clear and all is ready, the lever \( ab \) is raised and the pin \( f \) is inserted. The fly-wheel of the chronograph is now put in motion. The rapid interruptions of the current by the contact-breaker are recorded on the cylinder, till a pull at the lanyard, to fire the gun, simultaneously withdraws also the pin \( f \), and so shuts out the contact-breaker. The screen-records often follow so closely that it is difficult, before measurement, to say which is the first screen-record. Another improvement, represented in fig. 3, has recently been introduced. The two ends of the circuit are at A and B, near the chronograph. When the markers are being raised from the paper by a lever, the spring \( l \) is simultaneously brought into contact with \( m \), which turns the screen-current from the marker to the bell. Thus every interruption of the screen-current caused by the repair of the screens is signaled by the bell. When the bell rings continuously, it is known that the lever \( ab \) (fig. 2) is raised ready for firing. The fly-wheel is caused to spin, the markers are brought down upon the paper, and simultaneously the spring \( l \) is brought into contact with \( n \), when the bell is silenced and the marker registers the breaks of the contact-breaker till the pin \( f \) is withdrawn and the gun is fired. The clock is allowed to make a few beats; and then the markers are raised from the paper, and contact is reestablished between \( l \) and \( m \). Thus the galvanic current only circulates for eight or ten seconds about the screen electro-magnet for each experiment, and the current is always being rapidly interrupted quite up to the firing of the gun; so that there is no opportunity for the development of a varying strength of remaining magnetism. The whole arrangement is found to work so satisfactorily that on one occasion nine rounds (23 to 31) were fired in forty-five minutes. After all possible precautions have been taken, it is found that there are small corrections required in order to make the successive records of both clock and screen consistently regular. The unit of the scale used in measuring is about half an inch, and the scale is read off to two places of decimals, or to the \( \frac{1}{200} \) of an inch. The corrections are carried to three places of decimals of the scale. The final calculations are carried to four places of decimals of a second. This is done to secure accuracy to the nearest \( \frac{1}{1000} \) of a second of time, giving an opening for an error of \( \pm \frac{1}{2000} \) of a second of time, or 6 or 8 inches of space, in finding the time occupied by the shot in passing from the first to any succeeding screen. These corrections of the readings of the scale are rendered necessary because the screens cannot be practically maintained at perfectly equal distances. The point of the shot may strike fairly upon a thread at one screen, and between two threads at the next screen. One spring may act more promptly than another. One string may bend more than another before breaking. These corrections are often merely nominal, but there are some sufficiently large to warn us to beware of trusting implicitly to any measurement of a velocity by two screens only. Shortly after the publication of the description of my chronograph, my attention was directed to a chronograph with a cylinder, the invention of Captain Schultz*, of the French Artillery, which had been tried in France and America. The instrument is adapted for making any number of records; and, like my own, its success does not depend upon the uniformity of rotation of the cylinder. My instrument makes the clock-and screen-records side by side, on glazed paper which covers the cylinder, so that the original records of the experiments can be preserved for future reference. Captain Schultz makes his records on the slightly smoked metal surface of his cylinder, which are effaced when they have been read off. Captain Schultz uses a large tuning-fork, usually called a diapason, the vibrations of which are sustained by electro-magnetism, to effect the mechanical division of the second into 250 or more equal intervals. The diapason, vibrating as the cylinder turns, traces a sinuous spiral line. The pendulum in swinging interrupts a galvanic circuit once a second, and causes a spark from a Ruhmkorff's coil to strike the cylinder and make a record. Thus it is found how many vibrations the diapason makes per second. The clock is then taken out of the circuit, and the current is made to pass through the Ruhmkorff coil and the first screen. When the first screen is broken the coil gives a spark, and the galvanic current is made to pass through the * Colonel Benet's 'Electro-ballistic Machines,' 1866, p. 32. second screen*. When the second screen is broken, a recording spark is given by the coil, and the current is passed on to the third screen, and so on to the end. All the time, the diapason is tracing its sinuous spiral. Unless a careful system of compensation be provided, this method of working the screens would cause great variation in the resistance of the circuit. In the arrangement of my screens, I was careful to maintain a constant resistance to the current, which end is secured by making the current pass through all the screens simultaneously, and by providing for it to be interrupted, but not broken. I am not aware that Captain Schulz' instrument has been tried with more than two screens; but Colonel Benet has given a Table showing the number of vibrations per second made by the diapason as determined at the Frankfort Arsenal. In this Table there are striking variations in the numbers of vibrations made in successive seconds, as in the second trial, extending to twenty-five seconds, we find 249·1, 252·0, 249·5, 248·5, 246·0, 249·0, &c. The result of the trials is stated as if the constancy of the mean number of vibrations per second was all that was required. This is not the case. A succession of equal intervals of time must be marked out by the clock, or diapason, so that, when the gun is fired, the time of passing the screens may be noted by the side of a correct scale of time. Now it would make an important difference in the resulting velocity, if the gun were fired when the diapason was making 252·0, 248·5, or 246·0 vibrations per second. If the vibrations of the diapason be maintained by the vibrations of a second fork which alternately makes and breaks contact, as described by Helmholtz†, it is hardly to be expected that the number of vibrations per second can be maintained with sufficient constancy. There is another question. The point from which the spark is discharged cannot be allowed to touch the smoked surface of the cylinder; and it may be asked whether the spark is not liable to deviate in its passage. After my chronograph had passed its first trial (in December 1865), it appeared to be desirable to institute experiments with a view to find the resistance of the air to various forms of heads of elongated shot, but such as were likely to be of practical utility. The proposed experiments received the sanction of the Right Honourable the Secretary of State for War, and ten of each kind of the following elongated shot were prepared for the 40-pounder M.L. gun: (1) Solid. Hemispherical head. (2) Solid. Hemispheroidal head (ratio of diameters 1:2). (3) Solid. Ogival, struck with radius equal to a diameter of shot. (4) Solid. Ogival, struck with radius equal to two diameters of shot. (5) Hollow. Ogival, having precisely the same external form and length as (3). (6) Hollow. Ogival, having precisely the same external form and length as (4). * "In working the instrument it is essential that the current pass only through one target at a time, there being but one coil and one battery no matter how many targets may be used. After the first target is ruptured, the current must be transferred to the succeeding one before the projectile reaches it, and so on throughout the series."—Benet, p. 36. See also the Practical Mechanic's Journal, Oct. 1, 1867, p. 195, to the same effect. † Tonempfindung, p. 584. Rounds 1–13 were fired on the 25th September 1866. 14–31 ” ” ” 26th September 1866. 32–43 ” ” ” 27th September 1866. A barge having anchored in the range, there was no opportunity of firing a single round during the whole of the following day, which interfered with the completeness of the experiment with the solid shot. As a uniform charge of 5 lbs. of powder was used for all the rounds fired, and as the solid were nearly double the weight of the hollow shot, the hollow shot had a much higher initial velocity than the solid. Hence the hollow shot had also a much higher initial angular velocity than the solid shot; and it is probable that the initial angular velocity would be preserved nearly unimpaired throughout the observed range. This is manifested in the greater steadiness of the hollow ogivals. For the first round, the method of forming the Time-table, and the mode of using it in the calculation of the times of arrival of the shot at each successive screen, are indicated at full length. For further details I must refer the reader to the published description of the chronograph. The screens were placed 150 feet apart, which distance is denoted by $l$. The first screen was 75 feet from the gun. $t$ denotes the time occupied by the shot in passing from the first screen to a distance $s$ feet, when $t = as + bs^2$; and $0, t_2, t_3, \ldots t_n$ will denote the time when the shot passes the first, second, third, $\ldots n$th screen, or the times corresponding to the particular values $0, l, 2l, \ldots n-1 l$ of $s$. AIR TO THE MOTION OF ELONGATED PROJECTILES. (1) Hemispherical-headed Shot. Diameter 4·7 inches. Round 1. Weight of shot 39·34 lbs. | Clock. | Time-table obtained by interpolation. | |--------|-------------------------------------| | Reading | Correction | Corrected reading | 2·5 | 28·748 +1·584 | | 1 | 4·91 | 0 | 2·6 | 30·332 +1·583 | | 2 | 20·82 | 0 | 2·7 | 31·915 +1·582 | | 3 | 36·65 | +0·01 | 2·8 | 33·497 +1·582 | | 4 | 52·44 | -0·007 | 2·9 | 35·079 +1·581 | | 5 | 68·14 | +0·002 | 3·0 | 36·660 +1·580 | | 6 | 83·79 | 0 | 3·1 | 38·240 +1·580 | | | | | 3·2 | 39·820 +1·579 | | | | | 3·3 | 41·399 +1·578 | | | | | 3·4 | 42·977 +1·578 | | | | | 3·5 | 44·555 +1·577 | | | | | 3·6 | 46·132 +1·576 | | | | | 3·7 | 47·708 +1·576 | | | | | 3·8 | 49·284 +1·575 | | | | | 3·9 | 50·860 +1·574 | | | | | 4·0 | 52·433 +1·574 | Interpolation. | 2·0 | 20·820 | Δ₁ | | 2·5 | 28·748 | +7·928 | Δ₂ | | 3·0 | 36·660 | 7·912 | -16 | | 3·5 | 44·555 | 7·895 | -17 | | 4·0 | 52·433 | +7·878 | -17 | Screens. | No. | Reading | Correction | Corrected reading | 2·5 | 28·260 +2·002 Δ₁ | |-----|---------|------------|-------------------|-----|-----------------| | 1 | 28·26 | 0 | | 2·6 | 30·262 +2·030 Δ₂ | | 2 | 30·26 | +0·002 | | 2·7 | 32·292 +2·058 Δ₃ | | 3 | 32·30 | -0·008 | | 2·8 | 34·350 +2·086 Δ₄ | | 4 | 34·35 | 0 | | 2·9 | 36·436 +2·114 Δ₅ | | 5 | 36·43 | +0·006 | | 3·0 | 38·550 +2·141 Δ₆ | | 6 | 38·54 | +0·010 | | 3·1 | 40·691 +2·167 Δ₇ | | 7 | 40·76 | * | | 3·2 | 42·858 +2·193 Δ₈ | | 8 | 42·86 | -0·002 | | 3·3 | 45·051 +2·219 Δ₉ | | 9 | 45·06 | -0·009 | | 3·4 | 47·270 +2·245 Δ₁₀ | | 10 | 47·26 | +0·010 | | 3·5 | 49·500 +2·271 Δ₁₁ | Having thus obtained the space described by the clock marker at the end of every tenth of a second, from 2⅔ up to 4⅔, we can calculate the times when the screens were passed as follows: | Screen | Reading | Correction | Corrected reading | 2·5 | 28·260 +2·002 Δ₁ | |--------|---------|------------|-------------------|-----|-----------------| | 1 | 28·26 | 0 | | 2·6 | 30·262 +2·030 Δ₂ | | 2 | 30·26 | +0·002 | | 2·7 | 32·292 +2·058 Δ₃ | | 3 | 32·30 | -0·008 | | 2·8 | 34·350 +2·086 Δ₄ | | 4 | 34·35 | 0 | | 2·9 | 36·436 +2·114 Δ₅ | | 5 | 36·43 | +0·006 | | 3·0 | 38·550 +2·141 Δ₆ | | 6 | 38·54 | +0·010 | | 3·1 | 40·691 +2·167 Δ₇ | | 7 | 40·76 | * | | 3·2 | 42·858 +2·193 Δ₈ | | 8 | 42·86 | -0·002 | | 3·3 | 45·051 +2·219 Δ₉ | | 9 | 45·06 | -0·009 | | 3·4 | 47·270 +2·245 Δ₁₀ | | 10 | 47·26 | +0·010 | | 3·5 | 49·500 +2·271 Δ₁₁ | MDCCCLXVIII. ### Round 13. Weight of shot 39-33 lbs. | Clock | Reading | Correction | Corrected reading | Screen | Reading | Correction | Corrected reading | Time of passing each screen | |-------|---------|------------|-------------------|--------|---------|------------|-------------------|-----------------------------| | 1 | 15-51 | 0 | 15-510 +24-050 | 1 | 64-69 | 0 | 64-690 +3-025 | 0-0000 +1266 Δ₁ | | 2 | 39-56 | 0 | 39-560 +23-953 | 2 | 67-72 | -005 | 67-715 +3-061 | 1-986 +1282 Δ₂ | | 3 | 63-32 | -007 | 63-518 +23-857 | 3 | 70-77 | +006 | 70-776 +3-097 | 2-548 +1298 Δ₃ | | 4 | 87-37 | 0 | 87-370 +23-760 | 4 | 73-87 | +003 | 73-873 +3-131 | 3-816 +1312 Δ₄ | | 5 | 111-13 | 0 | 111-130 | 5 | 76-17 | -006 | 76-170 +3-166 | 5-183 +1328 Δ₅ | | | | | | | 80-17 | 0 | 80-170 +3-201 | 6-466 +1343 Δ₆ | | | | | | | 83-71 | * | 83-371 +3-237 | 7-729 +1359 Δ₇ | | | | | | | 86-61 | -002 | 86-608 +3-274 | 9-188 +1375 Δ₈ | | | | | | | 89-89 | -008 | 89-882 +3-311 | 1-0563 +1392 Δ₉ | | | | | | | 93-19 | +003 | 93-193 +3-347 | 1-1955 +1402 Δ₁₀ | ### Round 34. Weight of shot 39-34 lbs. | Clock | Reading | Correction | Corrected reading | Screen | Reading | Correction | Corrected reading | Time of passing each screen | |-------|---------|------------|-------------------|--------|---------|------------|-------------------|-----------------------------| | 4 | 89-69 | 0 | 89-690 Δ₁ | 1 | 104-33 | 0 | 104-330 Δ₂ | 0-0000 +1267 Δ₃ | | 5 | 111-17 | -010 | 111-160 +21-470 Δ₄| 2 | 107-05 | -001 | 107-048 +2-718 Δ₅| 1-267 +1283 Δ₆ | | 6 | 132-55 | +010 | 132-560 +21-400 Δ₇| 3 | 109-80 | +003 | 109-799 +2-751 Δ₈| 2-550 +1299 Δ₉ | | 7 | 153-89 | 0 | 153-890 +21-330 Δ₁₀| 4 | 112-58 | +001 | 112-583 +2-784 Δ₁₁| 3-849 +1315 Δ₁₂ | | | | | | | 115-40 | -001 | 115-399 +2-816 Δ₁₃| 5-164 +1331 Δ₁₄ | | | | | | | 118-24 | +010 | 118-250 +2-851 Δ₁₅| 6-495 +1348 Δ₁₆ | | | | | | | 121-16 | -023 | 121-137 +2-887 Δ₁₇| 7-843 +1367 Δ₁₈ | | | | | | | 124-06 | +002 | 124-062 +2-925 Δ₁₉| 9-210 +1384 Δ₂₀ | | | | | | | 127-02 | +003 | 127-023 +2-961 Δ₂₁| 1-0594 +1402 Δ₂₂ | | | | | | | 130-02 | 0 | 130-020 +2-997 Δ₂₃| 1-1996 +1402 Δ₂₄ | ### Round 43. Weight of shot 39-34 lbs. | Clock | Reading | Correction | Corrected reading | Screen | Reading | Correction | Corrected reading | Time of passing each screen | |-------|---------|------------|-------------------|--------|---------|------------|-------------------|-----------------------------| | 1 | 24-05 | 0 | 24-050 Δ₁ | 1 | 75-83 | 0 | 75-830 Δ₂ | 0-0000 +1260 Δ₃ | | 2 | 45-51 | -005 | 45-505 +21-455 Δ₄| 2 | 78-50 | +015 | 78-515 +2-685 Δ₅| 1-260 +1276 Δ₆ | | 3 | 66-88 | +010 | 66-890 +21-385 Δ₇| 3 | 81-24 | -005 | 81-235 +2-720 Δ₈| 2-536 +1276 Δ₉ | | 4 | 88-21 | -005 | 88-205 +21-315 Δ₁₀| 4 | 83-99 | 0 | 83-990 +2-755 Δ₁₁| 3-829 +1293 Δ₁₂ | | 5 | 109-45 | 0 | 109-450 +21-245 Δ₁₃| 5 | 86-78 | 0 | 86-780 +2-790 Δ₁₄| 5-140 +1311 Δ₁₅ | | | | | | | 89-61 | -003 | 89-607 +2-827 Δ₁₆| 6-469 +1329 Δ₁₇ | | | | | | | 92-48 | -010 | 92-470 +2-863 Δ₁₈| 7-815 +1346 Δ₁₉ | | | | | | | 95-37 | 0 | 95-370 +2-900 Δ₂₀| 9-179 +1364 Δ₂₁ | | | | | | | 98-31 | -003 | 98-307 +2-937 Δ₂₂| 1-0561 +1382 Δ₂₃ | | | | | | | 101-28 | 0 | 101-280 +2-973 Δ₂₄| 1-1961 +1400 Δ₂₅ | ### Summary (1). Hemispherical-headed Shot. Time occupied by shot in passing from the first to each of the other screens. | No. of round | Screen 1 | Screen 2 | Screen 3 | Screen 4 | Screen 5 | Screen 6 | Screen 7 | Screen 8 | Screen 9 | Screen 10 | |--------------|----------|----------|----------|----------|----------|----------|----------|----------|----------|-----------| | 1 | 0-0000 | 0-1264 | 0-2546 | 0-3847 | 0-5166 | 0-6504 | 0-7860 | 0-9233 | 1-0623 | 1-2030 | | 13 | 0-0000 | -1266 | -2548 | -3846 | -5158 | -6486 | -7829 | -9188 | -10563 | -11955 | | 34 | 0-0000 | -1267 | -2550 | -3849 | -5164 | -6495 | -7843 | -9210 | -10594 | -11996 | | 43 | 0-0000 | -1260 | -2536 | -3829 | -5140 | -6469 | -7815 | -9179 | -10561 | -11961 | ### AIR TO THE MOTION OF ELONGATED PROJECTILES. #### Velocities at the following distances from the gun. | No. of round | 150 ft. | 300 ft. | 450 ft. | 600 ft. | 750 ft. | 900 ft. | 1050 ft. | 1200 ft. | 1350 ft. | |--------------|---------|---------|---------|---------|---------|---------|----------|----------|----------| | 1 | 1187 | 1170 | 1153 | 1137 | 1121 | 1106 | 1092 | 1079 | 1066 | | 13 | 1185 | 1170 | 1156 | 1143 | 1130 | 1117 | 1104 | 1091 | 1078 | | 34 | 1184 | 1169 | 1155 | 1141 | 1127 | 1113 | 1097 | 1084 | 1070 | | 43 | 1190 | 1175 | 1160 | 1144 | 1129 | 1114 | 1100 | 1085 | 1071 | #### (2) Hemispheroidal-headed Shot (Axes 1 : 2), Solid. **Round 2. Weight of shot 38·72 lbs.** | Clock. | Reading. | Correction. | Corrected reading. | Screen. | Reading. | Correction. | Corrected reading. | Time of passing each screen. | |--------|----------|-------------|--------------------|---------|----------|-------------|--------------------|-----------------------------| | 2 | 24·44 | 0 | 24·440 | 1 | 103·28 | ±0·005 | 103·285 | ±0·0000 | | 3 | 48·19 | 0 | 48·190 +25·750 | 2 | 106·28 | ±0·002 | 106·282 | ±2·997 +24 | | 4 | 71·90 | -0·004 | 71·896 +23·706 | 3 | 109·31 | ±0·007 | 109·303 | ±3·021 +29 | | 5 | 95·56 | +0·003 | 95·563 +23·667 | 4 | 112·35 | ±0·002 | 112·352 | ±3·049 +29 | | 6 | 119·20 | -0·004 | 119·196 +23·633 | 5 | 115·43 | ±0·002 | 115·432 | ±3·112 +32 | | 7 | 142·80 | 0 | 142·800 +23·604 | 6 | 118·55 | ±0·006 | 118·544 | ±3·146 +34 | | | | | | | | | | ±0·0000 | | | | | | | | | | ±1·267 +1267 | | | | | | | | | | ±1·273 +1273 | **Round 7. Weight of shot 38·69 lbs.** | Clock. | Reading. | Correction. | Corrected reading. | Screen. | Reading. | Correction. | Corrected reading. | Time of passing each screen. | |--------|----------|-------------|--------------------|---------|----------|-------------|--------------------|-----------------------------| | 3 | 60·24 | 0 | 60·240 | 1 | 82·59 | ±0·003 | 82·593 | ±2·748 +28 | | 4 | 81·88 | 0 | 81·880 +21·640 | 2 | 85·34 | ±0·001 | 85·341 | ±2·748 +28 | | 5 | 103·45 | -0·010 | 103·440 +21·560 | 3 | 88·13 | ±0·013 | 88·117 | ±2·776 +28 | | 6 | 124·91 | +0·010 | 124·920 +21·480 | 4 | 90·92 | ±0·001 | 90·921 | ±2·804 +30 | | 7 | 146·32 | 0 | 146·320 +21·400 | 5 | 93·75 | ±0·005 | 93·755 | ±2·834 +30 | | | | | | | | | | ±0·0000 | | | | | | | | | | ±1·273 +1273 | | | | | | | | | | ±1·273 +1273 | **Round 35. Weight of shot 38·69 lbs.** | Clock. | Reading. | Correction. | Corrected reading. | Screen. | Reading. | Correction. | Corrected reading. | Time of passing each screen. | |--------|----------|-------------|--------------------|---------|----------|-------------|--------------------|-----------------------------| | 3 | 52·97 | 0 | 52·970 | 1 | 68·09 | 0 | 68·090 | ±0·0000 | | 4 | 71·20 | 0 | 71·200 +18·230 | 2 | ** | * | 70·399 +2·309 | ±0·0000 | | 5 | 89·37 | -0·003 | 89·367 +18·167 | 3 | ** | * | 72·728 +2·329 | ±0·0000 | | 6 | 107·47 | 0 | 107·470 +18·103 | 4 | ** | * | 75·080 +2·352 | ±0·0000 | | | | | | | | | | ±0·0000 | | | | | | | | | | ±1·268 +1268 | | | | | | | | | | ±1·268 +1268 | | | | | | | | | | ±1·268 +1268 | --- \[ \text{3 N 2} \] ### Round 40. Weight of shot 38-69 lbs. | Clock | Reading | Correction | Corrected reading | Screen | Reading | Correction | Corrected reading | Time of passing each screen | |-------|---------|------------|-------------------|--------|---------|------------|-------------------|-----------------------------| | 4 | 87-30 | 0 | 87-300 | 1 | 113-53 | 0 | 113-530 | 0-0000 | | 5 | 107-05 | 0 | 107-050 | 2 | 116-02 | +001 | 116-021 | -1264 | | 6 | 126-75 | 0 | 126-750 | 3 | 118-55 | -012 | 118-538 | -2542 | | 7 | 146-40 | 0 | 146-400 | 4 | 121-08 | +001 | 121-081 | -3833 | ### Summary (2). Hemispheroidal-headed Shot. Time occupied by shot in passing from the first to each of the other screens. | No. of round | Screen 1 | Screen 2 | Screen 3 | Screen 4 | Screen 5 | Screen 6 | Screen 7 | Screen 8 | Screen 9 | Screen 10 | |--------------|----------|----------|----------|----------|----------|----------|----------|----------|----------|-----------| | 2 | 0-0000 | 0-1267 | 0-2545 | 0-3836 | 0-5139 | 0-6457 | 0-7789 | 0-9136 | ** | ** | | 7 | 0-0000 | -1273 | -2559 | -3859 | -5173 | -6502 | -7844 | -9200 | 1-0569 | ** | | 35 | 0-0000 | -1268 | -2548 | -3841 | -5147 | -6466 | -7709 | -9146 | 1-0507 | 1-1881 | | 40 | 0-0000 | -1264 | -2542 | -3833 | -5138 | -6457 | -7790 | -9137 | 1-0498 | 1-1873 | ### Velocities at the following distances from the gun. | No. of round | 150 ft. | 300 ft. | 450 ft. | 600 ft. | 750 ft. | 900 ft. | 1050 ft. | 1200 ft. | 1350 ft. | |--------------|---------|---------|---------|---------|---------|---------|----------|----------|----------| | 2 | f.s. | f.s. | f.s. | f.s. | f.s. | f.s. | f.s. | f.s. | f.s. | | 7 | 1184 | 1174 | 1162 | 1151 | 1138 | 1126 | 1114 | ** | ** | | 35 | 1178 | 1166 | 1154 | 1142 | 1129 | 1118 | 1106 | 1096 | ** | | 40 | 1183 | 1172 | 1160 | 1149 | 1137 | 1125 | 1114 | 1102 | 1092 | ### (3) Ogival-headed Shot (one diameter), Solid. #### Round 3. Weight of shot 39-56 lbs. | Clock | Reading | Correction | Corrected reading | Screen | Reading | Correction | Corrected reading | Time of passing each screen | |-------|---------|------------|-------------------|--------|---------|------------|-------------------|-----------------------------| | 1 | 21-09 | 0 | 21-090 | 1 | 114-12 | 0 | 114-120 | 0-0000 | | 2 | 46-94 | 0 | 46-940 | 2 | 117-48 | 0 | 117-490 | -1293 | | 3 | 79-69 | 0 | 79-690 | 3 | 120-78 | -005 | 120-775 | -2600 | | 4 | 98-36 | +001 | 98-361 | 4 | 124-15 | +013 | 124-163 | -3025 | | 5 | 123-97 | 0 | 123-970 | 5 | 127-60 | -001 | 127-599 | -5263 | | 6 | 149-53 | 0 | 149-530 | 6 | 131-09 | -006 | 131-094 | -6631 | | | | | | 7 | 134-62 | -005 | 134-615 | -8012 | | | | | | 8 | 138-18 | +005 | 138-185 | -9405 | ### Round 36. Weight of shot 39-56 lbs. | Clock | Reading | Correction | Corrected reading | Screen | Reading | Correction | Corrected reading | Time of passing each screen | |-------|---------|------------|-------------------|--------|---------|------------|-------------------|-----------------------------| | 3 | 52-79 | 0 | 52-790 | 1 | 99-25 | 0 | 99-250 | 0-0000 | | 4 | 72-49 | 0 | 72-490 | 2 | 101-29 | * | 101-716 | 1264 + 1279 + 15 | | 5 | 92-08 | +0-05 | 92-085 | 3 | 104-21 | -0-01 | 104-209 | 2543 + 1294 + 15 | | 6 | 111-58 | 0 | 111-580 | 4 | 106-73 | -0-02 | 106-728 | 3837 + 1307 + 13 | | 7 | 130-98 | 0 | 130-980 | 5 | 109-27 | +0-02 | 109-272 | 5144 + 1320 + 13 | ### Round 41. Weight of shot 39-56 lbs. | Clock | Reading | Correction | Corrected reading | Screen | Reading | Correction | Corrected reading | Time of passing each screen | |-------|---------|------------|-------------------|--------|---------|------------|-------------------|-----------------------------| | 2 | 53-16 | -0-06 | 53-154 | 1 | 123-37 | 0 | 123-370 | 0-0000 | | 3 | 76-64 | +0-08 | 76-648 | 2 | 126-30 | +0-01 | 126-301 | 1260 + 1260 + 14 | | 4 | 100-06 | -0-08 | 100-052 | 3 | 129-26 | +0-03 | 129-263 | 2534 + 1288 + 13 | | 5 | 123-36 | +0-06 | 123-366 | 4 | 132-26 | -0-04 | 132-256 | 3822 + 1301 + 13 | | 6 | 146-59 | 0 | 146-590 | 5 | 135-27 | +0-09 | 135-279 | 5123 + 1314 + 13 | ### Summary (3). Ogival-headed Shot (one diameter), Solid. Time occupied by shot in passing from the first to each of the other screens. | No. of round | Screen 1 | Screen 2 | Screen 3 | Screen 4 | Screen 5 | Screen 6 | Screen 7 | Screen 8 | Screen 9 | Screen 10 | |--------------|----------|----------|----------|----------|----------|----------|----------|----------|----------|-----------| | 3 | 0-0000 | 0-1293 | 0-2600 | 0-3925 | 0-5268 | 0-6631 | 0-8012 | 0-9409 | * | * | | 36 | 0-0000 | 1264 | 2543 | 3837 | 5144 | 6464 | 7799 | 9148 | 1-0510 | * | | 41 | 0-0000 | 1260 | 2534 | 3822 | 5123 | 6437 | 7765 | 9106 | 1-0460 | 1-1827 | Velocities at the following distances from the gun. | No. of round | 150 ft. | 300 ft. | 450 ft. | 600 ft. | 750 ft. | 900 ft. | 1050 ft. | 1200 ft. | 1350 ft. | |--------------|---------|---------|---------|---------|---------|---------|----------|----------|----------| | 3 | f.s. | f.s. | f.s. | f.s. | f.s. | f.s. | f.s. | f.s. | f.s. | | 36 | 1160 | 1147 | 1132 | 1117 | 1101 | 1086 | 1074 | * | * | | 41 | 1187 | 1173 | 1159 | 1148 | 1136 | 1124 | 1112 | 1101 | * | | No. of round | Weight of shot. | Values of $bl^2$ | |--------------|-----------------|------------------| | 3 | 39-56 | -00085 | | 36 | 39-56 | -00071 | | 41 | 39-56 | -00068 | Mean... 39-56 Mean of 2 rounds -00070 (4) Ogival-headed Shot (two diameters), Solid. Round 4. Weight of shot 38·56 lbs. | Clock | Reading | Correction | Corrected reading | Screen | Reading | Correction | Corrected reading | Time of passing each screen | |-------|---------|------------|-------------------|--------|---------|------------|-------------------|-----------------------------| | 1 | 17·50 | 0 | 17·500 | 1 | 63·78 | -0·01 | 63·779 | 0·0000 | | 2 | 40·62 | -0·05 | 40·615 | 2 | 66·67 | +0·19 | 66·869 | -1·266 | | 3 | 63·64 | +0·10 | 63·650 | 3 | 69·63 | +0·02 | 69·652 | 2·546 | | 4 | 86·63 | -0·12 | 86·618 | 4 | 72·61 | -0·02 | 72·608 | 3·841 | | 5 | 109·53 | 0 | 109·530 | 5 | 75·62 | -0·04 | 75·616 | -5·151 | | | | | | | 6 | 78·65 | +0·06 | 78·656 | 6·475 | | | | | | | 7 | 81·73 | -0·03 | 81·727 | 7·812 | | | | | | | 8 | 84·83 | -0·04 | 84·826 | 9·165 | | | | | | | 9 | 87·95 | +0·04 | 87·954 | 1·0527 | | | | | | | 10 | 91·11 | 0 | 91·110 | 1·1903 | Round 37. Weight of shot 38·48 lbs. | Clock | Reading | Correction | Corrected reading | Screen | Reading | Correction | Corrected reading | Time of passing each screen | |-------|---------|------------|-------------------|--------|---------|------------|-------------------|-----------------------------| | 4 | 51·40 | 0 | 51·400 | 1 | 71·99 | 0 | 71·990 | 0·0000 | | 5 | 68·04 | 0 | 68·040 | 2 | 74·10 | +0·03 | 74·103 | -1·275 | | 6 | 84·60 | 0 | 84·600 | 3 | 76·24 | -0·01 | 76·239 | 2·565 | | 7 | 101·08 | 0 | 101·080 | 4 | 78·40 | -0·04 | 78·396 | 3·868 | | | | | | | 5 | 80·57 | +0·04 | 80·574 | 5·184 | | | | | | | 6 | 82·77 | 0 | 82·770 | 6·513 | Round 42. Weight of shot 38·47 lbs. | Clock | Reading | Correction | Corrected reading | Screen | Reading | Correction | Corrected reading | Time of passing each screen | |-------|---------|------------|-------------------|--------|---------|------------|-------------------|-----------------------------| | 4 | 69·77 | 0 | 69·770 | 1 | 81·34 | -0·04 | 81·336 | 0·0000 | | 5 | 84·74 | -0·10 | 84·730 | 2 | 83·17 | +0·02 | 83·172 | -1·230 | | 6 | 99·61 | 0 | 99·610 | 3 | 85·03 | +0·02 | 85·032 | 2·476 | | 7 | 114·41 | 0 | 114·410 | 4 | 86·91 | +0·05 | 86·915 | 3·739 | | | | | | | 5 | 88·82 | +0·01 | 88·821 | 5·018 | | | | | | | 6 | 90·75 | 0 | 90·750 | 6·313 | | | | | | | 7 | 92·71 | -0·08 | 92·702 | 7·625 | | | | | | | 8 | 94·68 | -0·04 | 94·676 | 8·952 | | | | | | | 9 | 96·66 | +0·12 | 96·672 | 1·0295 | | | | | | | 10 | 98·69 | 0 | 98·690 | 1·1654 | Summary (4). Ogival-headed Shot (two diameters), Solid. Time occupied by shot in passing from the first to each succeeding screen. | No. of round | Screen 1 | Screen 2 | Screen 3 | Screen 4 | Screen 5 | Screen 6 | Screen 7 | Screen 8 | Screen 9 | Screen 10 | |--------------|----------|----------|----------|----------|----------|----------|----------|----------|----------|-----------| | 4 | 0·0000 | 0·1266 | 0·2546 | 0·3841 | 0·5151 | 0·6475 | 0·7812 | 0·9163 | 1·0527 | 1·1903 | | 37 | 0·0000 | 0·1275 | 0·2505 | 0·3868 | 0·5184 | 0·6513 | 0·7825 | 0·9152 | 1·0295 | * | | 42 | 0·0000 | 0·1230 | 0·2476 | 0·3739 | 0·5018 | 0·6313 | 0·7625 | 0·8952 | 1·0295 | 1·1654 | Velocities at the following distances from the gun. | No. of round | 150 ft. | 300 ft. | 450 ft. | 600 ft. | 750 ft. | 900 ft. | 1050 ft. | 1200 ft. | 1350 ft. | |--------------|---------|---------|---------|---------|---------|---------|----------|----------|----------| | 4 | f.s. | f.s. | f.s. | f.s. | f.s. | f.s. | f.s. | f.s. | f.s. | | 1185 | 1172 | 1158 | 1145 | 1133 | 1122 | 1110 | 1100 | 1090 | | 37 | f.s. | f.s. | f.s. | f.s. | f.s. | f.s. | f.s. | f.s. | f.s. | | 1176 | 1163 | 1151 | 1140 | 1129 | * | * | * | * | | 42 | f.s. | f.s. | f.s. | f.s. | f.s. | f.s. | f.s. | f.s. | f.s. | | 1220 | 1204 | 1188 | 1173 | 1158 | 1143 | 1130 | 1117 | 1104 | AIR TO THE MOTION OF ELONGATED PROJECTILES. | No. of round. | Weight of shot. | Values of bP. | |---------------|-----------------|---------------| | | lbs. | | | 4 | 38·56 | -0·0071 | | 37 | 38·48 | -0·0069 | | 42 | 38·47 | -0·0081 | | Mean... | 38·52 | Mean of 2 rounds -0·0070 | (5) Ogival-headed Shot (one diameter), Hollow. Round 14. Weight of shot 21·78 lbs. | Clock | Reading. | Correction. | Corrected reading. | Screen. | Reading. | Correction. | Corrected reading. | Time of passing each screen. | |-------|----------|-------------|--------------------|---------|----------|-------------|--------------------|-------------------------------| | 1 | 27·48 | -0·01 | 27·479 Δ₁ | 1 | 96·37 | +0·02 | 96·372 Δ₁ | 0·0000 Δ₁ | | 2 | 48·56 | -0·02 | 48·558 +21·079 Δ₂ | 2 | 98·46 | -0·01 | 98·459 +2·087 Δ₂ | -1·001 +1001 Δ₂ | | 3 | 69·55 | +0·07 | 69·557 20·999 Δ₃ | 3 | 100·61 | -0·12 | 100·598 +2·139 Δ₃ | -2·027 +1026 Δ₃ | | 4 | 90·47 | +0·08 | 90·478 20·921 Δ₄ | 4 | 102·79 | -0·01 | 102·789 +2·191 Δ₄ | -3·078 +1051 Δ₄ | | 5 | 111·33 | -0·11 | 111·319 20·841 Δ₅ | 5 | 105·02 | +0·02 | 105·032 +2·243 Δ₅ | -4·155 +1076 Δ₅ | | 6 | 132·08 | 0 | 132·080 +20·761 Δ₆ | 6 | 107·32 | +0·07 | 107·327 +2·295 Δ₆ | -5·258 +1103 Δ₆ | | | | | | 7 | 109·68 | -0·07 | 109·673 +2·346 Δ₇ | -6·385 +1127 Δ₇ | | | | | | 8 | 112·09 | -0·21 | 112·069 +2·396 Δ₈ | -7·537 +1152 Δ₈ | | | | | | 9 | 114·52 | -0·04 | 114·516 +2·447 Δ₉ | -8·713 +1176 Δ₉ | | | | | | 10 | 117·01 | +0·04 | 117·014 +2·498 Δ₁₀ | -9·911 +1198 Δ₁₀ | Round 16. Weight of shot 21·81 lbs. | Clock | Reading. | Correction. | Corrected reading. | Screen. | Reading. | Correction. | Corrected reading. | Time of passing each screen. | |-------|----------|-------------|--------------------|---------|----------|-------------|--------------------|-------------------------------| | 1 | 5·14 | 0 | 5·140 Δ₁ | 1 | 58·75 | +0·05 | 58·755 Δ₁ | 0·0000 Δ₁ | | 2 | 31·63 | 0 | 31·630 +26·490 Δ₂ | 2 | 61·37 | +0·21 | 61·358 +2·662 Δ₂ | -0·986 +986 Δ₂ | | 3 | 58·07 | 0 | 58·070 26·440 Δ₃ | 3 | 64·02 | 0 | 64·020 +2·724 Δ₃ | -1·994 +1022 Δ₃ | | 4 | 84·45 | -0·02 | 84·448 26·378 Δ₄ | 4 | 66·74 | +0·04 | 66·744 +2·790 Δ₄ | -3·026 +1058 Δ₄ | | 5 | 110·75 | 0 | 110·750 +26·302 Δ₅ | 5 | 69·52 | +0·14 | 69·534 +2·859 Δ₅ | -4·084 +1084 Δ₅ | | | | | | 6 | 72·39 | +0·03 | 72·393 +2·927 Δ₆ | -5·168 +1110 Δ₆ | | | | | | 7 | 75·33 | -0·10 | 75·320 +2·995 Δ₇ | -6·278 +1136 Δ₇ | | | | | | 8 | 78·33 | -0·15 | 78·315 +3·063 Δ₈ | -7·414 +1162 Δ₈ | | | | | | 9 | 81·37 | +0·08 | 81·378 +3·131 Δ₉ | -8·576 +1188 Δ₉ | | | | | | 10 | 84·50 | +0·09 | 84·509 +3·131 Δ₁₀ | -9·764 +1188 Δ₁₀ | Round 18. Weight of shot 21·81 lbs. | Clock | Reading. | Correction. | Corrected reading. | Screen. | Reading. | Correction. | Corrected reading. | Time of passing each screen. | |-------|----------|-------------|--------------------|---------|----------|-------------|--------------------|-------------------------------| | 1 | 28·12 | -0·05 | 28·115 Δ₁ | 1 | 101·10 | -0·02 | 101·098 Δ₁ | 0·0000 Δ₁ | | 2 | 49·99 | -0·11 | 49·979 +21·864 Δ₂ | 2 | 103·23 | +0·21 | 103·251 +2·153 Δ₂ | -0·993 +993 Δ₂ | | 3 | 71·77 | +0·08 | 71·778 21·799 Δ₃ | 3 | 105·46 | -0·01 | 105·459 +2·208 Δ₃ | -2·011 +1018 Δ₃ | | 4 | 93·51 | +0·07 | 93·517 21·739 Δ₄ | 4 | 107·73 | -0·08 | 107·722 +2·263 Δ₄ | -3·055 +1044 Δ₄ | | 5 | 115·21 | -0·09 | 115·201 +21·684 Δ₅ | 5 | 110·03 | +0·09 | 110·039 +2·317 Δ₅ | -4·124 +1069 Δ₅ | | 6 | 136·84 | -0·05 | 136·855 +21·634 Δ₆ | 6 | 112·41 | 0 | 112·410 +2·371 Δ₆ | -5·213 +1094 Δ₆ | | | | | | 7 | 114·86 | -0·25 | 114·855 +2·425 Δ₇ | -6·308 +1120 Δ₇ | | | | | | 8 | 117·32 | -0·06 | 117·314 +2·479 Δ₈ | -7·488 +1145 Δ₈ | | | | | | 9 | 119·85 | -0·03 | 119·847 +2·533 Δ₉ | -8·653 +1170 Δ₉ | | | | | | 10 | 122·43 | +0·04 | 122·434 +2·587 Δ₁₀ | -9·848 +1195 Δ₁₀ | ### Round 20. Weight of shot 21-83 lbs. | Clock | Reading | Correction | Corrected reading | Screen | Reading | Correction | Corrected reading | Time of passing each screen | |-------|---------|------------|------------------|--------|---------|------------|------------------|-----------------------------| | 1 | 11-95 | 0 | 11-950 | 1 | 94-50 | 0 | 94-500 | 0-0000 | | 2 | 33-74 | 0 | 33-740 | 2 | 96-64 | -005 | 96-635 | +2-135 | | 3 | 55-65 | +0-01 | 55-660 | 3 | 98-82 | -006 | 98-826 | +2-191 | | 4 | 77-11 | 0 | 77-110 | 4 | 101-08 | -007 | 101-073 | +2-247 | | 5 | 98-68 | +0-01 | 98-690 | 5 | 103-37 | +004 | 103-374 | +2-301 | | 6 | 120-20 | 0 | 120-200 | 6 | 105-73 | +003 | 105-727 | +2-353 | | 7 | 141-65 | -0-01 | 141-640 | 7 | 108-15 | -019 | 108-131 | +2-404 | | | | | | | 8 | 110-59 | -005 | +2-454 | | | | | | | 9 | 113-08 | +009 | +2-504 | | | | | | | 10 | 115-64 | +003 | +2-554 | ### Round 22. Weight of shot 21-81 lbs. | Clock | Reading | Correction | Corrected reading | Screen | Reading | Correction | Corrected reading | Time of passing each screen | |-------|---------|------------|------------------|--------|---------|------------|------------------|-----------------------------| | 1 | 66-77 | 0 | 66-770 | 1 | 97-44 | 0 | 97-440 | 0-0000 | | 2 | 87-23 | -0-05 | 87-225 | 2 | 99-45 | +006 | 99-456 | +2-016 | | 3 | 107-60 | +0-08 | 107-608 | 3 | 101-52 | -001 | 101-519 | +2-063 | | 4 | 127-92 | 0 | 127-920 | 4 | 103-63 | -001 | 103-629 | +2-110 | | 5 | | | | | 5 | +007 | 105-787 | +2-158 | | 6 | | | | | 6 | -004 | 107-994 | +2-207 | | | | | | | 7 | -010 | 110-250 | +2-256 | | | | | | | 8 | +005 | 112-555 | +2-305 | | | | | | | 9 | +010 | 114-910 | +2-355 | | | | | | | 10 | -005 | 117-315 | +2-405 | ### Round 24. Weight of shot 21-83 lbs. | Clock | Reading | Correction | Corrected reading | Screen | Reading | Correction | Corrected reading | Time of passing each screen | |-------|---------|------------|------------------|--------|---------|------------|------------------|-----------------------------| | 1 | 36-69 | 0 | 36-690 | 1 | 71-78 | 0 | 71-780 | 0-0000 | | 2 | 61-65 | -0-01 | 61-640 | 2 | 74-26 | -006 | 74-254 | +2-474 | | 3 | 86-49 | +0-02 | 86-492 | 3 | 76-79 | -001 | 76-789 | +2-535 | | 4 | 111-24 | -0-02 | 111-238 | 4 | 79-39 | -005 | 79-385 | +2-596 | | 5 | 135-87 | +0-02 | 135-872 | 5 | 82-04 | +003 | 82-043 | +2-658 | | 6 | | | | | 6 | +013 | 84-763 | +2-720 | | | | | | | 7 | -015 | 87-545 | +2-782 | | | | | | | 8 | 0 | 90-390 | +2-845 | | | | | | | 9 | -002 | 93-298 | +2-908 | | | | | | | 10 | -001 | 96-269 | +2-971 | ### Round 26. Weight of shot 21-81 lbs. | Clock | Reading | Correction | Corrected reading | Screen | Reading | Correction | Corrected reading | Time of passing each screen | |-------|---------|------------|------------------|--------|---------|------------|------------------|-----------------------------| | 1 | 38-40 | +0-01 | 38-401 | 1 | 78-16 | -005 | 78-155 | 0-0000 | | 2 | 56-51 | -0-03 | 56-507 | 2 | 79-94 | +010 | 79-950 | +1-795 | | 3 | 74-54 | +0-04 | 74-544 | 3 | 81-78 | +010 | 81-790 | +1-840 | | 4 | 92-52 | -0-01 | 92-519 | 4 | 83-68 | -005 | 83-675 | +1-885 | | 5 | 110-44 | 0 | 110-440 | 5 | 85-59 | +014 | 85-604 | +1-929 | | | | | | | 6 | +006 | 87-576 | +1-972 | | | | | | | 7 | -009 | 89-591 | +2-015 | | | | | | | 8 | -002 | 91-648 | +2-057 | | | | | | | 9 | +007 | 93-747 | +2-099 | | | | | | | 10 | -002 | 95-888 | +2-141 | ### Round 28. Weight of shot 21-83 lbs. | Clock | Reading | Correction | Corrected reading | Screen | Reading | Correction | Corrected reading | Time of passing each screen | |-------|---------|------------|-------------------|--------|---------|------------|-------------------|-----------------------------| | 1 | 27·41 | +·010 | 27·420 | 1 | 77·15 | +·010 | 77·160 | 0·0000 | | 2 | 47·93 | -·005 | 47·925 | 2 | 79·19 | -·010 | 79·180 | 0·0991 | | 3 | 68·37 | +·008 | 68·378 | 3 | 81·25 | 0 | 81·250 | 0·0005 | | 4 | 88·78 | 0 | 88·780 | 4 | 83·37 | 0 | 83·370 | 0·0044 | | 5 | 109·13 | 0 | 109·130 | 5 | 85·54 | 0 | 85·540 | 0·0419 | | | | | | | 6 | 85·76 | 0 | 87·760 | 0·5198 | | | | | | | 7 | 90·05 | -·020 | 90·030 | 0·6312 | | | | | | | 8 | 92·35 | 0 | 92·350 | 0·7452 | | | | | | | 9 | 94·72 | 0 | 94·720 | 0·8616 | | | | | | | 10 | 97·14 | 0 | 97·140 | 0·9804 | ### Round 30. Weight of shot 21-81 lbs. | Clock | Reading | Correction | Corrected reading | Screen | Reading | Correction | Corrected reading | Time of passing each screen | |-------|---------|------------|-------------------|--------|---------|------------|-------------------|-----------------------------| | 1 | 27·54 | 0 | 27·540 | 1 | 91·67 | 0 | 91·670 | 0·0000 | | 2 | 49·61 | -·005 | 49·605 | 2 | 93·83 | +·013 | 93·843 | 0·0995 | | 3 | 71·58 | 0 | 71·580 | 3 | 96·07 | 0 | 96·070 | 0·2014 | | 4 | 93·47 | 0 | 93·470 | 4 | 98·35 | +·002 | 98·352 | 0·3059 | | 5 | 115·28 | 0 | 115·280 | 5 | 100·69 | -·001 | 100·689 | 0·4130 | | | | | | | 6 | 103·08 | 0 | 103·080 | 0·5226 | | | | | | | 7 | 105·53 | -·004 | 105·526 | 0·6347 | | | | | | | 8 | 108·04 | -·014 | 108·026 | 0·7494 | | | | | | | 9 | 110·59 | -·011 | 110·579 | 0·8665 | | | | | | | 10 | 113·18 | +·004 | 113·184 | 0·9863 | ### Round 32+. Weight of shot 21-81 lbs. | Clock | Reading | Correction | Corrected reading | Screen | Reading | Correction | Corrected reading | Time of passing each screen | |-------|---------|------------|-------------------|--------|---------|------------|-------------------|-----------------------------| | 3 | 44·45 | 0 | 44·450 | 1 | 65·19 | 0 | 65·190 | 0·0000 | | 4 | 66·08 | 0 | 66·080 | 2 | ** | * | 67·429 | 0·1036 | | 5 | 87·68 | +·003 | 87·683 | 3 | 69·72 | +·002 | 69·722 | 0·2097 | | 6 | 109·26 | 0 | 109·260 | 4 | 72·07 | -·001 | 72·069 | 0·3183 | | | | | | | 5 | 74·46 | +·005 | 74·465 | 0·4292 | | | | | | | 6 | 76·91 | +·001 | 76·911 | 0·5424 | | | | | | | 7 | 79·41 | -·001 | 79·409 | 0·6581 | | | | | | | 8 | 81·97 | -·006 | 81·964 | 0·7764 | | | | | | | 9 | 84·58 | -·002 | 84·578 | 0·8974 | | | | | | | 10 | 87·25 | 0 | 87·250 | 1·0211 | ### Summary (5). Ogival-headed Shot (one diameter), Hollow. Time occupied by shot in passing from the first to each succeeding screen. | No. of round | Screen 1 | Screen 2 | Screen 3 | Screen 4 | Screen 5 | Screen 6 | Screen 7 | Screen 8 | Screen 9 | Screen 10 | |--------------|----------|----------|----------|----------|----------|----------|----------|----------|----------|-----------| | 14 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | | 16 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | | 18 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | | 20 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | | 22 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | | 24 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | | 26 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | | 28 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | | 30 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | | 32+ | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | 0·0000 | † Gun loaded all night. The initial velocity is therefore reduced. MDCCCLXVIII. ### Velocities at the following distances from the gun. | No. of round | 150 ft. | 300 ft. | 450 ft. | 600 ft. | 750 ft. | 900 ft. | 1050 ft. | 1200 ft. | 1350 ft. | |--------------|---------|---------|---------|---------|---------|---------|----------|----------|----------| | 14 | f.s. | f.s. | f.s. | f.s. | f.s. | f.s. | f.s. | f.s. | f.s. | | 16 | 1499 | 1463 | 1427 | 1393 | 1360 | 1331 | 1302 | 1276 | 1259 | | 18 | 1521 | 1488 | 1453 | 1418 | 1384 | 1351 | 1320 | 1291 | 1263 | | 20 | 1511 | 1473 | 1437 | 1403 | 1371 | 1339 | 1310 | 1282 | 1255 | | 22 | 1515 | 1475 | 1438 | 1403 | 1372 | 1343 | 1315 | 1288 | 1263 | | 24 | 1508 | 1469 | 1435 | 1401 | 1369 | 1337 | 1308 | 1278 | 1251 | | 26 | 1505 | 1465 | 1431 | 1398 | 1367 | 1336 | 1309 | 1283 | 1257 | | 28 | 1514 | 1479 | 1444 | 1408 | 1377 | 1347 | 1316 | 1289 | 1263 | | 30 | 1508 | 1472 | 1435 | 1401 | 1369 | 1338 | 1308 | 1281 | 1252 | | 32 | 1448 | 1414 | 1381 | 1353 | 1325 | 1296 | 1268 | 1240 | 1213 | ### (6) Ogival-headed Shot (two diameters), Hollow. #### Round 15. Weight of shot 21-92 lbs. | Clock | Reading | Correction | Corrected reading | Screen | Reading | Correction | Corrected reading | Time of passing each screen | |-------|---------|------------|-------------------|--------|---------|------------|-------------------|-----------------------------| | 1 | 3-55 | 0 | 3-550 Δ₁ | 1 | 44-59 | -003 | 44-587 Δ₁ | 0-0000 Δ₁ | | 2 | 26-00 | 0 | 26-000 +22-450 -60 | 2 | 46-80 | +008 | 46-808 +2-291 Δ₁ | -0993 +993 Δ₁ | | 3 | 48-39 | 0 | 48-390 22-390 -90 | 3 | 49-08 | +004 | 49-084 +2-276 Δ₁ | -2012 +1019 Δ₁ | | 4 | 70-69 | 0 | 70-690 22-300 -120 | 4 | 51-43 | -015 | 51-415 +2-351 Δ₁ | -3055 +1043 Δ₁ | | 5 | 92-87 | 0 | 92-870 +22-180 -120 | 5 | 53-79 | +011 | 53-801 +2-386 Δ₁ | -4124 +1069 Δ₁ | | | | | | | 56-25 | -009 | 56-241 +2-440 Δ₁ | -5217 +1093 Δ₁ | | | | | | | 58-74 | -006 | 58-734 +2-483 Δ₁ | -6334 +1117 Δ₁ | | | | | | | 61-27 | +008 | 61-278 +2-544 Δ₁ | -7475 +1141 Δ₁ | | | | | | | 63-87 | +002 | 63-872 +2-594 Δ₁ | -8640 +1165 Δ₁ | #### Round 21. Weight of shot 21-89 lbs. | Clock | Reading | Correction | Corrected reading | Screen | Reading | Correction | Corrected reading | Time of passing each screen | |-------|---------|------------|-------------------|--------|---------|------------|-------------------|-----------------------------| | 1 | 4-24 | 0 | 4-240 Δ₁ | 1 | 59-66 | -005 | 59-655 Δ₁ | 0-0000 Δ₁ | | 2 | 24-66 | 0 | 24-660 +20-490 -69 | 2 | 61-66 | +013 | 61-673 +2-055 Δ₁ | -0994 +994 Δ₁ | | 3 | 45-02 | -009 | 45-011 20-351 -65 | 3 | 63-74 | -002 | 63-738 +2-113 Δ₁ | -2014 +1044 Δ₁ | | 4 | 65-29 | +007 | 65-297 20-286 -61 | 4 | 65-85 | +001 | 65-851 +2-161 Δ₁ | -3058 +1067 Δ₁ | | 5 | 85-53 | -008 | 85-522 20-225 -57 | 5 | 68-00 | +012 | 68-012 +2-209 Δ₁ | -4125 +1091 Δ₁ | | 6 | 105-69 | 0 | 105-690 +20-168 -57 | 6 | 70-23 | -009 | 70-221 +2-256 Δ₁ | -5216 +1115 Δ₁ | | | | | | | 72-48 | -003 | 72-477 +2-302 Δ₁ | -6331 +1138 Δ₁ | | | | | | | 74-78 | -001 | 74-779 +2-348 Δ₁ | -7469 +1161 Δ₁ | | | | | | | 77-13 | -003 | 77-127 +2-393 Δ₁ | -8630 +1185 Δ₁ | | | | | | | 79-52 | 0 | 79-520 +2-438 Δ₁ | -9815 +1185 Δ₁ | ### Round 23. Weight of shot 21-94 lbs. | Clock | Reading | Correction | Corrected reading | Screen | Reading | Correction | Corrected reading | Time of passing each screen | |-------|---------|------------|-------------------|--------|---------|------------|-------------------|----------------------------| | 1 | 21-66 | 0 | 21-660 | Δ₁ | 1 | 92-55 | 0 | 92-550 | | 2 | 45-00 | 0 | 45-000 +23-340 | Δ₂ | 2 | 94-84 | -002 | 94-838 | | 3 | 68-28 | +0-010 | 68-290 +23-290 | Δ₃ | 3 | 97-18 | -001 | 97-179 | | 4 | 91-53 | 0 | 91-530 +23-240 | Δ₄ | 4 | 99-57 | +003 | 99-573 | | 5 | 114-73 | -0-010 | 114-720 +23-190 | Δ₅ | 5 | 102-01 | +012 | 102-092 | | 6 | 137-85 | +0-010 | 137-860 +23-140 | Δ₆ | 6 | 104-53 | -003 | 104-527 | | | | | | | | 107-11 | -023 | 107-087 | | | | | | | | 109-71 | -007 | 109-703 | | | | | | | | 112-37 | +005 | 112-375 +2-672 | ### Round 25. Weight of shot 21-97 lbs. | Clock | Reading | Correction | Corrected reading | Screen | Reading | Correction | Corrected reading | Time of passing each screen | |-------|---------|------------|-------------------|--------|---------|------------|-------------------|----------------------------| | 1 | 28-14 | 0 | 28-140 | Δ₁ | 1 | 97-88 | -002 | 97-878 | | 2 | 49-45 | +0-005 | 49-455 +21-315 | Δ₂ | 2 | 99-95 | +015 | 99-965 +2-087 | | 3 | 70-67 | -0-002 | 70-668 +21-112 | Δ₃ | 3 | 102-10 | -001 | 102-099 | | 4 | 91-78 | 0 | 91-780 +21-010 | Δ₄ | 4 | 104-28 | 0 | 104-280 | | 5 | 112-78 | +0-010 | 112-790 +20-910 | Δ₅ | 5 | 106-50 | +007 | 106-507 | | 6 | 133-70 | 0 | 133-700 +20-910 | Δ₆ | 6 | 108-78 | +001 | 108-781 | | | | | | | | 111-11 | -008 | 111-102 | | | | | | | | 113-46 | +010 | 113-470 | | | | | | | | 115-88 | +006 | 115-886 | | | | | | | | 118-35 | 0 | 118-350 +2-464 | ### Round 27. Weight of shot 21-97 lbs. | Clock | Reading | Correction | Corrected reading | Screen | Reading | Correction | Corrected reading | Time of passing each screen | |-------|---------|------------|-------------------|--------|---------|------------|-------------------|----------------------------| | 1 | 1-40 | 0 | 1-400 | Δ₁ | 1 | 45-61 | 0 | 45-610 | | 2 | 20-84 | -0-010 | 20-820 +19-430 | Δ₂ | 2 | 47-59 | +006 | 47-596 +1-916 | | 3 | 40-21 | +0-010 | 40-220 +19-350 | Δ₃ | 3 | 49-46 | +028 | 49-488 +1-962 | | 4 | 59-57 | 0 | 59-570 +19-310 | Δ₄ | 4 | 51-50 | -004 | 51-496 +2-008 | | 5 | 78-88 | 0 | 78-880 +19-310 | Δ₅ | 5 | 53-54 | -008 | 53-548 +2-052 | | | | | | | | 55-65 | -007 | 55-643 +2-095 | | | | | | | | 57-78 | -002 | 57-782 +2-139 | | | | | | | | 59-96 | +004 | 59-964 +2-182 | | | | | | | | 62-19 | 0 | 62-190 +2-226 | | | | | | | | 64-46 | 0 | 64-460 +2-270 | ### Round 29. Weight of shot 21-97 lbs. | Clock | Reading | Correction | Corrected reading | Screen | Reading | Correction | Corrected reading | Time of passing each screen | |-------|---------|------------|-------------------|--------|---------|------------|-------------------|----------------------------| | 1 | 16-26 | 0 | 16-260 | Δ₁ | 1 | 62-14 | 0 | 62-140 | | 2 | 39-53 | 0 | 39-530 +23-270 | Δ₂ | 2 | 64-44 | -009 | 64-431 +2-291 | | 3 | 62-73 | +0-010 | 62-740 +23-137 | Δ₃ | 3 | 66-77 | +008 | 66-778 +2-347 | | 4 | 85-88 | -0-003 | 85-877 +23-052 | Δ₄ | 4 | 69-20 | -019 | 69-181 +2-403 | | 5 | 108-93 | -0-001 | 108-929 +23-052 | Δ₅ | 5 | 71-64 | -002 | 71-638 +2-457 | | | | | | | | 74-15 | -002 | 74-148 +2-563 | | | | | | | | 76-81 | -009 | 76-711 +2-615 | | | | | | | | 79-32 | +006 | 79-326 +2-666 | | | | | | | | 81-92 | * | 81-922 +2-718 | | | | | | | | 84-71 | 0 | 84-710 +2-718 | ### Round 31. Weight of shot 21-91 lbs. | Clock | Reading | Correction | Corrected reading | Screen | Reading | Correction | Corrected reading | Time of passing each screen | |-------|---------|------------|-------------------|--------|---------|------------|-------------------|-----------------------------| | 1 | 13-22 | -0.010 | 13-210 | 1 | 57-06 | 0 | 57-060 | 0-0000 | | 2 | 32-46 | +0.010 | 32-470 | 2 | 58-96 | -0.003 | 58-957 | 0-0993 | | 3 | 51-66 | -0.010 | 51-650 | 3 | 60-89 | +0.005 | 60-895 | 1-938 | | 4 | 70-73 | +0.010 | 70-740 | 4 | 62-87 | +0.004 | 62-874 | 1-979 | | 5 | 89-73 | 0 | 89-730 | 5 | 64-89 | -0.006 | 64-896 | 2-022 | ### Round 33†. Weight of shot 21-94 lbs. | Clock | Reading | Correction | Corrected reading | Screen | Reading | Correction | Corrected reading | Time of passing each screen | |-------|---------|------------|-------------------|--------|---------|------------|-------------------|-----------------------------| | 3 | 45-67 | 0 | 45-670 | 1 | 59-32 | 0 | 59-320 | 0-0000 | | 4 | 64-25 | 0 | 64-250 | 2 | 61-21 | +0.003 | 61-213 | 1-893 | | 5 | 82-82 | 0 | 82-820 | 3 | 63-14 | +0.006 | 63-146 | 1-933 | ### Summary (6). Ogival-headed Shot (two diameters), Hollow. Time occupied by shot in passing from the first to each succeeding screen. | No. of round | Screen 1 | Screen 2 | Screen 3 | Screen 4 | Screen 5 | Screen 6 | Screen 7 | Screen 8 | Screen 9 | Screen 10 | |--------------|----------|----------|----------|----------|----------|----------|----------|----------|----------|-----------| | 15 | 0-0000 | 0-0993 | 0-2012 | 0-3055 | 0-4124 | 0-5217 | 0-6334 | 0-7475 | 0-8640 | ** | | 21 | 0-0000 | 0-0994 | 0-2014 | 0-3058 | 0-4125 | 0-5216 | 0-6331 | 0-7469 | 0-8630 | 0-9815 | | 23 | 0-0000 | 0-0986 | 0-1955 | 0-3028 | 0-4084 | 0-5164 | 0-6268 | 0-7397 | 0-8549 | ** | | 25 | 0-0000 | 0-0983 | 0-2008 | 0-3047 | 0-4108 | 0-5192 | 0-6298 | 0-7427 | 0-8580 | 0-9757 | | 27 | 0-0000 | 0-0989 | 0-2003 | 0-3041 | 0-4102 | 0-5185 | 0-6291 | 0-7420 | 0-8572 | 0-9747 | | 29 | 0-0000 | 0-0989 | 0-2002 | 0-3040 | 0-4101 | 0-5186 | 0-6294 | 0-7425 | 0-8578 | 0-9754 | | 31 | 0-0000 | 0-0992 | 0-2008 | 0-3044 | 0-4104 | 0-5188 | 0-6296 | 0-7429 | 0-8587 | 0-9770 | | 33 | 0-0000 | 0-1018 | 0-2058 | 0-3121 | 0-4207 | 0-5319 | 0-6456 | 0-7617 | 0-8804 | ** | Velocities at the following distances from the gun. | No. of round | 150 ft. | 300 ft. | 450 ft. | 600 ft. | 750 ft. | 900 ft. | 1050 ft. | 1200 ft. | 1350 ft. | |--------------|---------|---------|---------|---------|---------|---------|----------|----------|----------| | 15 | 1511 | 1472 | 1438 | 1403 | 1372 | 1343 | 1315 | 1288 | ** | | 21 | 1509 | 1471 | 1437 | 1406 | 1375 | 1345 | 1318 | 1292 | 1266 | | 23 | 1521 | 1487 | 1452 | 1420 | 1389 | 1359 | 1329 | 1302 | ** | | 25 | 1511 | 1478 | 1444 | 1414 | 1384 | 1356 | 1329 | 1301 | 1274 | | 27 | 1517 | 1479 | 1445 | 1414 | 1385 | 1356 | 1329 | 1302 | 1277 | | 29 | 1517 | 1481 | 1445 | 1414 | 1382 | 1354 | 1326 | 1301 | 1276 | | 31 | 1511 | 1478 | 1448 | 1415 | 1384 | 1354 | 1324 | 1295 | 1268 | | 33 | 1473 | 1442 | 1411 | 1381 | 1349 | 1319 | 1292 | 1264 | ** | † Round 32 had remained in the gun all night. This probably accounts for the reduced initial velocity of Round 33. Since the second differences of $0, t_2, t_3, \ldots, t_{10}$ are nearly constant in the preceding experiments, we are led to the equation $$t = as + bs^2,$$ connecting space and time. This shows that if $v$ be the velocity of the shot at time $t$, and $f$ the retarding force, then $$v = \frac{ds}{dt} = \frac{1}{a + 2bs};$$ and if $v = V$ when $s = 0$, we have $V = \frac{1}{a}$; $$\therefore v = \frac{1}{V + 2bs},$$ and $$f = \frac{d^2s}{dt^2} = -2bv^3.$$ The values of $bl^2$ for each experiment were obtained as follows: Since $$t = as + bs^2$$ If $$s = l,$$ then $$t_2 = al + bl^2,$$ $$s = 2l,$$ then $$t_3 = 2al + 4bl^2,$$ $$s = n-1l,$$ then $$t_n = n-1al + n-1^2bl^2,$$ or $$\frac{t_2}{1} = al + bl^2,$$ $$\frac{t_3}{2} = al + 2bl^2,$$ $$\frac{t_n}{n-1} = al + n-1bl^2.$$ Finding, then, the numerical values of $$\frac{t_2}{1}, \frac{t_3}{2}, \ldots, \frac{t_n}{n-1}$$ for each experiment, and taking the difference of two of these quantities, we find $$\frac{t_n}{n-1} - \frac{t_3}{2} = n-3bl^2.$$ where \( l = 150 \) feet, and the mean value of \( bl^2 \) so determined from each round may be taken to correspond to the mean velocity of that round. Since the retarding force is \( f' = -2bv^3 \), acting upon an experimental shot of weight \( W' \) lbs., the resistance of the air measured in pounds \[ = -2bv^3 \frac{W'}{g}; \] and the resistance will always be the same against a shot of the same external form when moving with the velocity \( v \). If \( W \) be the weight of a different shot, then the retarding force \[ f' = -2bv^3 \frac{W'}{g} \times \frac{g}{W} = -2 \left( b \frac{W'}{W} \right) v^3, \] or \( b \) varies inversely as the weight of a shot of given external form. If we vary the diameter of the shot, it is usual to assume that the resistance of the air varies as the square of the diameter for similar forms of heads of shot. The values of \( 2b \) are given below in terms of \( R \), the radius of the shot in feet, and \( W \), its weight in pounds. But inasmuch as the diameters of shot are generally given in inches, the same values of \( 2b \) have been expressed in terms of \( d \), the diameter of the shot in inches. Unfortunately there are only three successful rounds for each of the solid ogivals, and one value of \( bl^2 \) in each case is manifestly too large. This must have arisen from unsteadiness of the shot in its flight. If we reject these two, the values of \( bl^2 \) derived from the solid ogivals agree very well with those derived from the hollow shot of the like forms. For the hemispherical head we have \( bl^2 = 0.0084 \) for shot of mean weight 39.34 lbs., and 4.7 inches in diameter. To find the value of \( 2b \) adapted for hemispherical-headed shot of weight \( W \) lbs., and diameter \( = d \) inches \( = 2R \) feet, we have \[ 2b = \frac{2 \times 0.0084}{(150)^2} \times \left( \frac{d}{4.7} \right)^2 \left( \frac{39.34}{W} \right) = 0.000001329 \frac{d^2}{W}, \] or \[ = \frac{2 \times 0.0084}{150} \left( \frac{24R}{4.7} \right)^2 \frac{39.34}{W} = 0.000077 \frac{R^2}{W}, \text{ and so on for the rest.} \] Table of values of \( 2b \) for differently formed heads adapted for elongated shot of weight \( W \) lbs. and radius \( R \) feet, or diameter \( d \) inches. | Form of head | Experimental value of \( bl^2 \) | Mean weight of experimental shot | Value of \( 2b \) when diameter \( = 2R \) feet. | Value of \( 2b \) when diameter \( = d \) inches. | |--------------|---------------------------------|---------------------------------|-----------------------------------------------|-----------------------------------------------| | (1) Hemispherical | 0.0084 | 39.34 | -0.000077 \(\frac{R^2}{W}\) | -0.000001329 \(\frac{d^2}{W}\) | | (2) Hemispherical | 0.0067 | 38.70 | -0.000060 \(\frac{R^2}{W}\) | -0.000001043 \(\frac{d^2}{W}\) | | (3) Ogival (1 diameter) | 0.0070 | 39.56 | -0.000064 \(\frac{R^2}{W}\) | -0.000001114 \(\frac{d^2}{W}\) | | (4) Ogival (2 diameters) | 0.0070 | 38.52 | -0.000062 \(\frac{R^2}{W}\) | -0.000001085 \(\frac{d^2}{W}\) | | (5) Ogival (1 diameter) | 0.0125 | 21.81 | -0.000063 \(\frac{R^2}{W}\) | -0.000001097 \(\frac{d^2}{W}\) | | (6) Ogival (2 diameters) | 0.0118 | 21.94 | -0.000060 \(\frac{R^2}{W}\) | -0.000001042 \(\frac{d^2}{W}\) | Although the motion of a shot may be well represented by supposing a retarding force $-2bv^3$ to act through a range of 1400 feet, there is reason to suppose that for velocities ranging from 1500 to 900 feet per second the value of $b$ will be less for the lower velocities with equal degrees of steadiness. It unfortunately happens, however, that the angular velocity imparted to a shot, which most probably remains little changed during the time of flight, depends directly upon the initial velocity of the shot. Hence, when shot are fired with low initial velocities with a view to determine the value of $b$ for low velocities, the steadiness of the shot is diminished, and therefore there is an increase of the resistance of the air on this account. The only way to meet the difficulty is to place screens near the gun to find the initial velocity, and others at a distance of 2000 yards or more, and so compare theory and experiment. It is worthy of notice that if a body move in a straight line under the action of a force varying as the velocity cubed, the mean velocity obtained by dividing space by time is exactly the actual velocity at the middle point of that space. Thus $$\frac{\text{space } 2s}{\text{time of describing space } 2s} = \frac{2s}{2as + 4bs^3} = \frac{1}{a + 2bs} = \text{velocity at distance } s.$$ The date of the Report of the above experiments was October 23, 1866. I have long been aware that Major Otto had made trial of various laws of the resistance of the air in a work published in 1855. The law of the cube of the velocity was tried, but without any definite result*. It was in April 1867 that I first learnt that M. Hélie† had proposed the law of the cube of the velocity as the law of the resistance of the air to elongated projectiles in a work dated 1865, which law he had deduced from experiments made at Gâvre in 1859, 1860, and 1861‡. It will be convenient to quote his own statement of the best series of experiments made at Gâvre in 1859, in order to show the nature of the work done, and the perfect independence of my own methods and numerical results. M. Hélie used one of the electro-ballistic pendulums to measure his velocities, but he does not state distinctly which it was. “Si la résistance de l’air est réellement proportionnelle au cube de la vitesse, on doit avoir un résultat sensiblement constant en substituant, dans l’expression $\frac{v' - v''}{v'v''x}$, les valeurs de $v'$, $v''$ et $x$ correspondantes à la même charge. * Hilfsmittel für ballistische Rechnungen, 1855, p. 12. † Traité de Balistique, 1865. ‡ [In a Memoir, “Études de Balistique expérimentale,” presented to the Belgian Academy by Captain P. C. Boulenge, June 12, 1867, the author, having deduced the cubic law of resistance of the air from his experiments, proceeds to remark:— “Ce résultat est en accord complet avec les travaux les plus récents faits en France; en effet, les expériences exécutées par la commission des principes du tir, en 1856 et 1857, ont conduit M. le capitaine Welter, professeur à l’École d’application de l’artillerie et du génie de Metz, à reconnaître que la résistance de l’air sur les projectiles sphériques est simplement proportionnelle au cube de la vitesse. “Cette loi, admise depuis 1862, comme base des études balistiques à cette école, a fourni des formules très-simples et très-facilement calculables sans l’intervention de tables, se prêtant à des recherches que les anciennes formules balistiques ne permettaient pas d’aborder, et donnant des résultats plus conformes à la pratique” (p. 84).—Aug. 1, 1868.] "Résultats moyens des expériences. (Chaque vitesse est déduite de 30 coups.) | Charge du canon. | Première vitesse, \(v'\), à 33 mètres du canon. | Dernière vitesse, \(v''\). | Intervalle des points d'observation, \(x\). | Valeur de \(\frac{v' - v''}{v'v''x}\) ou \(c\). | |------------------|---------------------------------|-----------------|-----------------|-----------------| | kilog. | métr. | métr. | métr. | | | 1·5 | 225·1 | 215·6 | 464 | 0·000000422 | | 2·0 | 263·7 | 252·5 | 467 | 0·0000008360 | | 2·5 | 291·9 | 275·9 | 467 | 0·000000425 | | 3·0 | 309·6 | 291·8 | 467 | 0·000000422 | | 3·5 | 326·9 | 306·4 | 467 | 0·000000438 | "Sauf l'anomalie que présente la charge de 2k·0 la valeur de \(\frac{v' - v''}{v'v''x}\) se montre sensiblement constante. La formule \(r = cv^3\) est donc suffisamment justifiée"*. The diameter of the shot was 0m·1623 = 6ins.39, its weight 30k = 66lbs.14, and its form hollow ogival with head struck with a radius of nearly two diameters. In order to facilitate the comparison of this experiment with my own, I have converted the French into English measures. | Charge. | \(v'\). | \(v''\). | \(x\). | \(c = 2b\). | |--------|--------|--------|------|------------| | lbs. | f.s. | f.s. | ft. | | | 3·31 | 738·5 | 707·4 | 1529·4 | 0·000000391 | | 4·41 | 865·1 | 828·4 | 1532·2 | 0·000000334 | | 5·51 | 957·7 | 905·2 | 1532·2 | 0·000000395 | | 6·61 | 1015·8 | 957·4 | 1532·2 | 0·000000392 | | 7·72 | 1072·5 | 1005·3 | 1532·2 | 0·000000407 | Mean value of \(c\) or \(2b\) = 0·000000384 \(= \cdot0000358 \frac{R^2}{W}\) if R ft. = radius of shot. \(= \cdot00000069 \frac{d^2}{W}\) if d in. = diameter of shot. The values of \(2b\) given by my own experiments for shot of the same form were \(\cdot000060 \frac{R^2}{W}\) or \(\cdot0000001042 \frac{d^2}{W}\) for velocities ranging from 1520 to 1270 f.s.; so that the value of \(2b\) deduced from my own experiments might be expected to be sensibly greater than its value deduced from M. Hélie's experiments made with lower velocities. There are three or four other tabular statements of less complete experiments given by M. Hélie. In the present state of the question it is impossible to make any more exact comparison of the two systems of experimenting†. * Hélie, Traité, pp. 407, 408. † An extended series of experiments just completed at Shoeburyness, with 3, 5, 7, and 9-inch elongated shot, has shown conclusively that, although the motion of a shot may be well represented by supposing the resistance of the air to vary as the cube of the velocity, and to be equal to \(-2bv^3\) for a range of 1200 or 1300 feet, still AIR TO THE MOTION OF ELONGATED PROJECTILES. Meteorological Register, Shoeburyness. | Date | Hour | Barometer | Thermometer | Wind | Weather and Remarks | |----------|------|-----------|-------------|------|--------------------| | | A.M. | P.M. | Reading | Dry | Wet | Direction | Force | | | 1866 | | | | | | | | | | Sept. 25th | 10 | ... | 30-05 | 61 | 61 | S.W. by S.| 2 | Clear. | | | ... | 3 | 30-05 | 61 | 61 | S.W. by W.| 2 | Clear. | | Sept. 26th | 10 | ... | 30-05 | 61 | 61 | S.S.W. | 3 | Hazy. | | | ... | 3 | 30-00 | 61 | 60 | S.W. by S.| 3 | Ditto. | | Sept. 27th | 10 | ... | 29-95 | 59 | 56 | N.W. by W.| 2 | Ditto with rain. | | | ... | 3 | 29-95 | 60 | 59 | W. | 2 | Hazy. | Direction of firing from the North to South. $b$ is subject to considerable variations for large variations in $v$. For ogival heads struck with a radius of $1\frac{1}{2}$ diameter of the shot, fired with great steadiness, the following values of $v$ and $b$ have been found. | $v$ (f.s.) | $\frac{2b}{d^2}$ | |-----------|------------------| | 900 | 0-0000000604 | | 950 | 0-0000000640 | | 1000 | 0-0000000720 | | 1050 | 0-0000000868 | | 1100 | 0-0000001005 | | 1200 | 0-0000001085 | | 1300 | 0-0000001046 | | 1400 | 0-0000000977 | | 1500 | 0-0000000856 | | 1600 | 0-0000000925 | It thus appears that M. Hélin's value of $2b$ or $c$ is true only for velocities about 950 f.s.—Aug. 1, 1868.]