mm^ O^ ry IT -•SO or REESE LIBRARY UNIVERSITY OF CALIFORNIA?. /deceived . - . . .- „ ^^^:^ht^lJLC/ , i88 y A c cessions No. <3 ^ ^/S ' Shelf fJo. Of- JL -3fe jt Digitized by the Internet Archive in 2007 with funding from IVIicrosoft Corporation http://www.archive.org/details/exteriorballistiOOcoasrich department of Ballistics of the U. S. Artillery School. EXTERIOR BALLISTICS IN THE PLANE OF FIRE BY JAIVIKS ISA, INOAIvIvS, Cai'tain P'ikst Aktilleky, U. S. Army, Instructor, NEW YORK: D. VAN NOSTRAND, PUBLISHER, 23 MURRAY AND 27 WARREN STREETS, 1886, ■I^*'" ^V' HEADQUARTERS UNITED STATES ARTILLERY SCHOOL. Fort Monroe, Va., February, 1885. Approved and Authorized as a Text- Book. Pat. 26, Regulations U. S. Artillery School, appioved 1882, viz.: " To the end that the school shall keep pace with professional progress, it is made the duty of Instructors and Assistant-Instructors to prepare and arrange, in accordance with the Programme of Instruction, the subject-matter of the courses of study committed to their charge The same shall be sub- mitted to the Staff, and, after approval by that body, the matter shall become the authorized text-books of the school, be printed at the school, issued, and adhered to as such." _ -, ^y By order of Lieutenant-Colonel Tidball. Tasker H. Bliss, First Lieutenant ist Artillery, Adjutant. Copyright, 1886, By D. van NOSTRAND. PREFACE. This work is intended, primarily, as a text-book for the use of the officers under instruction at the U. S. Artiller}^ School, and the arrangement of the matter has been made with reference to the wants of the class-room. The aim has been to present in one volume the various methods for calculating range-tables and solving impor- tant problems relating to trajectories, which are in vogue at the present day, developed from the same point of view and with a uniform notation. The convenience of this is manifest. It is hoped, also, that the practical artillerist will find here all that he may require either for computing range- tables for the guns already in use, or for determining in advance the ballistic efficiency of those which may be proposed in the future. ERRATA Page 54, line 27 : For - read -. u V Page 64, line 4 : For (i) and {(f) read {i\ and (^X Page 72, line 18: 4 i For sec ^ read sec 5 f. / Page 73, line 22 : ' -^4- ^^^^ V* Page 93, line 11 : For g read j. Page 116, equation {78): For r— read cos'' (p 2 cos ip CONTKNTS INTRODUCTION. Object and Definitions, •AGE 5 CHAPTER I. RESISTANCE OF THE AIR. Normal Resistance to the Motion of a Plane, Oblique Motion, ....... Pressure on a Surface of Revolution, .... Applications, ....... Resistance of the Air to the Motion of Ogival-headed Projectiles, CHAPTER II. EXPERIMENTAL RESISTANCE. Notable Experiments, .... Methods of Determining Resistances, Russian Experiments with Spherical Projectiles, Mayevski's Deductions from the Krupp Experiments, Ilojel's Deductions from the Krupp Experiments, Bashforth's Coefficients, Law of Resistance deduced from Bashforth's K, . Comparison of Resistances, Example, ..... 7 9 9 10-13 13-16 17 19 23 28 29 31 35 37 39 CHAPTER III. DIFFERENTIAL EQUATIONS OF TRANSLATION — GENERAL PROPERTIES OF TRAJECTORIES. Preliminary Considerations, Notation, ...... DifFerential Equations of Translation, Minimum Velocity, ..... Limiting Velocity, ..... Limit of the Inclination in the Descending Branch, Asymptote to the Descending Branch, Radius of Curvature, .... 4.1 41 42 46 47 48 49 50 CONTENTS. CHAPTER IV. RECTILINEAR MOTION. Relation between Time, Space, and Velocity, Projectiles differing from the Standard, Formulas for Calculating the T- and ^'-Functions, Ballistic Tables, .... Extended Ranges, .... Comparison of Calculated with Observed Velocities, PAGE 52 53 54 57 59 60 CHAPTER V. RELATION BETWEEN VELOCITY AND INCLINATION. General Expressions for the Inclination in Terms of the Velocity Bashforth's Method, . High-Angle and Curved Fire, Siacci's Method, Niven's Method, Modification of Niven's Method, CHAPTER VI. HIGH-ANGLE FIRE. Trajectory in Vacuo, Constant Resistance, Resistance Proportional to the First Euler's Method, Bashforth's Method, . Modification of Bashforth's Method, Power of the Velocity, 64 65 66 68 73 75 77 80 81 91 95 97 CHAPTER VII. DIRECT FIRE. Niven's Method, . 102 Sladen's Method, . 106 Siacci's Method, . 108 Practical Applications, .118 Correction for Altitude, . 127 EXTERIOR BALLISTICS IN THE PLANE OF FIRE. INTRODUCTION. Definition and Object. — Ballistics, from the Greek l^aUw, I throw, is, in its most general signification, the science which treats of the motion of heavy bodies pro- jected into space in any direction ; but its meaning is usu- ally restricted to the motion of projectiles of regular form fired from cannon or small arms. The motion of a projectile may be studied under three different aspects, giving rise to as many different branches of the subject, called respectively Interior Ballistics, Ex- terior Ballistics, and Ballistics of Penetration, 1. Interior Ballistics.— Interior Ballistics treats of the motion of a projectile within the bore of the gun while it is acted upon by the highly elastic gases into which the powder is converted by combustion. Its object is to deter- mine by calculation the velocity of translation and rotation which the combustion of a given charge of powder of known constituents and quality is capable of imparting to a projectile, and the effect upon the gun. 2. Exterior Ballistics. — Exterior Ballistics considers the circumstances of motion of a projectile from the time it emerges from the gun until it strikes the object aimed at. Its data are the shape, caliber, and weight of the pro- jectile, its initial velocity both of translation and of rotation. 6 EXTERIOR BALLISTICS. the resistance it meets from the air, and the action of grav- ity. 3. Ballistics of Penetration. — This branch of the subject has reference to the effect of the projectile upon an object; the data being the energy and incHnation with which the projectile strikes the object, the nature of the re- sistance it encounters, etc. The above is not the order in which the three divisions of the subject are usually presented to the practical artil- lerist, but the reverse. He desires to penetrate or destroy a given object — say the side of an armored ship. Ballistics of penetration enables him to determine the minimum en- ergy which his projectiles must have on impact, and the proper striking angle, to accomplish the desired result. Exterior Ballistics would then carry the data from the ob- ject to be struck to the gun, and determine the necessary initial velocity and angle of elevation. Lastly, Interior Ballistics would ascertain the proper charge and kind of powder to be used to give the projectile the initial velocity demanded. The following pages treat only of Exterior Ballistics; and this subject will be limited, at present, to motion in the vertical plane passing through the axis of the piece. CHAPTER I. RESISTANCE OF THE AIR. Preliminary Considerations. — The molecular the- ory of gases is not yet sufficiently developed to be made the basis for calculating the resistance which a projectile experiences in passing through the air. We know, how- ever, that if a body moves in a resisting medium, fluid or gaseous, the particles of the fluid must be displaced to allow the body to pass through ; and hence momentum will be communicated to them, which must be abstracted from the moving body. From the assumed equality of momenta lost and gained Newton deduced the law of the square of the velocity to express the resistance of the air to the mo- tion of a body moving in it. The following, which is the ordinary demonstration, supposes the particles of air against which the body im- pinges to be at rest, and takes no account of the reaction of the molecules upon each other, nor of their friction against the surface of the body. The result will therefore be but an approximation, which must be estimated at its true value by means of well-devised and accurately-executed experiments. Normal Resistance to the Motion of a Body presenting a Plane Surface to the Medium.— Let a moving body present to the particles of a fluid against which it impinges, and which are supposed to be at rest, a plane surface whose area is 5, and which is normal to the direction of motion. Let w be the weight of the moving body, V its velocity at any time t, d the weight of an unit- volume of the fluid, and ^ the acceleration of gravity. The plane 5 will describe in an element of time dt a path v d t, and displace a volume of fluid Svdt ; therefore the mass of fluid put in motion during the element of time is- Svdt. 8 EXTERIOR BALLISTICS. And as this moves with the velocity v, its momentum is — Sv'dt; and this has been abstracted from the moving body, whose velocity has thereby been decreased by dv. Therefore dvzm-Sv'dt g g or IV dv b ^ ^ ~ - -y- = - 5 z/" g dt g The first member of this last equation is the momentum- decrement of the body, due to the pressure of the fluid upon the plane face 5, and is therefore a measure of this pressure. Calling this latter P, we have _, w dv <5 - , g dt g or, per unit of mass, w dt w As before stated, several circumstances have been omit- ted in this investigation ^vhich, if taken into account, would probably increase the pressure somewhat, at least for high velocities. We will therefore introduce into the second member of the above equation an undetermined multiplier k {k y i), and we have P = k-Sv' .. g (0 The pressure is, therefore, proportional to the area of the plane surface, to the density of the medium, and to the square of the velocity. If in equation (i) we make 5=1, the second member will then express the normal pressure upon an unit-surface moving with the velocit}^ v; calling this /o,' we have and P^AS EXTERIOR BALLISTICS. Oblique Motion. — If the surface 5 is oblique to the direction of motion, let f be the angle which the normal to the plane makes with that direction ; and resolve the velo- city 7.' into its components v cos f, perpendicular, and v sin f, parallel, to vS. This last, neglecting friction, having no re- tarding effect, we have for the normal pressure upon 5 the expression P=/^-'z^''5cos' 6=/o 5cos'f / Poncelet {Mecanique Industrielle, 403) cites the following empirical formula for calculating the normal pressure, viz. : i"fsec' e ^ ' derived by Colonel Duchemin from the experiments of Vince, Hutton, and Thibault. As this expression satisfied the whole series of experiments upon which it was based better than any other that was proposed, we will adopt it in what follows. Pressvire on a Surface of Revolution. — Let A D B, Fig. I, be the generating curve of a surface of revolution, which we will suppose moves in a resisting medium in the direction of its axis,^_(9 A. \{ m in' in" = <^5 be an element of the surface, inclined ^ to the direction of motion by the angle Ninv=^e, it will suffer a pressure in the direction of the normal N in, equal, by (2), to 2p,dS I + sec' e Resolving this pressure into two components. 2 p^d S cos> 8 2p^d Ssin s i°+sec'. ' P^""^'^"'' ^"^ i+^c^' Pe'-Pe"dic"la'-. lO EXTERIOR BALLISTICS. to OA, it is plain that this last will be destroyed by an equal and contrary pressure upon the elementary surface n n' n" situated in the same meridional section as ;;/ in' m" , and making the same angle with the direction of motion. It is only necessary, therefore, to consider the first component, 2p^d S cos £ I + sec' e It is evident that expressions identical with this last are applicable to every element of the zone ;// m' n n' described by the revolution of m in' ; and we may, therefore, extend this so as to include the entire zone by substituting its area for dS. If we take O A for the axis of A'', this area will be expressed by 2 it yds, in which ds is an element of the gene- rating curve ; therefore, the pressure upon any elementary zone will be , y ds cos f dx' Substituting — dy for ds cos f , and 2 -|- -r-, for i -\- sec* e, and integrating between the limits 4- = /, and x^o, we have /' y, we shall evidently have ~ p^P-P' It is evident that P' will be zero whenever the velocity of the projectile is greater than that of air flowing into a vacuum. In this case, and also when P' is so small rela- tively to P that it may be neglected, we have approxi- mately p^P Application to Ogival Heads struck witli Radii of one and a half Calibers. — Experiments have proven that for practicable velocities exceeding about 1300 f. s. the resistance of the air is sensibly proportional to the square of the velocity ; and a discussion of the published results of Professor Bashforth's experiments has shown that, within the above limits, the resistance to elongated projectiles having ogival heads struck with radii of one and a half cali- bers may be approximately expressed by the equation, pz=z- d'v' S in which d is the diameter of the projectile in inches, ^ the acceleration of gravity (32.19 ft.), and log A = 6,1525284 — 10. Whence p — o.o'44i37^'z;' Making b = 534.22 grains, which is the weight of a cubic foot of air adopted by Professor Bashforth, and F{n)=:F{^) = 0.3176, we find for the corresponding expression for P P = o.o%io6g k d' v' A comparison of the second members of these two equa- tions seems to warrant the conclusion that for velocities greater than about 1300 f. s., the rear pressure is either zero or so small relatively to the front pressure that it may be EXTERIOR BALLISTICS. 1 5 neglected without sensible error. Equating the two mem- bers, we find for velocities greater than 1300 f. s. k— 1.0747 In the following table the first and second columns give the velocities and corresponding resistances, in pounds, to an elongated projectile one inch in diameter and having an ogival head of one and a half calibers. They were deduced from Bashforth's experiments by Professor A. G. Greenhill, and are taken from his paper published in the Proceedings of the Royal Artillery Institution, No. 2, Vol. XIII. The third column contains the corresponding pressures upon the head of the projectile computed by the formula 576^ in which the constants have the values already given. The fourth and fifth columns are sufficiently indicated by their titles. These results are reproduced graphically in Plate I. A is the curve of resistance (^), drawn by taking the velo- cities for abscissas and the corresponding resistances, in pounds, for ordinates. This curve is similar to that given by Professor Greenhill in his paper above cited. B is the curve of front pressures (P), and is a parabola whose equa- tion is given above. It will be seen that while the velocity decreases from 2800 f. s. to 1300 f s., the two curves closely approximate to each other; the differences (P— />) for the same abscissas being relatively small and alternately plus and minus. As the velocity still further decreases, the curve of resistance falls rapidly below the parabola B, showing that the resistance now decreases in a higher ratio than the square of the velocity. This continues down to about 800 f. s., when the parabolic form of the curve is again resumed, but still below B. The differences P— p from z/= 1300 f. s. to 2/= 100 f s. are shown graphically by the curve (7, which may represent, approximately, the rear pressures iox decreas- ing velocities, and possibly account, in a measure, for the i6 EXTERIOR BALLISTICS. sudden diminution of resistance in the neighborhood of the velocity of sound. V p P P-P P-P V p P P-P P-P P P 2800 2750 2700 35.453 33.586 31.846 34.603 33.378 32.176 -0.850 -0.208 + 0.330 1080 1070 1060 3-999 3.756 3.478 5.148 5.053 4.959 + 1.149 1.297 1. 481 0.223 0.256 0.298 2650 2600 2550 30.241 28.613 27.243 30.995 29.836 28.700 + 0.754 + 1.223 + 1.457 1050 1040 1030 3.139 2.823 2.604 4.866 4.774 4.684 1.727 1. 951 2.080 0.355 0.409 0.444 2500 2450 2400 26.406 25.898 25.588 27.585 26.493 25.422 + 1.379 +-0.595 -0.166 1020 lOIO 1000 2.482 2.404 2.330 4.592 4.502 4.414 2. 114 2.098 2.084 0.459 0.466 0.472 2350 2300 2250 25.242 24.760 23.566 24.374 23.347 22.344 -0.868 -1. 413 — 1.222 990 980 970 2.261 2.193 2.127 4.326 4.239 4.153 2.065 2.046 2.026 0.477 0.483 0.488 2200 2150 2100 22.158 20.811 19.504 21.362 20.402 19.464 -0.796 -0.409 —0.040 960 950 940 2.061 1.998 1.935 4.068 3.983 3.900 2.007 1.985 1.965 0.493 0.498 0.504 2050 2900 1950 18.229 17.096 16.127 18.548 17.654 16.783 + 0.319 + 0.558 + 0.656 930 920 910 1.874 1. 814 1.756 3.817 3.736 3.655 1.943 1.922 1.899 0.509 0.515 0.520 1900 1850 1800 15.364 14.696 14.002 15.934 15.106 14.300 + 0.570 + 0.410 +0.298 900 850 800 1.699 1. 431 1. 212 3.575 3.189 2.825 1.876 1.758 1. 613 0.525 0.551 0.580 1750 1700 1650 13.318 12.666 12.030 13.517 12.766 12.016 + 0.199 + 0.100 —0.014 750 700 650 1.043 0.905 0.784 2.483 2. 163 1.865 1.440 1.258 1. 081 0.580 0.581 0.580 1600 1550 1500 II. 416 10.829 10.263 11.298 10.604 9.930 —0.018 -0.225 -0.333 -0.342 -0.273 — 0.141 600 550 500 450 400 350 0.674 0.572 0.473 0.381 0.294 0.221 1.589 1.335 1. 103 0.894 0.706 541 0.915 0.763 0.630 0.513 0.412 0.320 0.576 0.572 0.571 0.574 0.583 0.592 1450 1400 1350 9.622 8.924 8.185 9.280 8.651 8.044 1300 1250 1200 7.413 6.637 5.884 7.459 6.896 6.356 + 0.046 0.259 0.472 0.006 0.038 0.070 300 250 200 0.162 0.112 0.072 0.397 0.276 0.177 0.235 0,164 0.105 0.592 0.595 0.591 II50 IIOO 1090 5.179 4.420 4.221 5.837 5.340 5.244 0.658 0.920 + 1.023 0.113 0.172 0.195 150 100 0.040 0.018 0.099 0.044 0.059 + 0.026 0.594 0.591 CHAPTER 11. EXPERIMENTAL RESISTANCE. Notable Experiments. — Benjamin Robins was the first to execute a systematic and intelligent series of experi- ments to determine the velocity of projectiles and the effect of the resistance of the air, not only in retarding but in de- flecting them from the plane of fire. He was the inventor of the ballistic pendulum, an instrument for measuring the momenta of projectiles and thence their velocities. He also invented the Whirling Machine for determining the resistance of air to bodies of different forms moving with low velo- cities. His *' New Principles of Gunnery," containing the results of his labors, was published in 1742, and immediately attracted the attention of the great Euler, who translated it into French. The next series of experiments of any value were made toward the close of the last century by Dr. Hutton, of the Royal Military Academy, Woolwich. He improved the apparatus invented by Robins, and used heavier projectiles with higher velocities. His experiments showed that the resistance is approximately proportional to the square of the diameter of the projectile, and that it increases more rapidly than the square of the velocity up to about 1440 f. s., and nearly as the square of the velocity from 1440 f. s. to 1968 f. s. In 1839 ^"<^ 1840 experiments were conducted at Metz, on a hitherto unprecedented scale, by a commission ap- pointed by the French Minister of War, consisting of MM. Piobert, Morin, and Didion. They fired spherical projec- tiles weighing from 11 to 50 pounds, with diameters varying from 4 to 8.7 inches, into a ballistic pendulum, at distances of 15,40,65,90, and 115 metres; by this means velocities I8 EXTERIOR BALLISTICS. were determined at points 25, 50, 75, and 100 metres apart, the velocities varying from 200 to 600 metres per second. From these experiments General Didion deduced a law of resistance expressed by a binomial, one term of which is proportional to the square, and the other to the cube, of the velocity. This gave good results for short ranges ; but with heavy charges and high angles of projection the calculated ranges were much greater than the observed. Another series of experiments was made at Metz, in the years 1856, 1857, and 1858, by means of the electro-ballistic pendulum invented by Captain Navez, of the Belgian Artil- lery. This, unlike the ballistic pendulum, affords the means of measuring the velocity of the same projectile at two points of its trajectory. The results of these elaborate ex- periments may be briefly stated as follows: The resistance for a velocity of 320 m. s. does not differ sensibly from that deduced from the previous experiments at Metz; but the resistances decrease with the velocity below 320 m. s., and increase with the velocity above 320 m. s., more rapidly than resulted from the former experiments. The commission having charge of these experiments, whose president was Colonel Virlet, expressed the resistance of the air by a single term proportional to the cube of the velocity for all velocities. In 1865 the Rev. Francis Bashforth, M.A., who had then been recently appointed Professor of Applied Mathematics to the advanced class of artillery officers at Woolwich, began a series of experiments for determining the resistance of the air to the motion of both spherical and oblong projec- tiles, which he continued from time to time until 1880. As the instruments then in use for measuring velocities were incapable of giving the times occupied by a shot in passing over a series of successive equal spaces, he began his labors by inventing and constructing a chronograph to accomplish this object, which was tried late in 1865 in Woolwich Marshes, with ten screens, and with perfect success. It was afterwards removed to Shoeburyness, where most of his EXTERIOR BALLISTICS. I9 subsequent experiments were made. He employed rifled guns of 3, 5, 7, and 9-inch calibers, and elongated shot hav- ing ogival heads struck with radii of i^ calibers; also smooth-bore guns of similar calibers for firing spherical shot. From the data derived from these experiments he constructed and published, from time to time, extensive tables connecting space and velocity, and time and velocity, which for accuracy and general usefulness have never been excelled. The first of these tables was published in 1870, and his Final Report, containing coefficients of resistance for ogival-headed shot, for velocities extending from 2800 f. s. to JOG f. s., was published in 1880. These experiments will be noticed more in detail further on. General Mayevski conducted some experiments at St. Petersburg, in 1868, with spherical projectiles, and in the following year with ogival-headed projectiles, supplement- ing these latter with the experiments made by Bashforth in 1867 with 9-inch shot. An account of these experiments, with the results deduced therefrom, is given in his " Traite Balistique Exterieure," Paris, 1872. General Mayevski has recently (1882) published the re- sults of a discussion of the extensive experiments made at Meppen in 1881 with the Krupp guns and projectiles. These latter, though varying greatly in caliber, were all sensibl}^ of the same type, being mostly 3 calibers in length, with an ogive of 2 calibers radius. General Mayevski's results, together with Colonel HojeFs still more recent dis- cussion of the same data, will be noticed again. -7? Methods of Determining Resistances. — If a prO'-* jectile be fired horizontally, the path described in the first one or two tenths of a second may, without sensible error, be considered a horizontal right line ; and, therefore, what- ever loss of velocity it may sustain in this short time will be due to the resistance of the air, since the only other force acting upon the projectile, gravity^ may be disregarded, as it acts at right angles to the projectile's motion. For ex- ample, an 8-inch oblong shell, having an initial velocity of 20 EXTERIOR BALLISTICS. 1400 f. s., will describe a horizontal path, in the first two- tenths of a second after leaving the gun, of 278 ft., while its vertical descent due to gravity will be less than 8 inches. Moreover, if its velocity should be measured at the distance of 278 ft. from the muzzle of the gun, it would be found to be but 1380 f. s., showing a loss of velocity of 20 f. s., due to the resistance of the air. The relation between the horizontal space passed over by a projectile and its loss of velocity may be determined as follows : Let w be the weight of the projectile in pounds, V and V its velocities, respectively, at the distances^ and a' from the muzzle of the gun, in feet per second, and g the accele- ration of gravity. The vis viva of the projectile at the dis- . wV" zv V" tance a from the gun is , and at the distance a\ : ^ g consequently the loss of vis viva in describing the path vu a' —a^ is -( F^— V ^) ; and this, by the principle of vis viva, is equal to twice the work due to the resistance of the air. If the distance a'— a is not too great, say from 100 to 300 ft., according to the velocity of the projectile, it may be as- sumed that for this distance the resistance will not vary perceptibly ; and if p is the mean resistance for this short portion of the trajectory, we shall have '^{V'-V'^) = 2{a'-a)p whence P- 2g{a'-a) As, the resistance of the air is proportional to its density, which is continually varying, it is necessary, in order to compare a series of observations made at different times, to reduce them all to some mean density taken as a standard. If b is the density of the air at the time the observations are made, and b^ the adopted standard density to which the ob- /' = EXTERIOR BALLISTICS. 21 servations are to be reduced, the second member of the preceding equation shoidd be multipHed by ~^ which gives ' 2g{a' — a) d We may take for the value of (\ the weight of a cubic foot of air at a certain temperature and pressure; o will then be the weight of an equal volume of air at the time of mak- ing the experiments, as determined by observations of the thermometer, barometer, and hygrometer. As ft is the mean resistance for the distance a^ — a, it may VA-V be considered proportional to the mean velocity, v^ — ; and substituting this in the above expression, it becomes wv{V- V') d, , By varying the charge so as to obtain different values for Fand V, the resistance corresponding to different ve- locities may be determined, and thence the /aw of resistance deduced. In order to compare the results obtained with projec- tiles of different calibers, the resistance per unit of surface (square foot) is taken ; and, to make the results less sensible to variations of velocity, Didion proposed to divide the values of o by -J^ and compare the quotients (p') instead of />. Therefore, making ^t — — ^2-^, equation (4) becomes ^ gTzF^via' -a) 3 ^^^ It will be observed that since p is divided by ^'', the values of f/ will be constant when the resistance varies as the square of the velocity ; when this is not the case // will evidentl}^ be a function of the velocity; or f/ = A' f{v) (suppose), where the constant A', and the form of the fune- tion,/(2/), are both to be determined. 3 / 22 EXTERIOR BALLISTICS. Two assumptions have been made in deducing the ex- pression for (), neither of which is exactly correct: ist, that the resistance can be considered constant while the pro- jectile is describing the short path a' — a ; and, 2d, that this assumed constant resistance is that due to the mean velo- city, V. The nature of the error thus committed may be exhibited as follows: The exact expression for p is w dv wv dv ^'~~~g~dt~~gds Comparing this with (4), it will be seen that we have made \V- V _ _dv a' — a ds which is true only when the path described by the projec- tile is infinitesimal. To determine the amount of error committed, we can re- calculate the values of// by means of the law of resistance deduced from the experiments; and it will be found that in the most unfavorable cases the two sets of values of />' will not differ from each other by any appreciable amount. For example, suppose the law of resistance deduced by this method is that of the square of the velocity ; what is the exact expression for // in terms of F~ V and a — a? We have , p _ w dv ^' ~^:^"J~ '^g^^' vds and therefore , , w dv p' dsz=L — — whence, integrating between the limits Fand V , to which correspond a and a' , we have, since p' is constant in this case, ^' "^ gTzR^oT^) ^^^ Y' To test the two expressions for //, take the follow EXTERIOR BALLISTICS. 23 ing data from Bashforth's ''Final Report," page 19, round 486: F=2826 f. s. ; F' = 2777 f. s. ; 7e' — Solbs. ; 7? = 4 in. = ^ft.; F — F' = 49 ; ^^= 32.191 ; a' — a=^ i$o ft., and z^ = V+ V 2 = 2801.5. We find ■ — ^-i^^-y—, v=: 0.047463; and this is a factor in ^-rrR'ia -a) both expressions for />'. Therefore, by the approximate method, f/ = 0.047463 28^-T = 0.00083 and by the exact method, ^ 1 2826 f)' = 0.047463 log = 0.00084. For a second example, suppose the law of resistance to be that of the cube of the velocity. In this case f/ varies as the first power of the velocity, or f/ =^ A^ v. Therefore A, 1 2v dv ^^ TT R V whence II ^,^ ee. F^~"F gTzK' a' - a and .'-A'^^-- "^ v{V-V'y ' — "~ gT.k'ia' -a) W Comparing this with (5), it will be seen that (omitting the factor ^0 the two equations are identical, if we assume z;^ = VV ; and this is very nearly correct when, as in the present case, V — V is very small compared with either For v. As an example of this method of reducing observations, the experiments made at St. Petersburg in 1868 by General 24 EXTERIOR BALLISTICS. Mayevski, with spherical projectiles, have been selected. In these experiments the velocities were determined by two Boiilenge chronographs, and the times measured were in every case within the limits of o.''io and o.'' 15. X * < •f * it ♦ \ dq\. ' \ \> § w The experiments were made with 6 and 24-pdr. guns and 120-pdr. mortars, and the velocities ranged from 745 f. s. to 1729 f. s. At least eight shots were fired with the EXTERIOR BALLISTICS. 25 same charge; the value of// was calculated for each shot, and the mean of all the values of />' so calculated was taken as corresponding to the mean velocity of all the shots fired with the same charge. The values o^ a' — a varied from 164 ft. to 492 ft., the least values being taken for the heaviest charges, and the greatest values for the smallest charges. The greatest loss of velocity {V — V) was 131 ft., and the least 33 ft. The values of {/ deduced from these experiments are given in the following table. For convenience English units of weight and length are employed ; that is, the weights of the projectiles are given in pounds, the veloci- ties in feet per second, and the radii of the projectiles and the values of <^' — ^ in feet. Values of p for Si'Herical Projectiles, deduced from the Experi- ments MADE AT St. Petersburg in 1868. Mean Mean Kind of Gun. Velocity Values of P' Kind of Gun. Velocity Values of 6-pdr. gun 745 f. s. 0.000561 24-pdr. gun 1247 f. s. 0.001054 24-pdr. gun 768 " 508 0-pdr. gun 1260 " "45 120-pdr. mort. 860 " 687 120-pdr. mort. 1339 " 1117 6-pdr. gun 912 " 807 6-pdr. gun 1362 " 1189 24-pdr. gun 942 " 782 24-pdr. gun 1499 " 1138 120-pdr. mort. 1083 " 934 120-pdr. mort. I5I9 " 1 163 24-pdr. gun TII9 " 987 6-pdr. gun 1558 " 1189 6-pdr. gun II22 " 0.001107 24-pdr. gun 1729 " 0.001178 These results are reproduced graphically in Fig. 3, the velocities being taken for abscissas, and the corresponding values of// for ordinates. It will be seen that the trend ot the last seven points is nearly parallel to the axis of ab- scissas, and may, therefore, be represented approximately by the right line A, whose equation is />'z= 0.00116 in which the second member is the arithmetical mean of the last seven tabulated values of />'. 26 EXTERIOR BALLISTICS. It was found that the remaining points could be best represented by a curve B, of the second degree, of the form (/ =: p-\-q 7>^, containing two constants p and q whose values were determined by the method of least squares, each tabular value of // and the corresponding value of V furnishing one " observation equation." it was found that the most probable values of/ and q were^/ = 0.012 and ^ = 0.00000034686 ; or, reducing to English units of k k weight and length by multiplying / by - ^, and q by «, where k is the number of pounds in one kilogramme, and m the number of feet in one metre, we have ^>' = 0.00022832 -[-0.00000000061309 v" or, in a more convenient form, ^/ = 0.00022832 )i+(g-^ J [ To find the point of intersection of the right line A with the curve B, equate the values of />' given by their respective equations, and solve with reference to v. It will be found that v^ 1233 f. s., at which velocity we assume that the law of resistance changes. In strictness there is probably but one laiv of resistance^ and this might be, perhaps, expressed by a very complicated function of the velocity, having variable exponents and co- efficients, depending, upon the ever- varying density of the air, the cohesion of its particles, etc. ; but, however compli- cated it may be, we can hardly conceive of its being other than a continuous function. But, owing to the difficulties with which the subject is surrounded, both experimental and analytical, it is usual to express the resistance by in- , tegral powers of the velocity and constant coefficients, so chosen, as in the above example, as to represent the mean resistance over a certain range of velocity determined by experiment. * Mayevski, " Traite de Balistique Exterieure," page 41. EXTERIOR BALLISTICS. Expression for />. — The expression for /> in terms of which, since [/ is generally a function of 7', may be written The resistance per unit of mass, or the retarding force, will therefore be or, taking the diameter of the projectile in inches, The first member of this equation expresses the retarding force when the air is at the adopted standard density and the projectile under consideration is similar in every respect to those used in making the experiments which determined //. To generalize the equation for all densities of the at- mosphere we must introduce into the second member the factor IT ; and we will also assume, at present, that the equa- tion will hold good for different types of projectiles if d'^ be multiplied by a suitable factor {c), depending upon the kind of projectile used. For the standard projectile and for spherical projectiles, 6=1; for one offering a greater re- sistance than the standard, 6'>i; and if the' resistance offered is less, r < i. Making, then, 576 and ^~ d cd' we have for all kinds of projectiles p- dv A ^ , . ,^. C is called the ballistic coefficient, and c the coefficient of reduction. i. 28 EXTERIOR BALLISTICS. For the Russian experiments with spherical projectiles the standard density of air to which the experiments were reduced was that of air half saturated with vapor, at a tem- perature of 1 5° C, and barometer at o"'. 75. In this condition of air the weight of a cubic metre is 1^.206; and, therefore, the weight of a cubic foot ( = o) is 0.075283 lbs. = 526.98 grs. The value of ^ taken was 9"\8i = 32.1856 feet. Applying the proper numbers, we have the following working expres- sions for the retarding force for spherical projectiles. Velocities greater than 1233 f. s. : ^ /? = — 7/%- log A = 6.3088473 - 10 Velocities less than 1233 f. s. : f- r = c ''' V "^ ?' / ' ^^^ ^ ^ 5.6029333 - 10 r = 612.25 ^^• Oblong Projectiles: Oeneral Mayevski's For- mulas. — General Mayevski, by a method similar in its gen- eral outline to that given above, the details and refinements of which we omit for want of space, has deduced the fol- lowing expressions for the resistance when the Krupp pro- jectile is employed, viz. : '^' 700™ >V> 419™, /> = 0.0394 TT R^ -^ v^ 419'" >v> 375"\ ^o = 0.0^94 r R' -yv' 375"" > -^ > 295"^, p — o.o'67 7: R" -^v" 295^^ > z; > 240^ /> = 0.0^583 ;r /?^ y 7^^ 240™ > v> o™, /> = 0.014 7: T?'^ -^ v" Changing these expressions to the form here adopted * Revue d^Artilleriey April, 1883. EXTERIOR BALLISTICS. 29 [equation (6)], and reducing to English units of weight and length, they become 2300 ft. > z/> 1370 ft. : ir- = ^T/%- log yi =6.1192437 - 1370 ft. >^'> 1230 ft.: - ID i'- --^ 7>\- log ^ = 2.9808825 - 1230 ft. > •z/>97o ft.: ID ir- - J, v" ; log A = 6.8018436 - 970 ft. > z/> 790 ft.: •20 i"- ^ 3 , 790 ft. > 7' > ft. : • ID i" = ^7^- log ^=5.6698755 - ID Colonel Hojel's Deductions from the Krupp Ex- periments.— Colonel Hojel, of the Dutch Artillery, has also made a study of the Krupp experiments discussed by General Mayevski : and, as it is interesting and instructive to compare the resistance formulas deduced by each of these two experts, both using the same data, we give a brief syn- opsis of Colonel Hojel's method and results. He expresses the resistance by the following formula, easily deduced from equation (6): in which, from (4), It is assumed that the loss of velocit}^ V — V\ is some func- tion of the mean velocity v, which can be expressed approx- imately, for a limited range of velocity, by a monomial of the form 4 30 EXTERIOR BALLISTICS. in which A and n are constants to be determined. The method of procedure is analogous to that followed m deter- mining fj', and need not be repeated. Colonel Hojel has considered it necessary to employ fractional exponents, thereby sacrificing simplicity without apparently gaining in accuracy. The results he arrived at are as follows: " 700^ >v> soo'", /{v) = 2A 868 v'-'' 500™ > -6^ > 400™, / (tj) = 0.29932 z/'" 400™ >v> 350'", / (v) = o.o'205 524 7/'-'' 350°^ > ^ > 300"\ / (v) = o.o'2 1692 V* 300™ >v> I40"\ /{z') = 0.033814 v'-' Substituting these values oi /{v) in the equation w^ zv -^ ■' 4w -^ ^ ^ and reducing the results to English units, that is, taking w in pounds, v in feet, and d in inches, we have as the equiva- lents of Hojel's expressions, all reductions being made, the following : 2300 ft. > 7^ > 1640 ft. : a- A ±- p z=z ~ v'''\- log y4 =6.4211771 — ID 1640 ft. > 7^> 1310 ft. : -|.«-^^^-"; iog^ = 5.3923859- 10 1 3 10 ft. > 7/ > II 50 ft. : 0- A -|-^ = — 7/^«%- log ^ = 0.4035263 - 10 1150 ft. >7'>98o ft.: a- A ^p = —v\- log ^ = 6.8232495 - 20 980 ft. > 7.' > 460 ft. : = — -- ir — 2b V w * ds will be variable, and its value must be so determined by ex- periment as to satisfy this equation for each value of v, Bashforth's method of deducing these values is briefly as follows : Ten screens are placed at equal distances (150 feet) apart in the plane of fire, and the exact time of the passage of a projectile through each screen is measured by the Bashforth chronograph. The first, second, third, etc., dif- ferences of these observed times are taken, which call d,, d,,d,, etc. Let s be the distance the projectile has moved from some assumed point to any one of the screens, say the first ; EXTERIOR BALLISTICS. 33 /the constant distance between the screens; and /,^ /,+/^ /,+2/, etc., the observed times of the projectile's passing succes- sive screens. Then from a well-known equation of finite differences we have , , n(n— 1) . , n(n— i)in — 2) , . , ts.ni=^t,-\- ltd, + ^——-^ d, H ^^ ~\\ ^- d, + etc. 1.2 I • - • 3 in which ;/ is an arbitrary variable. Arranging the second member according to the powers of//, we have ts.ni^t,-\-n \d, — \d^-\-\d^ — -d,\ etc. ) \ 2 3 4 / -f etc., etc., terms multiplied by the cube and higher powers of ^/. Since / is a function of s, we have t^—f{s) and t,^„i:= f{s + 111). Expanding this last by Taylor's formula, we have , dt, nl , dU, n'l' ^ whence, equating the coefficients of the first and second powers of 71 in the two expansions of /^ + „/, we have /^^ = ^-i^,+ i-^3_i^^ + etc. ds ' 2 ' ' 3 4 and ,„ ^V, , , , II - 10 7 1 X :r-^^=:^,-^3 + — <- — < + etC. The first of these equations gives ds I dt, ^-d,-\d,-^^\d,-\d, and the second 7; ds-" ' where i\ is the velocity and - /> the resistance per unit of mass at the distance s from the gun. .J,,_-q^^_^^ + iL^._I|^. + etc.) 34 EXTERIOR BALLISTICS. As an example take the following experiment made with a 6.92-inch spherical shot, weighing- 44.094 lbs., fired from a 7-inch gun."^ The times of passing the successive screens were as follows : Screens. Passed at, Seconds. d. d^ ^3 I 2 . 90068 8431 306 10 2 2.98499 8737 316 TO •3 3.07236 9053 326 10 4 3. 16289 9379 336 10 5 3.25668 9715 346 10 6 3.35383 10061 356 I I 7 3-45444 10417 367 II 8 3.55861 10784 378 9 3.66645 11162 10 3.77807 To find, for example, the velocity at the first screen, we have 150 1 1.4 t. s., = 1465.3 f"- s. ' 0.08431—^0.00306-1-^0.00010 and at the seventh screen 150 ' 0.10417 — ^0.003674-^0.00011 The retarding forces at the same screens are as follows: or V^ ^ f)^=z - — ^- (o . 00306 — o . oooio) = o . ooooooi 3 1 56 Z^,' = 2<^, V,^ and - Pt= 7 — '-^(0.00367 — 0.00011) = 0. ooooooi 5822 z'/ = 2^, z;/. As these small numbers are inconvenient in practice, * Bashforth, page 43. EXTERIOR BALLISTICS. 35 Bashforth substituted for them a coefficient K, defined by the equation A-=24J(.ooo)'. In the experiment selected above the weight of a cubic foot of air was 553.9 grains = (?, while the standard weight adopted was 530.6 grains = d^. Therefore we have (150) (6.92) 553.9 and j^ 0.00356 ^^ ^„ A; = ^\K,— 139.6* 0.00296 ' ^^ That is to say, when the velocity of a spherical projectile is 1811.4 f. s., A"=ii6.i; and when its velocity is 1465.3 f. s., A'= 139.6. By interpolation the values of K, after having been determined for a sufficient number of velo- cities, are arranged in tabular form with the velocity as argument. Bashforth determined the values of K by this original and beautiful method for both spherical and ogival-headed projectiles ; and for the latter for velocities extending from 2900 f. s. down to 100 f. s. The experiments upon which they were based were made under his own direction at various times between 1865 and 1879, ^'ith his chronograph, probably the most complete and accurate instrument for measuring small intervals of time yet invented. Law of Resistance deduced from Bashforth's K. — It will be seen, by examining Bashforth's table of A" for ogival-headed projectiles, that as the velocity decreases from 2800 f. s. down to about 1300 f. s., the values of K gradually increase, then become nearly constant down to about ii3of. s., then rapidly decrease down to about 1030 f. s., become nearly constant again down to about 800 f. s., and then gradually increase as the velocity decreases, to the ♦ Bashforth's " Mathematical Treatise," page 97. 36 EXTERIOR BALLISTICS. limit of the table. These variations show that the law of resistance is not the same for all velocities, but that it changes several times between practical limits. We may use Bashforth's K for determining these different laws of resistance as follows : We have for the standard density of the air, ^ p = 2bv' 3= _ _— — (7) w ^ zv (looo) and from which we get ,_ S76Kv ^^ ;r^(ioooy The values of />' have been computed by means of this formula, for ogival-headed projectiles, from if — 2900 f. s. to V = 100 f. s., and their discussion has yielded the following results : Velocities greater than 1330 f. s. : o A ^p — —,7>\- log ^ =: 6.1525284— 10 1330 f. s. > 't^> II20 f. s. : ^/> = -^7^'; log ^=: 3.0364351 - 10 1 1 20 f . s. > 2/ > 990 f . s. : ^/> = -^-^"/ log yi = 3.8865079 -20 990 f. s. > 7^ > 790 f. s. : fv^^'c'^'' log ^=2.8754872 -10 790 f. s. > 7/ > 100 f. s. : p- A ^pz=.-v\- log 7^ = 5.7703827- 10 These expressions, derived as they are from Bashforth's EXTERIOR 15ATXISTICS. 37 coefficients, give substantially the same resistances for like velocities as those computed directly by means of equation (7). The agreement between the two for high velocities is shown graphically by Plate I., in which A is Bashforth's curve of resistance, while that part of the parabola, B, com- prised between the limits ^^2800 f. s. and 7'= 1330 f. s., is the curve of resistance deduced from the first of the above expressions. If, hovvever, we compare these expressions with those deduced by Mayevski or Hojel from the Krupp experiments, it will be found that these latter give a less resistance than the former for all velocities. This is undoubtedly due to the superior centring of the projectiles in the Krupp guns over the English, and to the different shapes of the projectiles used in the two series of experiments, particularly to the difference in the shapes of the heads. The English projectiles, as we have seen, had ogival heads struck with radii of i| calibers, while those fired at Meppen had similar heads of 2 calibres, and, therefore, suffered less resistance than the former indepen- dently of their greater steadiness. Comparison of Resistances. — Let f) and {>^ be the re- sistances of the air to the motion of two different projectiles of similar forms ; w and zv^ their weights ; 5 and S^ the areas of their greatest transverse sections; d and d^ their dia- meters ; and D and D^ their densities. Then, if we suppose, in the case of oblong projectiles, that their axes coincide with the direction of motion, we shall have from (6) for the same velocity, since 5 and S^ are proportional to the squares of their diameters. i" s w A V ^ ; and ^ = — i''~ that is, for the same velocity the resistances are proportional to the areas of the greatest transverse sections, while the retardations are directly proportional to the areas and in- 5 38 EXTERIOR BALLISTICS. versely proportional tiles we have to the ) A^eights. For spher ical projec- .5=i;^^^ S,. = i^ ^A , 7i>- = i;r^^A an d w^ = :t-;r ^;^.; therefore _d^D, ~ dD that is, for spherical projectiles the retardations are in- versely proportional to the products of the diameters and densities. This shows that for equal velocities the loss of velocity in a unit of time will be less, and, therefore, the range greater, cceteris paribus, the greater the diameter and density of the projectile. As the weight of an oblong projectile is considerably greater than that of a spherical projectile of the same caliber and material, it follows that the retardation of the former for equal velocities is much less than the latter, indepen- dently of the ogival form of the head of an oblong projectile which diminishes the resistance still more. Indeed, the re tarding effect of the air to the motion of a standard oblong projectile, for velocities exceeding 1330 f. s., is less than for a spherical projectile of the same diameter and weight, and moving with the same velocity, in the ratio of 14208 to 20358. As an example, if d and w are the diameter and weight of a solid spherical cast-iron shot which shall suffer the same retardation as an 8-inch oblong projectile weighing 180 lbs. and moving with the same velocity, we shall have, since we know that a solid shot 14.87 inches in diameter weighs 450 lbs., ,_ (14.87)' X 180X20358 and ^ „ — 29.65 inches 450X64X 14208 ^ •" 450 X (20.65)' . ,, w— ^^ . W, ^^ —3567 lbs. (14.87 ' ^^ ^ The retarding effect of the air to the motion of projectiles EXTERIOR BALLISTICS. 39 of different calibers but having the same initial velocity and angle of projection, is shown graphically in Fig. 4, which was carefully drawn to scale. A is the curve which a projectile would describe in vacuo, B that actually described b}^ a spherical projectile 14.87 in diameter weighing 450 lbs., and C that described by a spherical shot 5.9 inches in diameter weighing 26.92 lbs. The initial velocity of each is 1712.6 f. s., and angle of projection 30°. Example. — Calculate the resistance of the air and the re- tardation for a 15-inch spherical solid shot moving with a velocity of 1400 f. s. Here ^= 14.87 in., 7(^ = 450 lbs., and A ~ 20358X iQ-l Substituting these values in equation (6), we have and dv dt (14.87)' 20358 , ^-^-^ X -^ X (1400)^ 32.16 10 ^ -1- / (i4.87r 450 X '-'^^ X (.400)' 2743 lbs.. 196.07 f. s. ; that is, at the instant the projectile was moving with a velocity of 1400 f. s. it suffered a resistance of 2743 lbs. ; and if this resistance were to remain constant for one second the velocity of the projectile would be diminished by 196.07 ft. As, however, the resistance is not constant, but varies as the square of the velocity, it will require an integration to determine the actual loss of velocity in one second. We have from (6) dt IV 40 EXTERIOR BALLISTICS. or dv iV . , -^,=- Adt whence, integraling between the limits F, 7>, we have Now, making V= 1400 and t=zi, we find v — 1228 f. s. ; and the loss of velocit}^ in one second is 1400 — 1228 = 172 ft, CHAPTER III. DIFFERENTIAL EQUATIONS OF TRANSLATION — GENERAL PROPERTIES OF TRAJECTORIES. Preliiiiiiiary Considerations.— A projectile fired from a gun with a certain initial velocity is acted upon during its flight only by gravity and the resistance of the air; the former in a vertical direction, and the latter along the tan- gent to the curve described by the projectile's centre of gravity. It will be assumed, as a first approximation, that the projectile, if spherical, has no motion of rotation ; and, in the case of oblong projectiles, that the axis of the pro- jectile lies constantly in the tangent to the trajectory ; also that the air through which it moves is quiescent and of uni- form densit}'. xA.s none of these conditions are ever fulfilled in practice, the equations deduced will only give what may be called the normal trajectory^ or the trajectory in the plane of fire, and from which the actual trajectory will deviate more or less It is evident, however, that this deviation from the plane of fire is relatively small ; that is, small in comparison with the whole extent of the trajectory, owing to the very great density of the projectile as compared with that of the air. Notation.— In Figure 5, let (9, the point of projection, be taken for the origin of rectangular co-ordinates, of which let the axis of X be horizontal and that of F vertical. Let O A be the line of projection, and O B E the trajectory de- scribed. The following notation will be adopted: o denotes the acceleration of gravity, which will be taken at 32.16 f. s. ; IV the weight of the projectile in pounds; d its diameter in inches; (p the angle of projection, A O E ; 42 EXTERIOR BALLISTICS. V the velocity of projection, or muzzle velocity ; [/ the horizontal velocity of projection = Fcos (f ; V the velocity of the projectile at any point M of the trajectory ; 3 the angle included between the tangent to the curve at any point Jf and the axis of X, = T M H ; CO the angle of fall, CEO; Y D u the horizontal velocity = ^' cos ^; / the time of describing any portion of the trajectory from the origin ; s the length of any portion of the arc, as O in ; X the horizontal range, O E ; T the time of flight ; ft the resistance of the air, or the resistance a projectile encounters in the direction of its motion, in pounds. Dift'erential Equations of Translation. — The ac- celeration'^" in the direction of motion due to the resistance of the air is — //; and the correspondins^ acceleration due to gravity is ^ sin /> ; therefore the / (8) The velocities parallel to X and Y are, respectively, * The term "acceleration" is here used for retardation. To avoid multiplying terms re- tardation will be regarded as negative acceleration. EXTERIOR BALLISTICS. 43 V COS <>and v sin /> ; and the accelerations parallel to the same o g axes are -- /> cos /> and j^ + -- p sin />. Therefore ^ (^ cos '>) ^ g , , ^ ,, = - — /' COS /!/ (9) at zu ' ^^^ and d {v sin /5^) ^. dt "- - ' -^ ^ f> sin ^y 7£/ Performing- the differentiations indicated in the above equations, multiplying the first by sin & and the second by cos />, and taking their difference, gives — ^ = -^^cos/V (10) Introducing the horizontal velocity u = 7' cos /> in (9) and (10), and substituting for •_'^ ft its value from (6), they become, making/ (7/) = 7^", du A u"" • , rf7 "" "~ 6'^ cos""-^^^ ^^^^ and -^^ = — a- cos' fi (12) whence, eliminating <3^/, <^ /> g C du (13) cos"^'<> ^ ?/**^' Symbolizing the integral of the first member of (13) by (/>)„, that is, making ^^^" J "cos^'^"^ 71 k^ C and writing for the sake of symmetry, for -^, we shall have rdu k^ id-\ = n k' I = h C 44 EXTERIOR BALLISTICS. If (z) is the value of (f"^) when // is infinite, we have C=(0; and therefore whence and ={t\-{»). k k sec (5* From (ii) we have C ^ , Kidn (it ■-— cos"-' f>—- A ?/" and this substituted in the equations c/x = ?/ d/, dy = // tan ^ dt, ds = // sec li dt, gives ^,= __^cos«-,>_ ./j/= --^sln/^cos'-^/^f-, ^^ = From (12) w^e have C cos" ,>i^ dt= - It d /> ^^ cos'' ^> d tan ^y whence, as before, dx=^ d tan & g dy ds tan b- d tan /V — sec ^ ^ tan ^^ (14) (•5) (>-) (18) (19) (20) (21) (22) (23) (24) EXTERIOR BALLISTICS. 45 Eliminating- u from these last four equations by means of (15), they take the following eleg-ant forms : it= ~ ~ k_ d tan d^ (25) , ^ ^t an d^ (26) , k- tan ^y^/ tan /V (27) _ _ /P sec d d tan /> (28) 7?^' w<7r/'j-.— Subject to the conditions specified in the pre- liminary considerations, equations (16) to (20) or (25) to (28) contain the whole theory of the motion of translation of a projectile in a medium whose resistance can be expressed by an integral power of the velocity. Equation (16) gives the velocity in terms of the inclination; (18) and (19) or (26) and (27), could they be integrated generally, would give the co- ordinates of any point of the trajectory, while the time would depend upon the integration of (17) or (25). But, unfortu- nately, the ''laws of resistance" which obtain in our atmo- sphere do not admit of the integration of these equations ; we are, therefore, obliged to resort to indirect solutions giving approximations more or less exact. Of these many have been proposed by different investigators; but, with few exceptions, they are either too operose for practical use or not sufficiently approximate. General Didion, in the fifth section of his '* Traite de Balistique," gives a full and interesting ri^sumi^ o{ \\\q labors of mathematicians upon this difficult problem up to his time (1847), ^"<^ ^" the same work gives an original solution of his own of great value. Within the last quarter of a cen- tury much has been accomplished to improve and simplify 6 46 EXTERIOR BALLISTICS. the methods for calculating tables of fire and for the solution of the various problems relating to trajectories ; and we will endeavor in the following pages to present such of these methods as are of recognized value, developed after a uni- form plan and based upon the preceding differential equa- tions. Oeiieral Properties of Trajectories. — Though it is impossible with our present knowledge to deduce the equation of the trajectory described by a projectile, there are certain general properties of such trajectories which may be determined without knowing the law of resistance, if we admit that the resistance increases as some power of the velocity greater than the first, from zero to infinity; whence, making — = /(^)> we shall have f {v) > o, and /(x)= X. Variation of tlie Velocity — Miniiniim Velocity. — The acceleration in the dircctio7i of motion is [equation (8)] f =-."T/(^') + sin*] in which — ^sin d is the component of gravity in the direc- tion of motion; and, therefore, whether the velocity is in- creasing or decreasing with the time at any point of the tra- jectory, depends upon the algebraic sign of the second mem- ber; and this, since f {v) \ = — ) is considered positive, depends upon the sign of sin d^. In the ascending branch sin & is positive, and, therefore, from the point of projection to the summit the velocity is decreasing. At the summit sin ?^ = o, and at this point gravity, which has hitherto con- spired with the resistance to diminish the velocity, ceases to act for an instant in the direction of motion, and then, as sin d- changes sign in the descending branch, begins to act in opposition to the resistance ; that is, its action tends to increase the velocity. The component of gravity acting perpendicular to the projectile's motion (^ cos d), and which EXTERIOR BALLISTICS. 47 is a maximum at the summit, tends to increase the in- clination in the descending branch, and thus to increase (numerically) — sin &, until at a certain point of the de- scending branch where the inclination is (say) — &' the acceleration of gravity in the direction of motion has in- creased until it just equals the retardation due to the re- sistance of the air, which latter has continually decreased with the velocity. Beyond this point, as the component of gravity in the direction of motion still increases with the inclination while the resistance remains constant for an in- stant, the velocity also increases; and, therefore, at the point where w the velocity is a minimum, and — = o. Passing the point of minimum velocity, the acceleration of gravity and the retardation due to the resistance of the air both increase; but that there is no maximum velocity, properly speaking, may be shown as follows : Differentiating the above expression for the acceleration, we have d'^v ' dv ^ d& and putting in place of ^ its value from (10), we shall have d'v ^,. ^dv , .^-^cos^/> df '*'' ' V/ ' V dv and this is necessarily positive whenever — == o. The velo- city, therefore, can only be a minimum ; but it tends towards 1- • • 1 • ^ P 10 '^ a hmitmp: value, viz., when — - =: i, and //=^ . Liinitiiig" Velocity. — As the limiting velocities of all service spherical projectiles are less than 1233 f. s., we can 48 EXTERIOR liALLlSTlCS. determine these velocities by means of the expression for the resistance given in Chapter II., from which we get A (P o zv (.+o>) := I where ^4 = 0.000040048 and r = 610.25. Solving with re- ference to 7' we ore t Z'+^^f--' which gives tlie limiting velocity. The following table contains the limiting velocities of spherical projectiles in our service calculated by the above formula : Solid Shot. Inches. Lbs. Final Velocity. ' Feet. i Shells Unfilled. Inches. 7(' Lbs. Final Velocity. iFeet. 20-inch 19.87 1080 859 : 15-inch 14.87 330 1 726 1 15-inch 14.87 450 783 13-inch 12,87 216 682 13-inch 12.87 283 743 lo-inch 9.87 101.75 635 lo-inch 9.87 128 684 8-inch 7.88 45 561 i2-pdr. 4.52 12.3 526 i2-pdr. 4.52 8.34 458 Limit of the Iiieliiiatioii of the Trajectory in the Desceiidinj>- Braiicli. — We have assumed above that tlie descending branch of tlie trajectory ultimately becomes vertical. To prove this, take equation (10), viz. : and integrating from a point of the trajectory where ^y = if and ^ = o, we have As the velocity v, between the limits / = o and / = x , is XY ) EXTERIOR BALLISTICS. 49 finite and continuous, and cannot become zero, we have, since v is a function of />, where A^ is some value of v greater than its least, and less than its greatest value between the limits of integration. As (^ is negative in the descending branch, the above equation shows that, when / is infinite, /V is equal to — '-. 2 From (24) we have tan('' + ^' z^^/i \4 2 A log ot/s = —v\ . COS /> and, therefore I, when t is infinite. c — 2 C0 cos/y- : K' g dty g From (13) and (6) we have d{v cos d) ft ■fe V dd- w whence, differentiating and reducing, + sin & dv '' f (^ d& cos d- \ zv Substituting this in the expression for -^ gives dy zr —^ = — sec ^'^ g ^.>g + 3sin,y) This equation shows that beyond the summit --~r is posi- EXTERIOR BALLISTICS. 5 I tive up to the point where — + 3 sin d =r o, and then changes its sign. At this point, therefore, the radius of curvature becomes a minimum and afterwards increases to infinit}^ At the point of maximum curvature we have, in conse- 20 quence of the condition ^^ + 3 sin & zzz o, -777 = V tan If dt^ 2 and therefore, since 0- is negative in the descending branch, -777- is positive at that point, and v is decreasing with d-] au in other words, the velocity has not yet become a mini- mum. Therefore the point of maximum curvature is near- er the summit of the trajectory than the point of minimum velocity. CHAPTER IV. RECTILINEAR MOTION. Relation between Time, Space, and Velocity. — For many practical purposes, and especially with the heavy, elongated projectiles fired from modern guns, useful results may be obtained by considering the path of the projectile a horizontal right line, and therefore unaffected by gravity. Upon this supposition f'i- becomes zero, and equations (17), (18), and (20) become C dv and n dx ^=- (is ^=^ — — — A v^-^ whence integrating, and making / and ^ zero when 7^= F, we have t-c\ ^ L__l ((;/- \)Av^-' {n- \)A F^'M and ,^r S I L_ ^ \{n — 2)Av^-^ {71— 2) A F«-^f Writing, for convenience, T {%>) for ; r ^ ^ -, and 5 {v) for ; , . „_„ these equations become t^C\T{v)-T{.^\ 7\ (29) and rji ' ' s=C\S{v)-S{zt)\ ' (30) When n = 2, the above expression for s becomes inde- terminate. In this case we have EXTERIOR BALLISTICS. 53 , C dv whence s = —\\og V-\ogv I and therefore, when n = 2, Equations (29) and (30) (or their equivalents) were first given by Bashforth in his *' Mathematical Treatise," Lon- don, 1 873. He also gave in the same work tables of 5 (v) and T{7>) for both spherical and elongated shot; the former ex- tending from V =: 1900 f. s. to v = 500 f. s., and the latter from V = 1700 f. s. to v =: 540 f. s. In a " Supplement " to his work above cited, published in 1881, he extended the tables for elongated projectiles to include velocities from 2900 f. s. to 100 f. s. Projectiles differing from tlie Standard. — It will be seen that the value of the functions T{v)a.nd S (v) depend upon those of v and A, the former of which is independent of the nature of the shot, while the latter depends partly upon the form of the standard projectile, which in this country and England has an ogival head struck with a radius of ij calibers, and a body 2^ calibers long. The fac- tor 6^ ( or -^ ~~) depends upon the weight and diameter of \ o ca / • the projectile, the density of the air, and the coefficient c ; which latter varies with the type of projectile used. The factor^ varies, therefore, with c ; but by the manner in which A and c enter the expressions for / and s, it will be seen that the results will be the same if we make A constant, and give to ^ a suitable value determined by experiment for each kind of projectile. By this means the tables of the functions T{v) and 5(z/), computed upon the supposition that ^ = I, can be used for all types* of projectiles. We will now show how these tables may be computed for ob- long projectiles, making use of the expressions for the re- 7 54 EXTERIOR BALLISTICS. sistance derived from Bashforth's experiments given in Chapter I. Oblong Projectiles, Velocities greater than 1330 f. s. — For velocities greater than 1330 f. s. we have ^^ = 2 and log ^ =6.1525284 — 10; therefore r(.) = -iandr(F)=-i, or, since the value of t depends upon the difference of T{y) and T{V), we may, if convenient, introduce an arbitrary constant into the expression for T{v). Therefore we may take and, similarly, •S (^) = ^ (- log r + log 0',) = ^ log ^ To avoid large numbers and to give uniformity to the tables we will determine the constants Q, and Q\ so that the functions shall both reduce to zero for the same value of v; and it will be convenient to begin the table with the highest value of v likely to occur in practice, which we will assume (following Bashforth) to be 2800 f. s. We therefore have A \2800 ' ---V 2800 I 1... Q\ loR -S^ =0 G'l = 2800 ^ 2800 ' Substituting the above values of A, Q^, and Q\ in the expressions for T{v) and S (v), and reducing, we have for velocities between 2800 f. s. and 1330 f. s. T{v) = [3.8474716] -^- - 2.5137 and S{v) =155866.12 — [4.2096873] log V. The numbers in brackets are the logarithms of the nu- merical coefficients of the quantities to which they^^are EXTERIOR BALLISTICS. 55 prefixed ; and the factor lo^ v is the common logarithm of z', the modulus being included in the coefficient. Velocities between 1330 f. s. and 1130 f. s.— For velocities between 1330 f. s. and 11 20 f. s. we have n — i and log A = 3.0364351 — 10; therefore, as before, Arbitrary Constants. — To deduce suitable values for the arbitrary constants Q^ and ^'2, we must recollect that the function representing the resistance of the air changes its form abruptly when the velocity is 1330 f. s. ; and to prevent a correspondingly abrupt change in our table at the same point — that is, to make the numbers in the table a continuous series — we must give to g^ and Q^ such values as shall make the second set of functions equal in value to the first when z;=i330. They will, therefore, be deter- mined by the following relations: _L/ L_ - ^) = lf_J L_^ 2 A V(i33o)' ^ ^7 A V1330 2800/ and I / I I ^/ \ I 1 2800 in which the A in the first member must not be confounded with that in the second. Making the necessary reductions, we have and Tizi) = [6.6625349] A-+ 0.1791 V 5(z/)= [6.9635649]-^ - 1674.: Velocities between 1130 f. s. and 990 f. s.— For velocities between 1120 f. s. and 990 f. s. we have ;/ = 6 and log A — 3.8865079 — 20 ; therefore ^(-^ = 5i-(^+^") 56 EXTERIOR BALLISTICS. and . V The constants must be determined as before, by equating the above expressions to the corresponding ones in the case immediately preceding, making ^= 1120. The results are, all reductions being made, r(T^) = [15.4145221] ■;:^ + 2.3705 and 5(2;)=[i5.5ii432i] I- + 4472.7 Velocities between 990 f. s. and 790 f. s.— For velocities between 990 f. s. and 790 f. s. we have // = 3 and log A = 2.8754872 — 10; whence and Proceeding as before, we have r(2;) = [6.8234828] i,- 1.6937 and *^(^) = [7-i245i28] i- 5602.3 Velocities less than 790 f. s. — For velocities less than 790 f. s. we have « 1= 2 and log ^4 = 5.7703827 — 10 ; therefore ^(-)=z(^ + a) and whence, as before, T{v) = [4.2296173] -^ - 12.4999 S{v) — 124466.4— [4-59' 8330] log 2/. EXTERIOR BALLISTICS. 57 Ballistic Tables. — Table I. gives the values of the time and space functions for oblong projectiles, computed by the above formulas, and extends from v zizz 2800 f. s. to v = 400 f s. The first differences are given in adjacent columns; and as the second differences rarely exceed eight units of the last order, it will hardh^ ever be necessary to consider them in using this table. Table II. gives the values of these functions for spherical projectiles, and is based upon the Russian experiments dis- cussed in Chapter II. EXAMPLES OF THE USE OF TABLES I. AJ^D II. Example i. — The velocity of an 8-inch service projectile weighing 180 lbs. was found by the Boulenge chronograph to be 1398 f. s. at 300 ft. from the' gun. What was the muzzle velocity ? Here C == ^, v = 1398, and s = 300, to find V. From 64 (30) we have and from Table 1. yXW"* 5(1398) =4903.8 - ^^^^^'^ =4888.7 also s 64 ^ C = 300 X-^= 106.7 whence 5(F) = 4782.0 . •. F= 141 5 + ^1^ = 14^9-4 f- s. Example 2. — Determine the remaining velocity and the time of flight of the 12-inch service projectile, weighing 800 lbs., at 1000 yds. from the gun, the muzzle velocity being 1886 f. s. 58 EXTERIOR BALLISTICS. 1. Fand s are given, to find v; where , >, & V u >» / 13 C s 6 s s B 1 ^ c< 53 Is r ■0 c 2 } 1 '^ a S I 240 2.8 125 1-245 1450 467 380 379.9 380.7 380.6 2 240 2.8 161 1.245 1450 1 454.5 390 388.3 •387.7 387.5 3 172.6 2.8 61.5 1.226 1389 477 388 388.7 389.3 388.7 4 172.6 2 8 61.5 1.226 1429 514.7 416.6 417.^ 417.6 415.7 5 149. 1 2.8 39-3 1.260 1429 1 518 401.6 402.1 403.0 401.2 6 149. 1 2.5 33.5 1.240 1429 507.7 380 380.7 379.9 379.1 7 149. 1 2.8 31-3 1.265 924 475.8 387.8 388.2 387.7 387.3 8 355 2.8 525 1.200 1884 495-9 432.7 433-1 433.8 432.6 9 355 2.8 525 1.200 2384 490 415 411. 8 414.4 412.3 lO 355 2.8 525 1.200 2389 488.5 409.6 410.4 412.3 410.9 II 149. 1 2.8 31.3 1.265 1950 609- 394 393-9 395.4 392.7 12 149. 1 4 51 1.206 1929 505-2 394.6 393.3 393.4 392.3 13 152.4 4 51.5 1.205 1450 472.4 391.3 389-3 389.1 388.6 14 152.4 2.8 32.5 1.205 1450 577 422 422.0 424.2 421.5 15 149-1 2.8 31.3 1.230 1450 632.4 460.9 460.3 462.8 459.8 i6 240 3.8 215 1.208 1904 480.4 412.8 412.0 412.4 411. 1 17 400 2.8 777 1. 180 2384 499.4 433.7 432.1 433 -o 431.7 i8 400 2.8 643 1. 190 2384 533.4 443-8 447.0 448.2 446.6 19 400 2.8 643 I.I90 2384 531.5 444-5 445-4 446.6 445.0 20 84 2.8 6.55 1. 197 2447 446.9 266 267.2 259-7 267.4 21 120 2.8 16.4 1. 211 2447 463.3 284.1 289.2 281.6 289.3 22 149. 1 2.8 31-3 1.285 3448 536.6 294.8 290.6 283.7 290.5 23 105 3.5 16 1.300 3436 481.5 282 278-4 271.2 279.6 24 96 3.5 12 1.340 3439 425.8 256.2 .250.5 244.1 254.4 25 107 2.7 12.5 I. 218 777.5 205.1 188.2 189.8 187.7 189.8 20 152.4 2.8 31.5 1.206 966.5 203 188 187.4 185.9 188.0 27 105 3.5 16 1.222 950 514.2 426.9 421. 1 422.2 420.4 28 149. 1 2.8 39 I. 218 1429 470 369-5 370.4 369.1 369.3 29 283 2.5 234-7 1.206 4450 464.7 321.2 31S.9 311-3 317.6 30 283 2.5 234-7 1.205 |i879 465-3 403.9 403.3 404.6 403.7 31 283 2.5 234.7 1.200 11919 465.9 385.4 384.7 384-0 383.8 32 283 2.5 234.7 1.200 12425.5 466.5 370.6 368.0 366.6 367.0 33 283 2.5 234-7 1.220 2921.5 464.8 347-8 350.9 347-7 349.7 34 283 2.5 234-7 1.227 3426.0 463-7 336.0 337.6 331-4 336.6 35 283 2.5 234.7 1.220 I4446.5 460.0 316.6 316.6 308.6 315.0 36 283 2.5 234.7 1. 192 ,5945.0 1 455.8 295.0 293.9 285.6 293-0 37 283 2.5 234.7 1.206 5945.0 453.1 294.7 291.5 283.2 291.4 62 EXTERIOR BALLISTICS. as in'the last four rounds, and one or two others. Equation (30) is based upon the supposition that the path of the pro- jectile is a horizontal right line, and, of course, gives only approximate results when this path has any appreciable curvature. It will be shown subsequently that, to obtain the real velocity, the " v " computed by (30) should be mul- tiplied by the ratio of the cosines of the angles of projec- tion and fall. In No. 37, for example, it will be found that to attain a range of 5945 metres (3! miles) the angle of pro- jection would have to be 12° 37', and the angle of fall would be 17° 40'. Making the necessary correction, we should find the velocity to be 290.7 m. The last column gives the remaining velocities computed by Mayevski's formulas. They follow very closely those computed by Krupp. In the absence of tables we ma}^ determine remaining velocities which exceed 1300 f. s. as follows: We have found, when n = 2, C , V •^ = -X log — V ^ , As ^ UAs^ , ^ V . As . i/AsV . As As -yr is usually a small quantity, all its powers higher than the first may be neglected, and we may put V 6 V As V 1+4 For oblong projectiles having ogival heads of i J- calibers A ^0.000142. If the ogive is of 2 calibers, A =0.0001316. This method gives correct results for distances of a mile, or even more, especially for the heavy projectiles used with modern seacoast guns. If the data are given in French units — that is zv, d, and d^ in kilogrammes, din centimetres, and s and V in metres — the value of A will be 0.000030357. EXTERIOR BALLISTICS. 63 Example. Let dz=.io.^ cm., 2e/ = 455 kg., <5 = 1.274 kg., <5^ = 1.206 kg., F= 520.8 m., and .^=1900 m. [Krupp's Bulletin, No. 31.] We have ^ 455 X 1.206 , C = , \^ =■ 0.46301 (30.5r X 1.274 and 520.8 520.8 . 0.000030357 X 1900 1. 12457 ^ ~^ 0.46301 The measured velocity in this example was 465.5 m., while the velocity computed by Krupp was 460.1 m. CHAPTER V. RELATION BETWEEN VELOCITY AND INCLINATION. Expressions for the Velocity. — Equation (15), which, since {i) = --— -|- (^), may be written (f)»-(#)» = ^-{jr--^4 (31) gives the relation between the horizontal velocities ^and 11 and the corresponding inclinations ^ and d^\ and of these four quantities any three being given, the fourth can be ac- curately computed, provided, of course, that the value of k has been accurately determined by experiment. The func- tions (^)„ and (?^)„ are the integrals of ;^^-^-, and the fol- COS t/" lowing are the forms they take for the values of n here adopted : ('>), = i { tan » sec » + log tan (^ + y) } {»), = tan » + i tan' » + A|og.„g+|) It is evident that all these expressions become o when /> — o, negative when f'^ is negative, and infinite when ?^ =: ' ; or, in symbols, (o) = o,(— ^) = — (/5^), and r' j = x If there were buto^ie " law of resistance" — in other words, \^ n had but one value for all velocities — it would be easy to calculate the velocity for any given value of /> by means of EXTERIOR BALLISTICS. 65 (31). It would only be necessary to tabulate the values of (/>)„ for all practical values of d- as the argument, and to pro- vide a similar table of (-j with ?/ as the argument. But, as we have seen, ;i may change its value two or three times in the same trajectory ; and though it would be possible to ascertain by trial the exact point of the trajectory where this change occurred, yet the labor involved would be very great. Basliforth's Method. — Professor Bash forth overcomes this difficulty by giving to 71 the constant value 3, and making /r' to vary in such a manner as to satisfy (31) for all velocities. His method of procedure is as follows: making « = 3 and /> = o, (31) becomes C/' i^tan ^ + i tan>^ in which 6^ and (f are the horizontal velocity and inclination, respectively, at the beginning of an}^ arc of the trajectory we may be considering; and v^ the velocity at the summit. In Bash forth 's notation 3^ ^' ^(loooy w' substituting this in the above equation and multiplying by (1000)^ to avoid the inconvenience of very small numbers, we have /iooo\^ /iooo\' K d' i ^ , . 3 ) by means of which either z^^, [/, or (p can be determined when the other two are known. When the resistance can be taken proportional to the cube of the velocity, K is con- stant; but for all other velocities it is a variable, and we must take a certain mean of its values for the arc under con- sideration. Prof. Bashforth takes the arithmetical mean, which will generally give very accurate results for arcs of 66 EXTERIOR BALLISTICS. lo or 15 degrees in extent. In his work he gives the ne- cessar}^ tables for suitably determining — for all velocities from 100 f. s. to 2900 f. s., and also tables giving values of 3 tan ip -\- tan^ ip for all practical values of ^ . Other approximate methods involving less labor will be given further on. High Aiig:le and Curved Fire. — When the initial velocity does not exceed 800 f s., which includes nearly all mortar and howitzer practice, the law of resistance for oblong projectiles is that of the square of the velocity; whence, making n ^ 2, and dropping the subscript, (31) be- comes or, writing / (u) for (^)-(^)=^{/W-/(C/)) (32) The value of /(?/) for any given value of ti can be taken directly from Tables T. and II., the method of construction of which will be given further on. Table III. gives (^) and extends from ^ = o to (^ = 60°. To use (32) for computing low velocities (and also for high velocities, exceeding 1330 f. s.), we have /«=f I (?)-('?)} + /(f^) (33) 2 in which u and B are the only variables; -^, (^), and I{U)j having been determined, do not change their values for the same trajectory. To illustrate the ease with which velocities may be cal- culated by (33), take the following data from Bashforth's "Treatise," page 115: EXTERIOR BALLISTICS. 67 V^ 751 f. s. ; ^ == 30°; w = 70 lbs., and (i' = 6.27 inches. Here C/== 751 cos 30° ^ 650.385 f. s. ; and from Table I., /(r7) = a93354; -g. = -^ = 1. 12323. We will, following Bashforth, compute the velocities for ^ = 28°, 24°, 20^ 40°. The work may be conve- niently arranged as follows: {f) z= 0.60799 I{U) = 0.93354. e (») (<<,) - (0) ~({6) - (0)) /(«) (Table I.) u « sec 6 = V Bash- forth's Differ- ence. 30° 0.60799 . 00000 0.00000 0.93354 650.38 751-0 75I.O 0.0 28" .55580 .05219 .05862 0.99216 636.09 720.4 720.4 0.0 24° .45953 .14846 .16675 I . 10029 612.03 669.5 670.2 - -7 20° .37185 .23614 .26524 I. 19878 592.33 630.3 630.5 .2 16° .29063 .31736 .35647 I . 29001 575.69 598.9 598.9 0.0 12° .21415 .39384 •44237 1.37591 561.23 573.8 573.5 + .3 8° . 14100 .46699 .52454 1.45808 548.38 553.8 553.1 •7 4° . 06998 .53801 .60431 i^537S5 536.71 538.0 537.0 i.o o"^ .00000 .60799 .68291 I. 61645 525-91 525.9 524.6 1.3 -4" — .06998 .67797 .76151 1.69505 515.74 517.0 515.5 1.5 8° .14100 • 74899 .84129 1.77483 505.99 511.0 509.3 1.7 12° .21415 .82214 •92345 1.85699 496.52 507.6 505.7 1.9 16° . 29063 .89862 1.00935 1.94289 487.15 506.8 504.7 2.1 20° .37185 .97984 . I. 10056 2.03410 477.77 508.4 506.2 2.2 24" •45953 1.06752 I . 19906 2. 13260 468.22 *5i2.5 510.2 2.3 28° •55580 I. 16379 1.30720 2 . 24074 458.38 5i9^i 516.8 2.3 32° .66343 I. 27142 1.42809 2.36163 448.06 528.3 525.9 2.4 36° .78617 I. 39416 1.56596 2.49950 437.11 540.3 537.9 2.4 40° .92914 1.53713 1.72654 2.66008 425.32 555.2 552.8 2.4 The numbers in the second column are taken directly from Table III. for the values of f^ given in column i. Sub- tracting the numbers in column 2 from {(p) (=0.60799) gives 2 those in column 3; and these multiplied by -^ {= 1. 12323) are written in column 4. Adding I (U) (=0.93354) to these last gives the values of / (ti) in column 5. The values of u are then taken from Table I., and these multiplied by sec '^ give the velocities sought. For com- parison the velocities computed by Bashforth, by his method already explained, are also given ; and it will be seen that 68 EXTERIOR BALLISTICS. the differences between his velocities and those computed by (33) are practically nil. This method of determining velocities may be used without material error when the initial velocity is as great as 1000 f. s. Example. — The 8-inch howitzer is fired with a quadrant elevation of 23°; muzzle velocity, 920 f. s. ; weight of shell, 180 lbs.; diameter, 8 inches. What will be the velocity in the descending branch when /> = — 27° 54' ? (See Mac- kinlay's '' Text-Book," page 109.) Here F=920, Z7= 920 cos 23°= 846.86 /(/7) =0.40884; log ^ = 9.85 194 The computation is as follows: (23°) = 0.43690 (-27° 540=— 0.55327 log 0.99017 = 9.99571 C log - = 9.85194 log 0.70412 = 9.84765 I{U)= 0.40884 I {11) := I.I 1296 . • . ^27. 5^. = 609.4 f. S. Mackinlay gets by Niven's method, dividing the tra- jectory into two parts, 6^270 54' = 610.6 f. s. It will be seen that by the method developed above for calculating veloci- ties, the length of the arc taken makes no difference in the accuracy of the results. Siacci's Method. — Equation (13) may be written /^ dd- _gC r ^ sec' & d u Since ^ is a function of u, there must be some constant mean value of sec d- which will satisfy the above definite EXTERIOR BALLISTICS. 69 integral. Representing this mean value of sec d- by a, and writing U' and u' for af/and au respectively, we have n d& _agC_ f^ _duf__ Making ^(«') = ^7^ + e (34) becomes tan ^ - tan ,!' = ^{ /(«')- 7(^7)} (35) The values of I {ii') are given in Table 1. for oblong pro- jectiles, and in Table II. for spherical projectiles. The method of computing the /-function is entirely similar to that already described for the 5 and /-functions, and need not be repeated. For oblong projectiles the formulae areas follows, in which, for uniformity, / (z^) is employed as the general functional symbol: 2800 f. s. > 7^ > 1330 f. s. : / W = [5.3547876] ^ — 0.028872 1 330 f. s. > 7/ > II 20 f. s. : /(t/) =[8.2947896] ^ + 0.015293 1120 f. s. > T^ > 990 f. s.: 7(7') = [17.1436868] -^ +0.085087 990 f. s. > z/ > 790 f. s. : 7(2;) = [8.4557375] -L — 0.061373 790 f. s. > z^ > o : I{v)^ [5.7369333] -^ — 0.356474 70 EXTERIOR BALLISTICS. If we compare (34) with (31) it will be seen that „_ i (f).-w« I ^ ( tan (p — tan § ) and this value of a renders (34) and (35) exact equations; in fact, reduces them to (31). It would seem at first as if nothing had been gained by introducing a into (35), since its value depends upon that of ^2, and must, therefore, change when n changes. The following table gives the values of a for the arcs contained in the first column, when ;/ = 2, w = 3, and n=z6, computed by the above formula : Arc Mo* 30° to 20° 30^ 10^ ic- 30° '' —20^ 30^ 30^ I . 1066 1-0741 I. 0531 I. 0419 I .0409 I. 0531 I . 1069 1.0749 I. 0541 I .0429 I .0418 I. 0541 1079 0772 0573 0460 0443 0573 It is evident from this table that when the angle of pro- jection is as great as 30°, the velocity at any point of the tra- jectory may be computed with sufficient accuracy by using either set of values «; since the greatest difference between those in the second and fourth columns on the same line is but 0.0042, and this would make but a slight difference in the values of U' or u'. MoreoVer,"since U' — a Fcos ^, and u' =^ av cos ^?, it is apparent that U' and u' differ less from V and V respectively than do U and u; and this is important when, as is usually the case, the law of resistance is different for the initial and terminal velocities. If in the above expression for a we make n = 2, we have Didion's expression for «, viz. : ^^ (y)-W tan (p — tan ^ EXTERIOR BALLISTICS. 7 1 in which (i?) — i I tan ^ sec & + log tan ^- + — ) I Example. — 'A 12-inch service projectile, weighing 800 lbs., is fired at an angle of projection of 30° and a muzzle velocity of 1886 f. s. Required its velocity when (a) the inclination of the trajectory is 15°, and (b) when the inclination is — 15°. Here^=i: 12, w — 800, V = 1886, and ip = 30°. From (35) we get / {u') =: /(^') -f -^ I tan ^ - tan ?? i (a) ^ = 15°. From our data we have „^ (30°)-(.5°) ^^gd3g821^,.o888 tan 30° — tan 15° 0.30940 U' = a Fcos30° = 1778.34 .'./{[/') = 0.04270 ^ _ w 800 d^ 144 and tan 30° — tan 15° =0.30940 Tt>\ X 288 X 0.30940 .-./(.)= 0.04270 + 3^ ^ ^3^3^ - 0.14500 .-. «'= 1149.77. 1149.77 . . • . z/,.o == — ^^-^ = 1093.3 I- s. " a cos 15° ^^ ^ (b) x^= — 15°. We have ^^ (30°) + (15°) ^ 0.87911 ^ J ^.^ tan 30° + tan 15° 0.84530 "^ C/' — « Fcos 30°= 1698.65 .-. 7(^0 = 0.04958 tan 30° + tan 1 5" == 0.84530 .-. I {u') =0.04958 + 0.29260 = 0.34218 .-. ?/' = 891.14 .-. z;,,,. =887.1 f. s. The values of v^^^ and z/_,^, computed by (31) are 1097.6 and 892.9 respectively. 72 EXTERIOR BALLISTICS. Siacci's Modification of (35) for Direct Fire.— Since in direct fire the angle of projection does not exceed 15°, and is generally much less, the values o^ a for this kind of fire will not differ much from unity. For example, with 10° elevation, and an angle of fall of — 12°, we shall have for a ,,_ (10°) + (12°) _ Q.39i39 _^^^., ■" tan 10° + tan 12° ~ 0.38889 "~ "^ It is manifest, therefore, that for sucli small angles no material error would result in making «= i ; the following, however, is a closer approximation. If we consider that part of the trajectory lying above the horizontal plane passing through the muzzle of the gun, it will be seen that * a should be greater than unity and less than sec co. Siacci makes W-2 a = (sec^)«-i therefore, when « = 2, a =1 ; when « = 3, a = V sec ^, and when n =^ 6^ a =z sec f ; and the average value of a for the whole trajectory generally fulfils the above condition. This value of a substituted in (34) gives, by an easy reduction, •tan c^ - tan ^ =: /^ , \ . L-— - -L \ - — ^ n A cos (p { {u sec cpf F" ) or, writing u' for u sec ^, and proceeding as already ex- plained, ' Example. — Take the follow^ing data: 800 ^= 12 ; 7£/ = 800; 6'= ; F= 1886 ; ^ =1 10°. Compute 144 the remaining velocity in the descending branch when t?=^ 13°. We have / {u') = -^ cos" (f (tan ^ - tan ^) + / (F) EXTERIOR BALLISTICS. and the computation will be as follows: log (tan 10° + tan 13°) = 9.60980 log- -^ = 9.55630 2 log- cos 10° = 9.98670 log 0.142 1 7 = 9.15280 7(1886)1=0.03477 / (//') z= 0. 1 7694 // = 1 07 1 . 76 IO7T.76 COS lo"^ n COS 13° = 1083.2 f. s. The velocity at the same point computed by (31), divid- ing- the trajectory into three arcs, with the points of division corresponding- to velocities of 1330 f. s. and 1120 f. s. respec- tively, is 7' =: 1081.55 f. s. This agreement is very close; but if we make if = 30° and ^ == 15°, as in the preceding ex- ample, we should find by tiiis method ?''i5« = 1113.1; and if d- =z — 15°, we should find 7^_i^« = 859.3, which differ consid- erably from their true values. Mven's Method.— W. D. Niven, Esq., M.A., F.R.S., has given the following method for determining velocities in terms of the inclination : Equation (13) may be written J , ^ Aj , («sec.?r" "-A J „. «'"*■ in which, as before, a is some mean value of sec d^ between the limits sec

>• function for velocities extending- from 400 f. s. to 2500 f. s. (See Table VI. ii Mackinlay's "Text-Book.") It will be seen by comparing the expressions for D {v) and I [v) that we have the relation and, therefore, in terms of the /-function, (38) becomes .^ = t|'{^M-/(^0} (39) log ^==1.4570926 Comparing (37) with (31), it is apparent that to make (37) or (38) exact equations we must have -\^^r For direct fire Didion's value of a may be used ; but for high-angle firing- the following gives more accurate results, obtained from the above equation by making a/ = 2 : a = i'^f[ Example. — Take the following data: df=l2; w=r8oo; F=: 1886; ^ — 30° and <> = — 30° ; /? = 30° + 30° = 60° ; to find v,^.. % * If we use Niven's tables, in which the functions decrease with the velocity. (38) should be written i>«£j/?(i/0-^(«')[ EXTERIOR BALLISTICS. 75 We have from (38) D{u') = D{U') + ^D The computation may be conveniently arranged as fol- lows : log (ip) = 978390 constant = 1.758 12 c log 30 = 8.52288 log 3) 0.06490 log a = 0.02163 log D^ 177815 r log (7= 9.25527 11.3516= 1.05505 log F= 3.27554 log a = 0.02163 log cos ip = 9.93753 log U' = 3.23470 6^'= 1716.74 (Niven's Table) D {U') = 84.6090 C D{u') Dzzz II. 3516 73-2574 .*.«' = 827.12 =:« Z/ cos ?? . • . v.j^^. = 908.7 f. S. Siacci's method, using Table I. of this work, gives ^_3oo = 907.5 f. s. ; while equation (31) gives v_^^ = 913.2 f. s. Modificatioii of (38) for Direct Fire. — If we make a = (sec (fY^ we shall have, by a process similar to that already employed in Siacci's method, the following modified form of (38), which can be used in all problems of direct fire, viz.: C j9 = cos (f in which u' ^=^u sec ip. Example, — Let (1886) = 84.9966 Z> (1500)== 83^9359, log 1.0607 =: 0.02560 log 6^ = 0.74473 log D = 0.77033 />=5°.89 = 3°-'> .-. ??= - 2°.89 (b) By Table 1. : 7(1500)= 0.07173 /(i886) = 0.03477 log 0.03696 = 8.56773 log?= 145709 log (7 = 0.74473 log 7^ = 0.76955 D = 5°.88 .-. ^= -2°.88 CHAPTER VI. TRAJECTORIES— HIGH-ANGLE FIRE. As we have seen, the differential equations for x,y^ t, and s do not generally admit of integration in finite terms for any law of resistance pertaining to our atmosphere ; that is, for any recognized value of ?i. It is true that Professor Greenhill has recently* succeeded, by a profound analysis, in deducing exact finite expressions for x and y by means of elliptic functions, when ?^ = 3 ; but these results, though of great interest to the mathematician, are far too complicated for the practical use of the artillerist. When /^ = 2 the ex- pression for ds can be integrated and useful results deduced therefrom, as will be seen further on. For low velocities, such as are generally employed in high-angle and curved fire, the effect of the resistance of the air upon heavy projectiles is comparatively slight; and for a first (though rough) approximation we may, in such cases, omit the resistance altogether, or, better still, we may suppose the projectile subject to a mean constarit resistance. A still closer approximation may be obtained by taking a resistance proportional to the first power of the velocity. As the differential equations for the co-ordinates and time are susceptible of exact integration upon each one of these hypotheses, we will consider them in turn. TRAJECTORY IN VACUO. Making p = o, (9) becomes duzuzo and therefore, in vacuo, the horizontal velocity is constant, or /^= U Integrating (21), (22), (23), and (24) between the limits (p and d- gives, \i ti ^ U, * " Proceedings of the Roj^al Artillery Institution," Vol. XI. 10 78 EXTERIOR BALLISTICS. and / = —(tan f — tan d) (4O ;ir = — (tan ip — tan d) (42) 772 7 = — '(tan' ip ~ tan' />) (43) ((^) - ('>)) (44) Equation of Trajectory in Vacuo. — Eliminating tan d- from (42) and (43) gives y ^=L X tan (p 2W which is the equation of a parabola whose axis is vertical A parabola, therefore, is the curve a projectile would de- scribe in vacuo. Since a parabola is symmetrical with respect to its axis, the ascending branch is similar in every respect to the de- scending branch, the angle of fall being equal to the angle of projection ; and generally, for the same value of j, tan d^ has numerically the same value, but with contrary signs, in both branches; being positive in the ascending branch, negative in the descending branch, and zero at the vertex. If we make ?^ = — ^ in (42) it becomes ^- 2 C/' V sin 2ip X = tan

whence ^ log u = m log tan 1 - -] I -f- ^. Let v^ be the velocity when ?> = o, that is, at the summit of the trajectory ; then C — log v^, and we have (f+4) <«) r= 2/„ tan 4 Substituting this value oi u in equations (21) to (24), and integrating so that /, x, y, and s shall all be zero at the origin, that is, when -& =npy we have, making the necessary reductions, TT sm w — m sin i^ — m j^2 COS (p (sin ^ — 2m) ^ cos ?^ (sin d — 2m) ~ g{i — 4^') "" ^ <^ (i - 4^^') j^2 I + sin ^ (sin (p — 2m) ^9 1 + sin d- (sin ?^ — 2/;?) ^ ~ 4^(1 -Iff) "" 4^(1 -f^) ,j, cos^ (f -{-2m (sin ip — in) ^ ^ cos'' d- ~\-2m (sin z? — ??/) 4w^(i — ^^z'') 4^«^(i — ^^^) When 2?;^ = i, the differential expression for x becomes logarithmic, as do those for /, y, and s when in — \. The integrations are easily obtained for these values of m^ but are omitted on account of their length, and as being of no great practical importance. In the application of these for- mulae it will be necessary, since the resistance of the air is not constant, but varies with the velocity, to determine a proper mean value for m between the limits of integration ; and this we may do as follows : After having computed the horizontal velocities u^ and u^ by means of (33), corre- sponding to the inclinations a and /9, the value of in may be determined by the following equation deduced from the above expression for 71 : EXTERIOR BALLISTICS. 8 1 ^jj^ ^ log u^ - log U^ logta„(^ + f)-logta„(^+4) Example. — Compute the values of t, .r, y, and s, from (p = 30° to ?? = o, with the data given on pag-e 6y. We have ,,, _ log 75 1 + log cos 30" - log 525.9 1 _^^.^_ '"^ - log tan 60° - ^-^^^^^ Substituting in the above formulae, we find ^ = 3-I073 + 7.4295 = 10^537 X = 16908 — 10557 = 6351 ft. y = 4446 — 2526 = 1920 ft. ^=: III55 — 4578 = 6577 ft. Bashforth gets, by dividing the arc into 8 parts, t = io''.4i3, X = 6074 ft., and 7 = 1882 ft. It is easy to see how by suitable tables, the construction of which offers no difficulty, the time and co-ordinates ma}^ by this method be readily, and for arcs of limited extent accurately, computed. For example, we have x = A V~A' v' A being a function of m and ^, and A^ the same function of m and d^. RESISTANCE PROPORTIONAL TO THE FIRST POWER OF THE VELOCITY. Differential Equations. — When ;/ = i, the differential equations (13), (17), (18), and (19) become respectively, since tiA CO^ ^^ dt=- k du g u dx = k du dy = - k tan -& du 82 EXTERIOR BALLISTICS. Time and Co-ordinates. — The integration of the first three of these equations between the limits {cp, d) and {[/, 7i) gives (supposing k constant) tan

The projectile is still 1882 — 1856 — 26 ft. above the level of the gun = Ay. If Ax and At are the corresponding addi- tions to the range and time of flight, we shall have approxi- mately Ax Ax = 26 cot 37° =r 35 ft. ; and At = = o''.o8o. We therefore have ^=6069 + 5284 -f- 35 = 11388 ft. r=i 10.412 -[- II. 103 + 0.080 = 2i'^595 These values agree almost exactly with those deduced by interpolation from the table on page 117 of Bashforth's work. Example 2. — The 8-inch howitzer is fired with a quad- rant elevation of 23°. Muzzle velocity, 920 f. s. ; weight of shell, 180 lbs. ; diameter, 8 inches. Find the range and time of flight. (Mackinlay's "Text-Book of Gunnery," page 107.) Assuming the angle of fall to be — 27° 54', we find by the above method X=z 7886 + 7108— 13 = 14981 ft. T = 10.183 + 10.801 — 0.022 = 20^^.962 Mackinlay gets, using Niven's method, X— 14787 ft, and r=2o".8i3 He states that "the published range-table gives 15000ft. as the range, and 2i'^5 for the time of flight." EXTERIOR BALLISTICS. 89 Example 3. — Let V =. 2g% m. = 977.71 ft., d ^= \^ cm., w = io k.g-., f = 35° 21', o — 1.270 k.g-., and 0,= 1.206 kg. Find Xand T. (Krupp's Bulletin, No. 55, December, 1884.) For the Krupp projectiles and low velocities we will take for c the ratio of the coefficients of resistance deduced from the Krupp and Bashforth experiments respectively, and which are given in Chapter II. Let these coefficients be represented by A and A' . Then for velocities less than 790 f s. we have 10^^ = 5.6698755- 10 log ^'=5.7703827 — 10 log c = 9.8994928 .•.^=0.7934 To find cT, expressed in English units, when w and d are given in kilogrammes and centimetres respectively, we have ^ _ loooo k w ~ \^m^ c d' in which k is the number of pounds in one kilogramme, and in the number of feet in one metre. Reducing, we have C-= [1.2534887] J As the initial velocity in this example is considerable, we will take into account the density of the air at the time the shots were fired, and also the diminution of density due to the altitude attained by the projectile; and for this pur- pose we will assume the mean value of y for the whole tra- jectory to be 2000 ft. The complete expression for (7 is (Chapter VII.), from which we determine log 6" as follows: log w =: 1.4771213 ^ log ^^ = 7.6478175 constant log = 1.2534887 log 0, = 0.0813473 ^ log ^ = 9.8961963 z log eh =1:0.0312468 log (7=0.3872179 90 EXTEklOR BALLISTICS. Assuming the angle of fall to be — 44° 40', and proceed- ing as in the first example, we find X= 10408 + 8736 + 104 = 19248 ft. 7^r=: 15.088 + 16.324 + 0.221 == 3i''.633 Krupp gives the ranges of three shots fired with the initial velocity and angle of departure of this example, and the ranges reduced to the level of the mortar, as follows: NO. OF SHOT. RANGE IN FEET. 18 19039 19 19265 20 19364 Mean of the three shots = 19223 ft. Computed — mean = 25 ft. Example 4. — Given F= 206.6 m. — 677.834 ft., d = 21 cm., w = gi k.g., and (f = 60°, to find Jfand T. (Krupp's Bulletin, No. 31, Dec. 30, 1881.) It will be found that (assuming the angle of fall to be — 63° 30', and taking no account of atmospheric conditions) ^ = 5390 + 4945 + 67 = 10402 ft. T= 17.016+17.543+0.250 = 34''.8o9 Krupp gives the observed ranges of five shots, with the above data, as follows : NO. OF SHOT. OBSERVED RANGE. 22 10332 ft. 23 10305 " 24 10384 " 25 10463 " 26 10440 ** Mean of the five shots = 10385 ft. Computed — mean = 17 ft. Example 5. — Given F=: 204.1 m. = 669.63 ft., W), we shall have from (55) /=i:^— log ^ ^ - and whence s" — s' — As—'M — los- If??'' differs but little from ??' (say one degree), the cor- responding values of Ax and Ay can be calculated with sufli- «■ -(f) «- -(*") w- -(f) (0- - (^") EXTERIOR BALLISTICS. 93 cient accuracy by multiplying Js by cos ^ {d-' -\- d-") for the former, and sin ^ {&' -\~ &") for the latter; or, Ax^M— log ^f^.^ ~ ^/2 cos I (ir + ^^'0 = M—J^ (sav) Jj/ = M J log ll^^j sin i {^r + &") = M^A: (say) For the entire range we evidently have X= y Jx =r M~ I A^=M-^ K g the summation extending from ^ = ^ to ^ = w, w being the angle of fall. To determine the value of co we have, since the sum of the positive increments of ^'^ in the ascending branch is equal (numerically) to the sum of the negative increments in the descending: branch, Expression for the Time.— For the time of flight we have, when dx is small, u in which u is the mean horizontal velocity corresponding to Ax ; but, from (15), when n = 2, __ k whence \{i)-{^)\ I Ax ^ ( '^^ — ^^^'^ ^ ^ At = k or, substituting for Ax its value given above, M = ^AZS,.. ,.A\ If we put je=Jf{(0-w[* 12 94 EXTERIOR BALLISTICS. we may have log J0=log J^ + i log [(0 - i^)] The two values of log [{i) — (^)] corresponding to the extremities of the arc Js, are log [ (0 -( S j I I _ -^ (3 tan /> + tan^ if) \ v'' r^ tan d- d tan d- y-irf T^ — — T;T* = T^n and ^.. ^ tan d- d tan ^ _ t^^ -y{2> tan ,> + tan^ (>) U' ^ Professor Bashforth has published extensive tables of the definite integrals ''*7^,*X:J,and '^F^^ for values of ^ extending from +60° to — 60°, and of y from o to 100, calculated by quadratures; by means of which the principal elements of a trajectory may be accurately determined as follows: As the coefficient of resistance K generally varies with the velocity, the trajectory must be divided into arcs of such limited extent that the value of K for each arc may be con- sidered constant ; and it should be so taken as to give, as nearly as possible, its mean value for the arc under con- sideration. In the equation given on page 65, viz.: /loooX-' /iooo\' . K d" { ^ , ♦ . I suppose U and ^ to be the initial horizontal velocity and angle of projection respectively, and both known ; and let &, also known, be the inclination of the forward extremity \a^. — *• » TJ J. X EXTERIOR BALLISTICS. 97 of the first arc into which the trajectory is divided. Now, assuming a mean velocity for this arc, take out the corre- sponding value of K from the proper table and compute (I ooo\ ^ - — 1 ; then, in the same equation, changing (p to />, 17 be- comes tlie horizontal velocit}^ at the forward extremity ot the arc, which can also be computed. Next compute y by means of the equation given above, with which and the known values of ^ and ^ enter the tables and take out '^T^ , ^X^ , and ^Y^-^ lastly, multiplying 2 the first by — ^, and each of the others by -— , we have the ^ g ^ g time of describing the first arc of the trajectory and the co- ordinates of its for vvard extremity. By repeating the process with the second and following arcs into which the trajectory may be divided, the whole trajectory becomes known. Professor Bashforth gives various other tables in his work, besides those we have mentioned, for facilitating the calculation of trajectories by his method, with examples of their application and full directions for their use. Modiflcatioii of Bashfortli's Method for low Velo- cities. — When the initial velocity. does not exceed 790 f. s. the law of resistance is that of the square of the velocity for the entire trajectory; and even when the initial velocity is as great as 1000 f. s. examples show that no material error results if we still retain the law of the square in our calcu- lations ; and this furnishes a very easy method for calcu- lating trajectories for high angles of projection and for the initial velocities usually employed in high-angle fire, and which, it is believed, gives as accurate results as by any other method, and with less labor. Making ;/ =2, equation (25) becomes k d tan & dt in which 6 {(,-)_(^)|4 ((?) = !{ tan d sec & + log tan g + |^) | 98 EXTERIOR BALLISTICS. We also have from (15), when « = 2, and ^ = 0, k^ I ( = — = — (say) and this substituted in the above expression for ^/ gives Vc. d tan /> dt— — ? ji_^(#)|i whence tan ^ V, In the same wa}^ we obtain from (26) and (27) the follow- ing expressions for x and y : d tan d- v^ < Pi r(^>) "^x,' ^"^ - . ^ tan ^ ^ tan & It will be seen that this method depends upon tables of definite integrals which must be calculated by quadratures as in Bashforth's method, and with the same number of arguments; but the great advantage of these formulas over Bashforth's is in the fact that y is constant for a given tra- jectory, and, therefore, the labor of calculation is the same for all angles of projection. To determine the value of k^ for oblong projectiles of the standard type we have 2A Taking the value of A derived from the Bashforth experi- ments for velocities less than 790 f. s., and making ^^=1 32.16, -« fi"d k' ^ [5.4359033] c For the Krupp projectiles we should have, taking May- evski's value of A, '^' = [5-5367564] c The numbers between brackets are the logarithms of the factors by which C is to be multiplied. EXTERIOR BALLISTICS. 99 For computing- v^ we have from (32), when ^ = 0, /(t/„)=-J(^)+/(f/) (56) in which ^ may be the inclination at any point in either branch, and U the corresponding horizontal velocity. The values of (^) are given in Table III. To show the practical working of this method, we will take the example from Bashforth already given (see page 6j). The data are: V:^j^\ f. s. ; ^ = 30°; ^ = 6.27 inches, and w:=z'/o lbs.; whence ^=650.385 f. s., and 70 C^^-rp — - == 1.78059. Determine the range, time of flight, angle of fall, and terminal velocity. First compute v^. We have from Table III. (30°) = 0.60799 whence, from (56), ^^^S~ "^ ""(^50.385) = 0.68291 + 0.93354 = I.61645 therefore, from Table I., ' ^0 = 525.91 f. s. Computation of y : log (7=0.2505630 constant log = 5 -435903 3 log /^'' = 5.6864663 log z/q' = 5.4418228 log r = 97553565 r = 0.56932 As general tables of the definite integrals '^7'^, '^X^ , and ** V^ have not yet been prepared, the following table has been calculated for this particular example, merely to illus- trate the method : lOO EXTERIOR BALLISTICS. r = = 0.56932 T X V 30° 0.63676 . 70486 0.21775 24 .47838 .51493 .12039 18 .34169 .35965 .06045 12 .21944 .22662 . 02460 + 6 + .10673 + .10838 + .00575 .00000 . 00000 . 00000 - 6 - .10358 — . 10208 + .00531 12 .20647 .20061 .02091 18 .31104 .29793 .04701 24 .41977 . 39620 .08479 30 •53551 .49759 .13656 36 .66179 .60449 .20615 37 .68417 .62303 .21987 The value of ^°°V° by the above table is 0.21775, and as this must be equal to °F" we see at a glance that co lies between — 36° and — 37° ; and by interpolation we get w=r— 36°5i'; and therefore °X'J = 0.62025 and ^T'J 0.68081. Adding to these the numbers corresponding to the argument 30°, we get "PX- = 1.32511, and *^7:y^ = I.3I757- Lastly, multiplying the first of these by -^,and the second by — , we obtain X= 11396 ft. and r= 21^546 which agree with Bashforth's calculations. The terminal velocity is found from (32), viz.: and V... = «,.. sec CO We find and ^o. = 434.7 f. s. ^«o = 543-2 f. s. It will be seen that the inverse problem, namely, Given EXTERIOR BALLISTICS. lOI the terminal velocity and angle of fall, to determine the initial velocity, angle of projection, range, and time, can be solved by this method with the same ease and accuracy as the direct problem. We should first compute the summit velocity by the equation /W = /(0-|=H (57) and then all the other elements would be determined, as already explained. In calculating trajectories by this method with the help of tables of the definite integrals '^T^ , etc., it will generally be necessary, as in Bashforth's method, to interpolate with reference to y as well as d-, and for this purpose the integrals must be tabulated for different values of y proceeding by constant diff'erences, and including the highest and lowest values of y likely to be needed in practice, which are, ap- proximately, I and O.2. 13 CHAPTER VII. TRAJECTORIES CONTINUED — DIRECT FIRE. Niven's Method. — If a is some mean value of sec d between the limits of integration ; that is, if we make a = sec ^? (say) then equations (17) to (20) may be written as follows: _ 6; d {a li) ^ A (a uf C -T-d (a m) ^^=-^cos^-^^^^^ (58) C . -. d{au) dy— — -7- smd . .„_\ A {a lif ' C d (a u) A (auf-' Making a u=iu\ and integrating so that t, x, y, and s shall each be zero when u' = U\ we have t = —£_\_l L_l C cos i)^ y = {n-2)A "-''^^ ( u'^-' — U'""-- c 11 — 2) A \ u"-' U'"^ ) Comparing these equations with those deduced in Chap- ter IV. for rectilinear motion, it will be evident that we have as follows: t = ciT{u^)- r{U')-] (59) x=Cco?>^[S {u') - S ( U')~j (60) y=C sin J [S (?/) - S{U'\^x tan 5 (61) s=ClS{u')-S{U')'] (62) EXTERIOR BALLISTICS. IO3 The first three of these equations (or their equivalents) were first published by Mr. Niven in 1877, and in connection with equation (38), viz.: D=Ccos.J[D {u') - D ( U')] (63) constitute what is known as "Niven's Method." If we use the /-function instead of the Z>-function, equa- tion (63) becomes /} = 25_^ cos &U W) - I ( U')-\ (64) or, better still, for direct fire (see Chapter V.), D = 25_^ sec (f [/ (u sec f) - I { F)] (65) in which log ^=:l. 45 70926* The values of ^ adopted by Mr. Niven are as follows: For the /^-integral -— tan

/-^tan^} -^I{U')x=- \a{u')-A{U')\ From (60) we have ^x = S{u')-S{U') whence, by division, ^ \y nn<.l nu'\- A{u')-A{U') _|__tan^}-/(f/)_- ^-(„,)_5(f;,) aC iA(u')-A(C/') ,,rml iA^\ or z X EXTERIOR BALLISTICS. IO9 Calculation of the ^-Function.— We have (Chap- ter V.) and therefore _ g Q , ^ l2\n- I) A' u"^'^ + {n-2)Au'^-' + ^' which becomes, when ;/ = 2, The constants Q, corresponding to the five different ex- pressions for the resistance, are given in Chapter V., and the values of Q' are to be determined as explained in Chapter IV. Making the necessary substitutions, and using A {v) as the general functional symbol, we have for standard oblong projectiles the following expressions for calculating the A- functions : 2800 f. s. > -t^ > 1330 f. s. : A {v) = [8 9012292] -^, + [2.6701589] log v - 1714-55 1330 f. s. > z' > II 20 f. s. : A (7;) = [14.6562945] ^ + [5.1480576] i - 53.13 1 1 20 f. s. > z' > 990 f . s. : A (v) = [32.2571789] ^„ + [14.4412953] ^ + 126.68 990 f. s. > z^ > 790 f . s. : A (v) = [14.9781903] — , — [5.9124902] ^ + 449.89 790 f. s. > -z/ > 100 f. s. : A {v) = [9.6655206] ^ + [4.1438598] log V - 45916.40 The values of A {v) calculated by the above formulae are given in Table I. 14 no EXTERIOR BALLISTICS. Equation {66), together with (35), (59), and (60), are the fundamental equations of *' Siacci's method." This method, by Major F. Siacci, of the Italian Artillery, was published in the Revue d' Artillerie for October, 1880. A translation of this paper by Lieutenant O. B. Mitcham, Ordnance De- partment, U. S. A., was printed in the report of the Chief of Ordnance for 1881. Lieutenant Mitcham added to his translation a ballistic table adapted to English units, and based upon the coefficients of resistance deduced by Gene- ral Mayevski from the Russian and English experiments noticed in Chapter IL In this table he gives for the first time the values of T{v). We will, for convenience, collect thesd equations to- gether and renumber them : They are : tan ^ - tan ?? = ^ I / (;/) — I{U') | {^7) x^^-\s{u')-S{U')\ (68) : — : I >i 1 1 in — ^ S{u') — i,{U') t =ClT{u')-T{U')^ (70) u' — av cos d^ (71) As the origin of co-ordinates is at the point of departure, y is zero at the origin and also at the point in the descend- ing branch where the trajectory pierces the horizontal plane passing through the muzzle of the gun. Calling the velo- city at this point v^, we shall have, making — d^ z:z w, u'o> = « ^ ^ cos CO (72) From (69) we have aC \A (u'^)-A (U') .. ^„^ ) ._s ^ 2 [ S {u'^) — S {[/') ^ ) and from {6y) a C ~ ^ 2 \s{u') — :^{U') ^'^^M ^ ^^ tan ^ =^ "^ I / {ti'^) - 7(^0 I - tan CO (74) EXTERIOR BALLISTICS. Ill Eliminating^ tan ^ from these last two equations gives tan ^^ = — I ^ (^^ ^) - ■5l/?.)-5(^0 ^ From (68) and (70) we have X=z— \s{2/^)-S{U')\ (76) and a I ^ ^ ' ) T= C[T(u'^)^T{U')] {77) By means of equations {67) to {77) all problems of ex- terior ballistics in the plane of fire may be solved. If we wish to compute the co-ordinates of the extremities of any arc of a trajectory having the inclinations (f and d^, we should make use of equations {67) to (71). If the object is to deter- mine the elements of a complete trajectory lying above the horizontal plane passing through the muzzle of the gun, at one operation, we should employ equations (72) to {77). We will give an example of each, using Didion's value of a. Example i. — Given F= 1886 f. s. ; <^= 12 in.; 2£; = 800 lbs., ^ = 10°, and ?? = — 13°; to find z/^, ;ir0, je, and /e. (See example i, Niven's method.) We have first (10°) + (13°) a = — ^ i \ I o = 1.00723 1 Next ^^" 10° + tan 13° ^ ^ U' = 1886 « cos 10° = 1870.78 From Table I., 5(^0=-2838.3;zJ (^0=44.06; /(^0=O-O358i; r<^U')^i.2t,o From {67^ we have /(^/.) =^ 1 ^"'^^ '""^ + ^""^ '^° 1 +^(^') = 0.14554 + 0.03581 =0.18135 .-. 2^'=: 1063.42; 5(/0==7Oii<4; -^K)=440.44; 7 V) =4-282. These values substituted in (68), (69), and (70) give xq^=. 23017 ft. yQ = 248.06 ft. tQ z= i6''.844 112 EXTERIOR BALLISTICS. From (71) we have Ve = K = 1083.6 f. s. a cos a These results are quite as accurate as those deduced by Niven's method by two steps. Example 2. — Required the horizontal range, time ot flight, and striking velocity, with the data ot Example i. In computing « we will assume an angle of fall of — 14° 30', which gives «== 1.008645 .-. ^'=1873.40 5(^0=2828.5; A{U')=Al-7^\ 7(^/0=0.03563; n^0=i-243- From (73) we have ^^^^g|i=^^tan, + /(t/') = o.09856 from which to calculate ic'^. As the relation between the S-function and y^-function does not admit of a direct solu- tion of this equation, it will be necessary to determine the value of z/o, by successive approximations; and for this pur- pose the rule of ** Double Position" is well adapted. This rule is deduced as follows : Let u^ and u^ be two near values of?/ (or the quantity to be determined), one greater and the other less ; and e^ and e^ the errors respectively, when n^ and u^ are substituted for u in the equation to be solved. Tiien, upon the hypothesis that the errors in the results are pro- portional to the errors in the assumed data, we have e^\ e^W u — //j : u — u^ whence, by division, e^— e^\ e^W u^ — u^'. u — u^ or e^ — e^\ e^\ : u^ — u^\ ii — ti^ from which is derived the following rule: As the difference of the errors is to the difference of the assumed numbers, so is the lesser of the two errors (numerically) to the correc- tion to be applied to the corresponding assumed number. EXTERIOR BALLISTICS. II 3 If 11^ and u^ are selected with judgment, the resulting value of II will generally be sufficiently correct by a single application of the rule, or, at most, by two trials. In our example assume it^ =z 1050, for a first trial ; whence 5 (1050) = 7143.7, and A (1050) =464.94 ; and these in the above equation give 464.94 — 43.71 ^ ^ ^ ^ -to / _ 0.09762 7143.7-2828.5 If we had taken for 71^ the correct value of u^^, the second member would have been 0.09856, and hence ^, = — 0.00094. Whenever ^, is negative the assumed value of u' ^ is too great; we will, therefore, next suppose 2/2= 1040, and pro- ceeding in the same way we find ^^ = +0-00128. The cor- rect value of u^^ is, then, between 1050 ft. and 1040 ft. Ap- plying the rule, we have the following proportion : 222 : 10 :: 94 : 4.23 consequently u'^ = 1050 — 4.23= 1045.77 f- s. : and this satis- fies the above equation. We next find 5(?/^) = 7i87.i; ^ (//^)=473.2o; 7(2^'^) =0.1 9 154; T{u'^)=4.44S We now have from (75) tan io = j o. 191 54 — 0.09856 i = 0.2605 1 .-..(>= 14" 36^ (By Table III.) From {26) and (yy) I 7187.1 — 2828.5 I =24007 ft. r= c:[ 4.448 - 1.243] = i7".8o6 From (72) ^« = ^=i07i.4f. s. a cos Various other problems may be solved by a suitable com- bination of equations {6y) to (71). Indeed, if a velocity, a 114 EXTERIOR BALLISTICS. either initial or terminal, and one other element be given, all the other elements may be computed, though in certain cases this can only be accomplished by successive approxi- mations. Most of these problems, for direct fire, will be solved further on. Api>licatioii of Siacci's Equations to Mortar- Firing. — For low velocities, such as are used in mortar- firing, we may take for a in all cases the following value : tan ^ This simplifies the calculations, and gives results sufficiently accurate for most practical purposes, as the fbllowing ex- amples will show : Example i.— Given F=75i f. s. ; ^ = 30°; and log C zizz 0.25056. Required X, T, w, and v^. (See Example i, Chap- ter VI.) We have, Table III., {(f) = 0.60799. log {(f) = 9.78390 log tan

^ cos ^6» 1 A{v)-A(V)l , . sin 2<. = C- I / (.•) - s{v)-S{V) \ ^^7) For the larger angles of projection employed in direct ^5 Il8 EXTERIOR BALLISTICS. fire, if accurate results are desired, we must determine (o by the equation tan CO = tan w ^ \ I {v) - I (V) \ ^ 2 COS if \ ^ ■' ^ ^ \ using ?/ instead oi v, as already explained. Practical Applications. — We will now apply Siacci's equations to the solution of some of the most important problems of direct fire. Problem i. — Given the initial velocity and angle of pro- jection, to determine the range, time of flight, angle of fall, and terminal velocity. We have [equation (86)] A{:v)-A {y) _ sin 2^ S{v)--S{l^) C "T" ^^^ from which to calculate v by '' Double Position," as already explained. Having found v^ the remaining elements are computed by the equations x=c[se.o-5(F)] r=— ^ I T{v)-T{V)\ COS ^ \ ' ' ) For curved fire we may proceed as follows: We have, from the origin to the summit, Now, if we assume tiiat the time from the point of pro- jection to the summit is one-half the time of flight, we shall have, from the above expressions for 7' and 4, r(7;) = 2 T{v^-T{y) which gives z^ by means of the 7^-functions, v^ being computed bv the equation derived from (82). Example i. — The 8-inch rifle (converted) fires an ogival- EXTERIOR BALLISTICS. II9 headed shot weig-hing 183 lbs. If the angle of projection is 10°, and the initial velocity 1404 f. s., find the range, time of flight, angle of fall, and terminal velocity. We have F^ 1404 f. s. ; ^=zio°; ze; = 183 lbs.; d=Z inches, whence log C — 0.45627 : to find X, T, od, and v. From Table I. we find 5(F) = 4878.6 -0.8 X 25.1 =4858.5 A {V) = 163.96 — 0.8 X 2.16 = 162.23 / (F) = 0.08661 — 0.8 X 0.00082 = 0.08599 T{V) = 2.514—0.8 X 0.018=2.500. Next compute v: log sin 2^ = 9.53405 log (7=0.45627 log o. II 96 1 = 9.07778 /(F) = 0.08599 0.20560 The value of v satisfying this equation is found to be V = 873.8 ft., whence 5 {v) = 9641.8 A (z/^) = 1 145.65 / (v) = 0.36668 T {v'^) = 7.030 X, T, (0, and z' are now computed as follows : log C = 0.45627 log[S(2.)- 5(F)] = 3^7973 log X= 4.13600' X= 13677 ft. ==4559 yc^s. \og[T{vy- r(F)] = 0.65610 log sec

-r -F777V = 0.26807 11633.6 — ^ (F) ^ For the first trial assume V ^z 1500, and, substituting in the first member of the above equation, it reduces it to 0.26691, which is too small by 0.00116 = ^j. Next make F= 1480, and we shall find that the first member now be- comes too great by 0.00140 =: e^; then 256 : 20 : : 116 : 9.1 The correct value of Fis therefore 1500 — 9,1 =: 1490.9 f. s., from which are easily found ^ = 9° 51^ X= 12440 ft. ; T=i2".72. Example 2. — " In attacking a place with curved fire it was required to drop shell into the place with an angle of 122 EXTERIOR BALLISTICS. descent of 12°, and terminal velocity of 600 f. s., using the 8-inch howitzer and a projectile of 180 lbs.; find the requi- site position of the battery, and the requisite elevation and charg-e of powder."'^ Here <3f=8 inches; zv =: iSo lbs.; 7^=600 f. s., and co^ 12°; to find X, V, and (p. We have log sin 2C0 = 9.60931 log (7 = 0.44909 log 0.14462 = 9.16022 / (7;) = 1. 15929 I{v^ = 1. 01467 v^ = 630.85 f. s. whence we find T{V) = 2X 14-396— 15779 = 13-012 F:= 665.1 f. S. 5 (v) = 15926.6 5(F) 3^ 141 78.9 log 1747.7 = 3-24247 log A"=: 3.69156 X=49i5 ft. = 1638 yds. I{z'o)= 1. 01 467 /(F) = 0.87708 log 0.13759 = 9.13859 log sin 2(p = 9.58768 2

to obtain the correct striking velocities. In our example the results are as follows: yds a, T 500 o°44' o°47' 1303 I^IO 1000 i°33' i°43' I2I3 2^30 1500 2° 2/ 2° 50^ II35 3''-59 2000 3° 2/ 4° 08' 1070 4^96 2500 4° 32' 5° 38' I02I 6^40 3000 5° 43' 7° 14' 982 7^92 3500 6° 59' 9° 01^ 947 9^52 4000 8° 21^ 10° 58^ 916 11^19 4500 9° 49' 13° 06' 888 12^94 5000 11° 24' 15° 25' 862 14^78 By interpolation, using first and second differences, the interval between successive values of the argument {X) may be reduced from 500 yards to 100 yards. Example 3. — Given d — 20 93 cm. ; if> = 140 kg. ; V = 521 m. s. ; d^ = 1.206; d= 1.233 ; X=4097 m.; angle o( jump = 8'; required the angle of elevation == ^ — 8', the angle of fall, the striking velocity, and the time of flight.^ Making the ballistic coefficient {c) =0.907, we have for * '■ Ballistische Formeln-von Mayevski nach Siacci. Fur Elevationen unter 15 Grad," Essen, Fried. Krupp'sche Buchdruckerei, 1883, page 22. Also quoted by Siacci in " Rivista di Artiglieria e Genio," vol. ii. page 414, who solves the example, using Mayevski's table. EXTERIOR BALLISTrcS. 125 computing C in English units, when <^ is expressed in centi- metres and w in kil(3grarames, the following expression : C-[i..953743]f ^ The following are the results obtained by experiment, by Mayevski's calculations, by Siacci's calculations, and by Table I. of this work : T Angle of Elevation. Angle of Fall. Striking Velocity, f. s. By experiment Mayevski... Siacci Table I 9"-7 9".6 9".675 9".66 5° 30' 5° 32' 5° 31' 5° 29' 30" 7° 16' I 176 I 169 Example 4. — Given ^=24 cm.; 7e/ = 2i5 kg.; F= 529 m. s. = 1735.6 f. s. ; required the angle of departure for each of the horizontal ranges contained in the first column of the followintr table : Horizontal Range. in 5/ J Computed by Table I. Observed value of Values of computed by Mayevski's Table. Hojel's Table. 2026 0.9569 2°.;' 2° .9' 2° 18' 2° 14' 3000 0.9407 3° 36' 3° 41' 3° 37^ 3° 35' 4000 0.9756 5° 5' 5° 10' 5° 6' 5° 5' 5964 0.9560 8" 41' 8° 35' 8° 44^ 8° 44' 7600 0.9461 12° 31' 12° 5' 12° 31' 12° 32' The data in the first, second, and fourth columns are taken from Krupp's Bulletin, No. 56 (February, 1885), page 4. The values of

s{v)-s(y)S sm 2(« X=C[5(^)-5(F)] 7-=-^ I T{v)- T{V)\ cos ] log ^ A (v) = [9.6655206] -^ -f [4.1438598] log V - 45916.40 /(^) = [5.7369333] ^ - 0.356474 T{v) = [4.2296173] ^ - 12.4999 Example i.— Let (^=8 in., w = 180 lbs., F= 700 f. s., and ^ = 60°. Find V when ^ = — 60°. We have from (33) and U ^=. joo cos 60° = 350, which is below the limit of 2 BALLISTIC TABLES. the table. The operation may be concisely arranged as follows : const. log=:: 57369333 2 log f/= 5.0881360 0.6487973 = log 445448 (60) = 2.39053 log 4 (60°) = 0.9805542 log C= 0.4490925 0.5314617 = log 3.39987 0.895 1103 = log 7-85435 2)4.8418230 2 42091 15 =: log 263.6 . • . 7/ =: 263.6 X 2 = 527.2 f. S. Example 2. — Given 5 {v) = 25496.8, to find v. We proceed as follows: 1 24466.4 25496.8 log 98969.6 = 4.9954886 const, log = 4.5918330 log (log z/) = 0.4036556 .-. log 7;=2.533i2 £^=341.3 Table II. is the ballistic table for spherical projectiles, and extends from z^= 2000 to ^^ = 450. It is based upon the Russian experiments discussed in Chapter II., and is be- lieved to be the only ballistic table for spherical projectiles yet published. Table III. is abridged from Didion's " Traite de Bal- istique." Forniulse for Interpolation. — To find the value of f{z^ when V lies between v^ and v^, two consecutive values of V, in Tables I. and II. Let v^ — v^r=^ h. Then, if d^ and d^ BALLISTIC TABLES. are the first and second diflferences of the function, we shall have, since y(?7) increases while v decreases, 2 by means of which f{v) can be computed. Conversely, if f{7>) is given, and our object is to find v, we have 7\ — v\ d^ 2 In using this last formula, first compute —^ — by omit- Ti ting the second term of the second member (which is usually very small), and then supply this term, using the approxi- mate value of-^-^^ — already found. Ii ^ If the second differences are too small to be taken into account, the above formulae become /(z,)=/(t;,) + ^S-^rf, and which expresses the ordinary rules of proportional parts. Example i. — Find from Table I. S{v) when z/= 1432.6. We have v, = 1435, f{v^ = 4704.8, h — 5, and d, = 24.6. .•.S{v) = 4704.8 + 1 435 - 1432.6 ^ ^^^^ ^ ^^j^^^ Example 2. — Given A (7/) = 229.89, to find v. Here 7/^ = 1274, /(7/,) = 229.29, ^^ = 5» ^, = 158.2, and ^,= 7.8. 2.2 2.2 / 2.2X7.8 .-. ^ (^) = 3755.9+ -X 158.2 ---(i--)^ = 3755-9 + 69-60 — 0.96 = 3824.5 Example 4.— Find from Table HI. the value of (^) when ^ = 54° 32'. Here ?^, == 54° 2o\ (^,) = 17619 1» h=z2o',d,z^ .02971, d^ = .00074. .-. (^)z= 1.76191 +0.6X 0.02971 —0.6 X 0.4 X 0.00037 = 1.76191 +0.01783 —0.00009= 1.77965 TABLE I. Ballistic Tabic for Ogival-Hcaded Projectiles. V 6- (7') Diflf. A iv) Diff. 1 Diff. T{v) Diff. 2800 2750 2700 j 000.0 126.8 [ 256.0 1268 1292 1315 0.00 0.07 0.28 7 21 36 0.00000 0.00106 0.00218 106 112 118 0.000 0.046 0.093 46 47 49 2650 2600 2550 387.5 521.6 658.3 1341 1367 1393 0.64 1. 18 1.89 54 71 93 0.00336 0.00461 0.00594 125 140 0.142 0.193 0.246 51 53 56 2500 2450 2400 797.6 939.8 1085.0 1422 1452 1481 2.82 3.97 5.37 115 140 166 0.00734 0.00883 0.01043 149 160 169 0.302 0.359 0.419 57 60 62 2350 2300 2250 I233.I IJ84.5 1539.2 '514 1547 1582 7.03 9.00 11.31 197 231 266 O.OI2I2 0.01392 0.01584 180 192 205 0.481 0.546 0.614 65 . 68 72 2200 2190 2180 1697.4 1729.5 I76I.7 321 322 323 13.97 14.55 15.15 58 60 62 0.01789 0.01832 0.01876 43 44 44 0.686 0.700 0.715 14 ^5 15 2170 2160 2150 1794.0 1826.5 1859.2 325 327 328 15.77 16.40 17.05 65 67 0.01920 0.01964 0.02010 44 46 46 0.730 0.745 0.760 15 15 15 2140 2130 2120 1892.0 1924.9 1958.0 329 331 17.72 18.40 19.10 70 73 0.02056 0.02102 0.02149 46 47 48 0.775 0.791 0.806 16 15 16 2IIO 2100 2090 I99I.3 2024.8 2058.4 335 336 337 19.83 20.57 21.33 74 76 79 0.02197 0.02246 0.02295 49 49 50 0.822 0.838 0.854 16 16 16 2080 2070 2060 2092.1 2126.0 2I60.I 339 341 343 > 22.12 22.92 23.74 80 82 85 0.02345 0.02396 0.02447 51 51 52 0.870 0.886 0.903 16 17 17 2050 2040 2030 2194.4 2228.8 2263.4 344 346 348 24.59 25.46 26.35 [ 87 89 91 0.02499 0.02552 0.02606 53 54 54 0.920 0.937 0.954 17 17 17 2020 2010 2000 2298.2 2333.1 2368.2 349 351 353 27.26 28.20 29.16 94 96 98 0.02660 0.02715 0.02772 55 57 57 1 0.971 0.988 1.005 17 17 18 TABLE L— Continued. V S{v) Diff. A {V) Diff. 7(7') Diff. T{v) Diff. 1990 1980 1970 2403-5 2439.0 2474.6 355 ^ 356 358 30.14 31-15 32.19 lOI 104 107 0.02829 0.02886 0.02945 57 59 60 1.023 1. 041 1-059 18 18 18 i960 1950 1940 2510.4 2546.4 2582.6 360 362 363 33-26 34-35 35-48 109 113 115 0.03005 0.03066 0.03127 61 61 62 1.077 1.096 1. 114 19 18 19 1930 1920 I9I0 2618.9 2655.5 2692.2 306 367 370 36.63 37-81 39.02 118 121 124 0.03189 0.03253 0.03318 64 65 65 I-I33 1. 152 1. 171 19 19 20 1900 1890 1880 2729.2 2766.3 2803.7 371 374 375 40.26 41-53 42.83 127 130 0.03383 0.03450 0.03517 67 69 1. 191 1. 210 1.230 19 20 20 1870 i860 1850 2841.2 2878.9 2916.9 377 380 382 44.16 1 45-53 46.93 137 140 143 0.03586 0.03656 ; 0.03727 70 71 72 1.250 1.270 1. 291 20 21 20 1840 1830 1820 2955-1 2993-4 3032.0 386 388 48.36 49-83 51-34 147 151 155 0.03799 0.03872 0.03946 73 1 74 76 1. 311 1-332 1-353 21 21 22 I8I0 1800 1790 3070.8 3109.8 3149.0 390 392 394 52.89 54-47 56.09 158 162 167 ' 0.04022 0.04099 10.04177 77 78 80 1-375 1.396 1.418 21 22 22 1780 1770 1760 3188.4 3228.0 3267.9 396 399 401 1 57-76 1 59-47 61.21 171 174 179 i 0.04257 0.04338 0.044.20 81 1 821 84! 1.440 1.463 1-485 23 22 23 1750 1740 1730 3308.0 3348.3 3388.9 403 406 409 63.00 64-83 66.71 183 188 193 0.04504 0.04589 0.04676 85 1 87 88! 1.508 1-531 1-555 23 24 23 1720 I7I0 1700 3429.8 3470-8 3512. 1 410 413 415 ! 68.64 : 70.61 72.63 1 197 202 207 0.04764 0.04854 0.04945 90 9r\ 1-578 1.602 1.626 24 24 25 1690 1680 1670 3553-6 3595-4 36374 418 420 423 1 74-70 76.83 79.01 213 218 223 0.05038 0.05133 0.05229 95 96 1 98 1 1. 651 1.676 1. 701 25 25 25 1660 1650 1640 3679-7 3722.2 3765-0 425 428 430 81.24 83-52 85.86 228 234 241 6 0.05327 0.05427 ,0.05529 100 102 103 1 1.726 1-752 1.778 26 26 26 TABLE I.— Continued. 3808.0 3851-3 3894.9 3938.7 3960.7 3982.8 4005.0 4027.3 4049.6 4072.0 4094.4 4116.9 4139-5 4162.2 4185.0 4207.8 4230.7 4253-6 4276.7 4299.8 4323-0 4346.2 4369.6 4393-0 4416.5 4440.1 4463-8 4487-5 4511-3 I 4535-2 4559-2 4583.2 4607.4 4631.6 4655-9 4680.3 Diff. I I 433 I 436 I 4381 220 1 221 j 222 I 223 223 224 224 225 226 227 228 228 229 229 231 231 232 232 234 234 235 236 237 237 238 239 240 240 242 242 243 244 245 A {7') 88.27 90-73 93-25 95-84 97.16 98.49 99.84 IOI.2I 102.60 104.00 105.42 106.86 108.32 109.79 111.29 112.80 114-33 115.88 117-45 119.04 120.65 123-93 125.60 127.29 129.01 130.75 132.50 134.28 136.09 137.92 139-77 141.65 T43-54 T45-47 147.42 Diff. 246 252 259 132 133 135 I 137 139 140 142 144 146 147 150 151 153 155 157 159 i6i 163 165 167 169 172 174 175 178 181 183 185 188 193 195 197 7 /{v) 0.05632 0.05738 0.05845 0.05955 0.06010 0.06066 0.06123 0.06180 0.06238 0.06296 0.06355 0.06414 0.06474 0.06534 0.06595 0.06657 0.06719 0,06782 0.06846 0.06910 0.06975 0.07040 0.07106 0.07173 0.07241 0.07309 0.07378 0.07447 0.07517 0.07588 0.07660 0.07732 0.07805 0.07879 0.07954 0.08029 Diff. 106 107 55 56 57 57 58 58 59 59 60 60 61 62 62 63 64 64 65 65 66 67 68 68 69 69 70 71 72 72 73 74 75 75 76 T{v)' Diff. 1.804 1-831 1.858 27 27 27 1.885 1.899 14 14 1.913 14 1.927 14 1. 941 14 1-955 14 1.969 1.983 1.998 14 15 14 2.012 15 2.027 15 2.042 15 2.057 15 2.072 2.086 14 15 2.101 16 2. 117 15 2.132 15 2.147 2.162 15 16 2.J78 16 2.194 16 2.210 16 2.226 16 2.242 16 2.258 16 2.274 16 2.290 17 2.307 16 2.323 17 2-340 17 2-357 17 2.374 17 TABLE I. -Continued. V S{v) Diff. A {j^ Diff. 7(7.) Diff. r{v) Diff. 1435 1430 1425 4704.8 1 4729-4 i 4754-1 246 247 247 149-39 151-39 153-42 200 203 205 0.08105 0.08182 0.08260 77 78 78 2.391 2.408 2.425 18 1420 I4I5 I4I0 1 4778.8 1 4803.6 , 4828.5 248 249 250 155-47 T57-55 159.66 208 211 214 0.08338 0.08418 0.08498 80 81 81 2.443 2.460 2.478 1 17 18 18 1405 1400 1395. ! J 4853-5 i 4878.6 49P3-8 251 252 253 1 161.80 1 163.96 j 166.15 216 219 222 0.08579 0.08661 0.08744 82 83 84 2.496 2.514 2^-532 18 18 18 1390 1385 1380 4929-1 4954-5 j 4979-9 254 254 256 168.37 170.62 172.90 225 228 231 0.08828 0.08913 0.08999 85 86 87 2.550 2.568 2-587 18 19 18 1375 1370 '365 5005.5 ■ 5031-1 5056.8 256 257 258 175.21 177-55 179.92 234 237 241 0.09086 0.09173 0.09262 87 89 89 2.605 2.624 2.643 19 19 19 1360 1355 1350 5082.6 ! 5108.6 5134.6 260 260 261 182.33 184.76 187.23 243 247 250 0.09351 c. 09442 0-09533 91 91 93 2.662 2.681 2.700 19 19 .19 1345 1340 1335 5160.7 5186.9 5213-2 262 i 263' 263 , 1 189.73 192.27 194.84 254 257 260 0.09626 0.09719 0.09813 94 94 95 2.719 2-739 2-758 20 •9 20 1330 1325 1320 5239-5 5265.8 5292.0 263! 262 j 106 , 197.44 200.06 202.69 262 263 107 0.09908 0.10004 o.idioi 96 1 97 39 2-778 2.798 2.818 20 20 8 I3I8 I3I6 I3I4 5302.6 53^3-2 5323-8 106 : 106 107 203.76 204.84 205.92 108 108 109 0.10140 0.10179 0.10219 39 1 40 1 40' 2.826 2.834 2.842 8 8 8 I3I2 I3I0 .1308 5334-5 5345-2 5355-9 107 107 108 207.01 208.11 209.22 I 10 I r I III! 1 0.10259 0.10299 0.10339 40 40 41 2.850 2.858 2.866 8 8 9 1306 1304 1302 5366.7 5377-5 108 108 109 210.33 211.45 •212.58 1 112 113 114 0.10380 0.10421 0.10462 41 41 41 2.875 2.883 2.892 8 9 8 1300 1298 1296 5399-2 5410.1 5421.0 109 109 no 213.72 214.87 216.02 1 115 115 117 0.10503 0.10544 0.10586 41 42 42 2.900 2.908 2.917 8 9 8 TABLE I.— Continued. V Six,) 1 Diff. 1 A{v) Diff. I{v) Diff. I : T{v) Diff. 1294 I 292 1290 5432.0 5443-0 5454.0 no no III 1 1 217.19 1 218.36 ! 219.54 1 117 118 119 0.10628 0.10670 0.10713 42 43 43 1 2.925 2.934 1 2.942 9 8 8 1288 1286 T284 5465.1 5476.2 5487.3 III III 112 220.73 221.93 223.13 120 120 122 0.10756 0.10799 0.10842 43 43 44 ! 2.950 1 2.959 2.968 9 9 9 1282 1280 1278 549«-5 5509-7 5521.0 112 113 113; 224.35 225.57 226.80 122 123 124 0.10886 0.10930 0.10974 44 44 45 2.977 2.985 2.994 8 9 9 1276 1274 1272 5532.3 5543-6 5554-9 113 i 113! 114 228.04 229.29 230.54 125; 125 127 ! 0.I10I9 0.11064 0.11109 45 45 45 3.003 3.012 3.021 9 9 9 1270 1268 1266 i 5566.3 5589-1 114 114 115 231.81 234.37 127 129 J29 0.11154 0.11200 0.11246 46 46 46 3-030 3-039 3.048 9 9 9 1264 J262 1260 5600.6 5612.1 5623.7 115 116 116! 1 235-66 236.97 238.28 13.1 1 131 1 132 1 0.11292 0.11338 O.I 1385 46 47 47 3-057 3.066 3-075 9 9 9 1258 1256 •1254 5635-3 5647.0 5658.6 117 116 117 239.60 240.94 242.28 134: 134 i 136! 0.11432 O.II479 0.11527 47 48 48 3.084 3-094 3-103 10 9 10 1252 1250 1248 5670.3 5682.1 5693-9 118 118 118 1 243-64 245.00 246.37 136 1 1371 139' O.II575 O.I1623 0.11671 48 48 49 3-1^3 3.122 9 9 10 1246 1244 1242 5705-7 5717-6 5729-5 119 119 119 247.76 249-15 250-55 139; 140 i 142 O.II72O O.II769 0.11819 49 50 50 3-141 3-150 3.160 9 10 9 1240 1238 1236 5741.4 5753-4 5765.4 120 120 121 251.97 253-39 254.83 142 144 j 144 0.11869 O.II919 0.11969 50 50 5^ 3.169 3-179 3-189 10 10 9 1234 1232 ; 1230 5777-5 5789.6 5801.7 121 121 122 256.27 257.73 259.20 146! 147 1 148 ■ 0.12020 O.T2071 0.12123 51 52 52 3-198 3.208 3.218 10 10 10 TABLE L— Continued. V S{v) Diff. j 1228 1226 1224 5813.9 5826.1 5838.4 i 122 123 123! 1222 1220 I218 5850.7 5863.0 5875-4 123 124 124 I216 I 2 14 I2I2 5887.8 5900.3 1 5912.8 125 i 125 1 125 I2IO 1208 1206 5925.3 5937-9 1 5950.5 126 126 127 I 204 1202 1200 5963.2 5975-9 5988.6 127I 127 128 1 II98 II96 II94 6001.4 6014.2 6027.1 128 129 129 1 II92 I 1 90 I188 6040.0 6053.0 6066.0 130 130 131 I 186 I 184 I182 6079.1 6092.2 6105.3 131 131 132 ! 1 I180 II78 II76 6118.5 6131-7 6145.0 132 II74 II72 II70 6158.3 6171.7 6185. I 134 134! 135 ! I168 I166 1 164 6198.6 6212. 1 6225.6 135 135 136' A{v) 260.68 262.17 263.67 265.18 266.71 268.24 269.79 271-35 272.92 274.51 276.11 277.72 279.34 280.97 282.62 284.28 285.95 287.63 289.33 291.04 292.76 294.50 296.25 298.02 299.80 301.59 303-40 305.22 307.06 308.91 310.77 312.65 314-55 Diff. i 149 150 151 I{v) Diff. r(z/) O.I2I75 0.12227 0.12280 52 53 53 ! 3.228 3-238 3-M8 '53 153 155 0.12333 0.12386 0.12439 53 53 54 3-258 3.268 3-278 156 \ 157 159 0.12493 0.12547 O.T2602 54 55 55 3.288 1 3-299 3-309 1 160 i 161 1 162 0.12657 O.I 27 I 2 0.12768 55 56 56 1 3-319 i 3-329 3-340 163 165 166 0.12824 0.12881 0.12938 57 57 57 3-350 3-361 3-371 167 168 170 0.12995 O.T3053 O.I3III 58 58 58 3-382 3-393 3.404 171 172 1.74 O.I3169 0.13228 0.13287 59 59 60 3.415 3.426 3.437 175 177 178 0.13347 0.13407 0.13467 60 60 61 3.448 3.459 3.470 179 181 182 0.13528 0.13589 0.13651 61 62 62 3.481 3.492 3.504 i 184 185 186 O.I3713 0.13776 0.13839 63 63 63 3.515 3-527 3-538 ! 188 190 191 0.13902 0.13966 O.T403O 64 64 65 3-550 3-561 3-573' Diff. TABLE 1.— Continued. S{v) Diff. 162 160 159 158 157 156 1531 152 151 150 149 148 147 146 145 144 143 142 141 140 139 138 137 136 135 134 133 132 131 130 129 6239.2 6252.8 6259.7 6266.6 6273.4 6280.3 6287.2 6294.1 6301.0 6307.9 6314.8 6321.8 6328.8 6335-7 6342.7 63497 6356.7 63637 6370.7 6377.8 6384.8 6391.9 6399.0 6406. T 6413.2 6420.3 6427.4 6434.6 6441.7 6448.9 6456.1 6463.3 6470.4 136 69 69 68 69 69 69 69 69 69 70 70 69 70 70 70 70 70 71 70 71 71 7T 71 72 71 72 72 72 71 72 A {7') 316.46 318.39 31936 320.34 321.32 322.30 323.28 324.27 325.26 326.26 327.26 328.27 329.28 330.29 331-31 33^-33 333-3^ 334-39 335-43 336.47 337-51 338.56 339-61 340.67 341.73 342.79 343-^6 344-94 346.02 347.10 348.19 349.28 350.38 )iff. 1(7') 1 Diff. 193 97 98 i I 0.14095 O.I4160 O.I4I92 65 33 98 98 98 0.14225 0.14258 O.I429I 33 33 33 99 99 1 0.14324 1 0.14358 34 33 ! T(v) Diff. 100 lOI lOI lOI 102 102 103 103 104 104 104 105 105 106 106 106 107 108 108 108 109 109 1 10 109 II -.--too-! 10.14391 0.14425 0.14458 \ 0.14492 i 0.14526 1 0.14560 1 0.14594 i O.T4628 ; 0.14662 0.14697 0.14731 0.14766 0.1 480 1 0.14836 0.14871 i 0.14906 0.14942 0.14977 0.15013 0.15049 0.15085 0.15121 ^0.15157 io.15193 ' 0.15229 34 33 34 34 34 34 34 34 35 34 35 35 35 35 35 36 35 36 36 36 36 36 36 36 3-584 3-596 3.602 3.608 3.614 3.62c 3.626 3-632 3-^3^ 3-644 3-650 3656 3.662 3.668 3-674 3-680 3.686 3-693 3-699 3-705 3. 711 3-717 3-723 3-730 3-736 3-742 3-748 3-755 3-761 3-767 3-774 3-780 3.786 TABLE I.— Continued. V S{v) Diff. Ah) Diff. I{z) Diff. T (7') Diff. II28 1.27 II26 6477-6 6484.8 6492.1 72 73 72 351 47 352-57 353-68 no III 111 0.15265 0.15302 0.15338 37 36 37 3-793 3-799 3.806 6 7 6 II25 II24 6499.3 6506.6 73 73 354-79 355-90 II I j 113 0.15375 O.I54I2 ^7 37 3.812 3-818 6 7 II23 65139 73 357-03 113 1 0.15449 38 1 3-825 6 II22 i II2I 1 II20 6521.2 6528.6 6536.0 74 74 74 358.16 35930 36045 114 115 115 0.15487 0.15524 15562 37 1 38; 38 1 3.831 3-838 3-844 7 6 7 III9 II18 I I 17 6543-4 6550-8 6558-3 74 75 75 361.60 362.76 36392 116 116 117 0.15600 0.15638 0.15676 38 38 39 3-851 3.858 3.864 7 6 7 II16 11 14 6565.8 ^573-3 6580.8 75 75 76 365-09 366.28 367-47 119 119 120 0.15715 0.15754 0.15793 39 39 39 1 3.871 3.878 3-885 7 7 7 III3 II12 6588.4 6596.0 76 77 368.67 369.88 121 121 0.15832 0.15872 40 40 3.892 3.898 6 7 IIII 6603.7 77 37109 123 0.15912 40 3905 • 7 TIIO 6611.4 77 372.32 123 0.15952 41 3912 7 1 109 1108 6619. 1 6626.9 78 78 373-55 374-79 124 125 O.T5993 0.16033 40 41 3.919 3926 1 7 7 ITO7 I 106 1 105 6634.7 6642.5 6650.3 78 78 79 376.04 377-30 i 37857 126 127 128 0.16074 0.16115 0.16157 41 42 41 3-933 3940 3-947 7 7 8 1 104 I 103 II02 6658.2 6666.2 6674.1 80 79 80 379-85 381.14 382.44 129 130 131 0.16198 0.16240 0.16282 42 42 43 3-955 3-962 3-969 7 7 7 IIOI I 100 1099 6682.1 6690.2 6698.3 81 81 81 j 383-75 38506 T31 132 0.16325 0.16367 0.16410 42 43 43 3-976 3-983 3-991 7 8 7 1098 1097 1096 6706.4 6714-5 6722.7 81 82 83 387-71 389.06 ' 39041 135 135 137 0.16453 0.16497 0.16541 44 44 44 3-998 4.006 4013 8 7 8 TABLE I.— Continued. V S{v) Diff. A^zi) Difif. nv) Diff. T{v) Diff. 1094 1 1093 i 6731.0 6739.2 6747-5 82 83 84 391-78 393-15 394-53 137 138 140 0.16585 0.16629 0.16674 44 45 45 4.021 4.029 4-036 8 7 8 1092 I09I 1090 1 6755-9 6764.3 6772.7 84 84 85 395-93 397.34 398.75 141 141 142 0.16719 0.16764 0.16810 45 46 46 4-044 4.051 4.059 7 8 8 1089 1088 1087 6781.2 6789.7 6798.2 85 85 86 400.17 401.60 403-05 143 145 145 0.16856 0.16902 0.16948 46. 46 47 4.067 4.075 4.083 8 8 8 1086 1085 j 1084 6806.8 6815.4 6824.1 86 1 87 87 1 404-50 405.97 407-45 147 148 149 0.16995 0.17042 0.17089 47 47 48 ' 4.091 4.098 4.106 7 8 8 1083 T082 I08I 6832.8 6841.5 6850.3 87 88 88 408.94 410.44 411.95 150 151 152 0.17137 0.17185 0.17233 48 48 49 4.114 4.122 4.130 8 8 8 1080 1079 1078 6859.1 6867.9 6876.8 88 89 90 413-47 415.00 416.54 153 154 156 0.17282 0.17331 0.17380 49 49 49 4.138 4.146 4.155 8 9 8 1077 1076 1075 6885.8 6894.7 6903.7 89 90 91 418.10 419.66 421.24 156 158 159 0.17429 0.17479 0.17529 50 50 51 4.163 4.172 4.180 9 8 9 1074 ro73 1072 6912.8 6921.9 6931. 1 91 92 92 422.83 424.44 426.06 161 162 163 0.17580 0.17631 0.17682 51 51 51 4.189 4.197 4.206 8 9 8 1071 1070 1069 6940.3 6949.5 6958.8 92 93 93 427.69 429.33 430.98 164 165 166 0.17733 0.17785 0.17837 52 52 53 4.214 4.223 4.232 9 9 9 1068 1067 1066 6968.1 6977-5 6986.9 94 94 94 432.64 434.32 436.01 168 169 171 0.17890 0.17943 0.17996 53 53 53 4.241 4.250 i 4.259 9 9 9 1065 1064 1063 6996.3 7005.8 7015-4 95 96 96 437.72 439-44 441.17 172 173 175 0.18049 0.18103 0.18158 54 55 55 4.268 4.277 . 4.286 9 9 9 ^3 TABLE I.— Continued. V S{v) Diff. A (v) Diff. I{v) Diff. 1 T{v) 1 Diff. 1062 1061 1060 7025.0 7034.6 7044-3 96 97 97 442.92 444-68 446.45 176 177 178 0.18213 0.18268 0.18323 55 55 56 1 1 4-295 i 4-304 4-313 1 1 9 9 i 9 1059 1058 1057 7054.0 7063.8 7073.6 98 98 99 448.23 450-03 451.84 180 181 182 0.18379 0-18435 0.1 849 1 56 56 57 '] 4-322 4-332 4.341 10 9 9 1056 1055 1054 1 7083.5 7093-4 7103.4 99 100 100 453.66 455.50 457-36 184 186 187 0.18548 0.18605 0.18663 57 58 58 4-350 4-360 4-369 10 9 9 1053 1052 105 1 7113-4 7123.4 7133-5 100 lOI I02 459.23 461.12 463.02 189 190 192 O.18721 0.18779 0.18838 58 59 59 4.378 4.387 4.397 9 10 9 1050 I049 1048 7143.7 7153-9 7164.1 102 I02 103 464-94 466.87 468.81 193 194 196 0.18897 0.18956 O.I 90 1 6 59 60 61 4.406 4.416 4.426 10 10 10 1047 1046 1045 7174-4 7184.7 7195-I I03 104 105 470-77 472-74 474-73 197 199 201 0.19077 0.19138 0.19199 61 61 61 4.436 4.446 4-455 10 9 10 1044 1043 1042 7205.6 7216.1 7226.6 105 105 106 476.74 478.77 480.81 203 204 206 0.19260 0.19322 0.19385 62 63 4-465 4-475 4-485 10 10 10 1041 1040 1039 7237.2 7247.9 7258.6 107 107 107 482.87 484.95 487.04 208 209 211 0.19448 0.19511 0-19575 ^3 64 64 4-495 4-505 4-516 10 II 10 1038 1037 1036 7269.3 7280.1 7291.0 108 109 109 489-15 491.28 493.42 213 214 216 0-19639 0.19703 0.19768 64 65 66 4-526 4.537 4.547 II 10 II 1035 1034 7301.9 7312.9 73239 no no III 495.58 497.76 499-95 218 219 222 0.19834 0.19900 0.19966 66 66 67 4.558 4-569 4-579 II 10 II 1032 1031 1030 7335.0 7346.1 7357.3 III 112 112 502.17 504.40 506.65 ' 223 225 226 1 0.20033 0.20100 0.20168 67 68 68 4-590 4.600 4.611 10 II II 14 TABLE I.— Continued. V S{v) Diff. A{v) Diff. I{v) Diff. T{.v) Diff. 1029 1028 1027 7368.5 7379-8 739I-I 113 113 114 i 508.91 511.20 513-50 229 230 232 1 0.20236 0.20305 0.20374 69 69 69 4.622 4-633 4.645 II 12 II 1026 1025 1024 7402.5 7414.0 7425-5 115 115 116 515-82 518.17 520.54 235 237 238 0.20443 0.20513 0.20584 70 71 71 4.656 4.667 4.678 II II II 1023 1022 I02I 7437-1 7448.7 7460.4 116 117 117 522.92 525-32 527-75 240, 243 245 0.20655 0.20726 0.20798 71 72 73 4.689 4.701 4.712 12 II II 1020 IOI9 IO18 7472.1 7483.9 7495.7 118 118 ii9| 530.20 532.66 535-14 ! 246 248 251 0.20871 0.20944 O.21017 73 73 74 4.723 4.735 4-747 12 12 12 ICI7 IO16 IOI5 7507.6 7519-6 7531-6 120 120 121 537-65 540.17 542.72 252 255 258 O.21091 0.2 1 165 0.21240 1 74 75 76 4.759 4.771 4-782 12 II 12 IOI4 IOI3 IOI2 7543-7 7555-8 7568.0 121 122 123 545-30 547-89 550.51 259 262 265 O.21316 0.21392 0.21468 76 76 77 4.794 4.806 4.818 12 12 12 lOII lOIO 1009 7580.3 7592.6 7605.0 123 124 124 553-i6 555-82 558-51 266 269 272 0.21545 0.21623 O.21701 78 78 79 4.830 4.842 4.855 12 13 12 1008 1007 1006 7617.4 7629.9 7642.5 125 126 1.6 561.23 563.96 566.71 273 275 278 0.21780 0.21859 0.21939 79 80 80 4.867 4.880 4.892 13 12 13 1005 1004 1003 7655-1 7667.8 7680.6 127 128 128 569-49 572.29 575.11 280 282 285 0.22019 0.22100 0.22182 81 82 82 4.905 4.918 4-930 13 12 13 1002 lOOI 1000 7693-4 7706.3 7719-3 129 130 131 577-96 580.83 583-72 287 289 292 0.22264 0.22347 0.22430 83 83 84 4.943 4.955 4.968 12 13 13 999 998 997 7732.4 7745-6 7758.8 132 132 133 586.64 589-59 592.56 295 297 300 0.22514 0.22599 0.22684 85 85 86 4.981 4.995 5.008 14 13 14 15 TABLE I.— Continued. V S\v) Diff. A{t^ Diff. /(zO Diff. T{v) Diff. 996 995 994 7772.1 7785.4 7798.7 134 595.56 598.59 601.65 303 306 Z^9 0.22770 0.22857 0.22944 87 87 87 5.022 5.035 5.048 T3 13 14 993 992 991 7812. 1 7825.5 7839.0 134 135 135 604.74 607.85 610.99 311 314 317 0.23031 0.23118 0.23206 87 88 89 5.062 5-075 5.089 13 14 13 990 989 988 7852.5 7866.1 7879.7 136 136 137 614.16 617.33 620.52 317 319 321 0.23295 0.23384 0.23474 89 90 90 5.102 5. 116 5-130 14 14 14 987 986 985 7893.4 7907.1 7920.8 137 ^37 137 623.73 626.96 630.21 323 325 327 0.23564 0.23655 0.23746 91 91 91 5-144 5.158 5-171 14 13 14 984 983 982 7934-5 7948.3 7962.1 138 138 138 633-48 636.77 640.08 329 0.23837 0.23929 0.24021 92 92 92 5.185 5.199 5.213 14 14 14 981 980 979 7975-9 7989.8 8003.7 139 139 139 643-41 646.76 650.12 335 339 0.24T13 0.24206 0.24299 93 93 93 5.227 5-241 5-255 14 14 15 978 977 976 8017.6 8031.5 8045.5 139 140 140 653-51 656.92 660.35 341 343 345 0.24392 0.24486 0.24580 94 94 95 5.270 5.284 5-299 14 15 14 975 974 973 8059.5 8073-5 8087.6 140 141 141 663.80 667.26 670.75 346 349 351 0.24675 0.24770 0.24865 95 95 96 5-3^3 5-327 5.342 14 15 14 972 971 970 8101.7 8115.8 8129.9 141 141 142 674.26 677.80 681.35 354 355 357 0.24961 0.25057 0.25154 96 97 97 5.356 5.371 5-385 15 14 15 969 968 967 8144. 1 8158.3 8172.5 142 142 143 684.92 688.51 692.12 359 361 363 1 0.25251 0.25348 0.25446 97 98 98 5.400 5.415 5-429 15 14 15 966 965 964 8i86.8 8201. 1 8215.4 143 143 144 695.75 699.41 703-09 366 368 370 0.25544 0.25643 0.25742 99 99 99 5-444 5-459 5-474 15 15 , 15 16 TABLE 1.— Continued. V i Diff. A(v) Diff. /{v) Diff. T(v) Diff. 963 962 961 : 8229.8 8244.2 8258.6 144 144 144 706.79 710.51 714.26 372 375 377 0.25841 0.25941 0.26041 100 100 lOI 5-489 5-503 j 5-518 1 14 15 15 960 959 958 1 8273.0 \ 8287.4 i 8301.9 144 145 145 718.03 721.81 1 725.62 378 381 384 0.26142 0.26243 0.26344 lOI 101 102 5-533 ' 5-548 1 5-564 15 16 15 957 956 955 8316.4 ! 8331.0 ! 8345.6 146 146 146 729.46 1 733.32 737.20 386 3SS 390 0.26446 0.26549 0.26652 103 103 103 5-579 5-594 5.609 15 15 16 954 953 952 i 8360.2 8374.8 i 8389.5 146 147 J47 741.10 745-03 748.98 393 395 398 0.26755 0.26858 0.26962 103 T04 105 5-625 5-640 5-655 15 15 16 951 950 949 8404.2 i 8419.0 8433.8 148 T48 148 752.96 756.96 760.98 400 402 404 0,27067 0.27172 0.27277 105 105 106 5-671 5.686 5-702 T5 16 948 947 946 8448.6 1 8463.4 , 8478.3 148 149 149 765.02 769.09 773.18 407 409 412 0.27383 0.27489 0.27596 106 107 107 5-718 ■5-733 5-749 15 16 16 945 944 943 8493.2 8508.1 ! 8523.1 149 150 150 777-30 781.45 785.62 415 417 420 0.27703 0.278II 0.27919 108 108 108 5-765 5-781 5-797 16 16 15 942 941 940 1 8538.1 \ 8553.1 \ 8568.2 1 150 151 151 789.82 794.04 798.29 422 425 427 0.28027 0.28136 0.28246 109 no no 5-812 5-828 5-844 16 16 16 939 938 937 8583.3 8598.4 8613.6 151 152 152 802.56 806.85 811. 17 429 432 1 435 0.28356 0.28467 0.28578 III III III 5.860 5-877 5-893 17 16 16 936 935 934 8628.8 8644.0 8659.2 152 152 153 815-52 819.89 824.30 437 441 443 0.28689 0.28801 0.28913 112 112 113 5-909 5-926 5-942 17 16 16 933 932 931 8674.5 8689.8 8705.2 153 ^54 154 . 828.73 837-67 445 449 451 0.29026 0.29140 0.29254 114 114 114 5-958 5-974 •5-991 16 17 16 T7 TABLE I.— Continued. V S{v) Diff. A{v) Diff. /(Z') Diff. Tiv) Diff. 930 929 928 8720.6 8736.0 8751.5 154 155 155 842.18 846.71 851.27 453 456 459 0.29368 0.29483 0.29598 ^15 115 116 6.007 6.024 6.041 17 17 16 927 926 925 8767.0 8782.5 8798.0 155 155 156 855.86 860.48 865.13 462 1 465 468 0.29714 0.29830 0.29947 116 117 117 6.057 6.074 6.091 17 17 17 924 923 922 8813.6 8829.2 8844.9 156 157 157 869.81 874-51 879-25 470 474 477 0.30064 0.30T82 30300 118 118 119 6.108 6.125 6. 141 17 16 17 921 920 919 8860.6 8876.3 8892.0 157 157 158 884.02 888.81 479 1 4821 485 j 0.30419 0.30538 0.30658 119 120 120 6.158 6.175 6.192 17 17 18 918 917 916 8907.8 8923.7 8939-5 159 158 159 898.48 903.36 908.27 488 491 494 0.30778 0.30899 0.31020 121 121 122 6.210 6.227 6.245 17 18 17 915 914 913 8955-4 8971-3 8987-3 159 160 160 913.21 918.18 923.19 497 501 503 0.3II42 0.31264 0.31387 122 123 124 6.262 6.279 6.297 17 18 17 912 911 910 9003-3 9019.3 9035-4 160 161 161 928.22 933-28 938.37 506 509 513 0.3I5II 0.31635 0.31760 124 125 125 6.314 6.332 6.349 18 17 18 909 908 907 9051-5 9067.6 9083.8 161 162 162 943-50 948.65 953-84 515 519 522 0.31885 0.320II 0.32137 126 126 127 6.367 6.385 6.403 18 18 18 906 905 904 9100.0 9116.2 9132.5 162 163 163 959.06 964.31 969.60 525 529 532 0.32264 0.32392 0.32520 128 128 129 6.421 6.439 6.457 18 18 18 903 902 901 9148.8 9165.2 9181.6 .64 164 164 974-92 980.27 985-65 535 538 541 0.32649 0.32778 0.32908 129 130 130 6.475 6.493 6. 511 18 18 18 900 899 898 9198.0 9214-5 9231.0 165 165 165 991.06 996.51 1001.99 545 548 552 0.33038 0.33169 0.33300 131 131 132 6.529 6.548 6.566 19 18 19 iS TABLE I.— Continued. S(v) 9247-5 9264.1 9280.7 9297.3 9314-0 9330-7 9347.5 9364.3 9381. 1 9398.0 9414.9 9431-9 9448.9 9465.9 9483.0 9500.1 9517.2 9534.4 9551.6 9568.9 9586.2 9603.5 9620.9 9638.3 9655.8 9673.3 9690.8 9708.4 9726.0 9743.7 9761.4 9779.T 9796.9 Diff. 166 166 166 167 167 168 168 168 169 169 170 170 170 171 171 171 172 172 173 173 173 174 174 175 175 175 176 176 177 177 177 178 178 A{v) 007.51 013.06 018.65 024.27 029.92 035-61 041.34 047.10 052.90 058.73 064.60 070.52 076.47 082.45 088.47 094-53 100.62 106.75 112.92 119.13 125-38 131.67 138.00 144-37 150.78 157-23 163.72 170.25 176.82 183.44 190.09 196.79 203.54 Diff. 555 559 562 565 569 573 576 580 583 587 592 595 598 602 606 609 613 617 621 625 629 ^33 637 641 645 649 653 657 662 665 670 675 678 I{v) 0.33432 0.33565 0.33698 0.33832 0.33966 0.34101 0.34237 0.34373 0.34510 0.34647 0.34785 0.34924 0.35063 0.35203 0.35344 0.35485 0.35627 0-35770 0.35913 0.36057 0.36202 0.36347 0.36493 0.36639 0.36786 0.36934 0.37083 0.37232 0.37382 0.37532 0.37683 0.37835 0.37988 Diff. 133 ^33 134 134 135 136 136 137 137 138 139 139 140 141 141 142 143 143 144 145 145 146 146 147 148 149 149 150 150 151 152 ^53 153 T{v) 6.585 6.603 6.622 6.640 6.659 6.677 6.696 6.714 6.733 6.753 6.772 6.791 6.811 6.830 6.849 6.868 6.888 6.907 6.927 6.947 6.966 6.986 7.006 7.026 7.046 7-065 7.085 7.105 7.126 7.146 7.167 7.187 7.208 Diff. 19 TABLE I.— Continued. 864 863 862 861 860 859 858 857 856 855 854 853 852 851 850 849 848 847 846 845 844 843 842 841 840 839 838 837 836 835 834 832 Siv) 9814.7 9832.6 9850-5 •4 9886.4 9904.4 9922.5 9940.6 9958.7 9976.9 9995-2 10013.5 10031.8 10050.2 10068.6 10087. 1 10105.6 10124.1 10142.7 10161.3 10180.0 10198.8 10217.5 T0236.3 10255.2 10274.1 10293.0 10312.0 10331.0 10350.1 10369.2 10388.4 10407.6 Diff. 179 179 1791 I 180 180 181 181 181 182 183 183 183 184 184 185 185 185 186 186 187 188 187 188 189 189 189 190 190 191 191 192 192 193 A{v) 210.32 217.15 224.02 230.93 237.89 244.89 251-94 259.04 266.18 273.36 280.59 287.87 295-19 302.56 309-98 317-44 324.96 332.52 340.13 347.79 355-50 363-26 371.07 378.93 386.84 394.80 402.82 410.89 419.01 427.18 435-41 443.69 452.02 Diff. 683 687 691 696 700 705 710 714 718 723 728 732 737 742 746 752 756 761 766 771 776 781 786 791 796 802 807 812 817 823 828 ^33 839 I{v) I0.38I4T : 0.38295 ; 0.38450 0.38606 0.38762 0.38919 0.39077 0.39235 0.39394 0.39554 0.39715 0.39877 0.40039 0.40202 0.40366 0.40530 0.40695 0.40861 0.41028 0.41196 0.41364 0.41533 0.41703 0.41874 0.42046 0.42218 0.42392 0.42566 0.42741 0.42917 0.43093 0.43271 0.43449 Diff. 154 155 156 156 157 158 158 159 160 161 162 162 163 164 164 165 166 167 168 168 169 170 171 172 172 174 174 175 176 176 178 178 180 T(v) 7.229 7.249 7.270 7.290 7-311 7-332 7-354 7-375 7-396 7.418 7-439 7.460 7.481 7.503 7-524 7-546 7-568 7-590 7.612 7-635 7-657 7-679 7.701 7-723 7-745 7-768 7-790 7-813 7.836 7-858 7.881 7-904 7.928 TABLE I.— Continued. S{v) Diff. 0426.9 0446.2 0465.6 [0485.0 [0504.4 0523-9 0543-4 0563.0 0582.7 [ 0602. 4 0622,1 [0641.9 0661.7 0681.6 0701.6 0721.6 0741.6 0761.7 :o78i.8 :o8o2.o :o822.2 :o842.5 0862.8 : 0883. 2 10903.6 0924.1 0944.6 0965.2 0985.8 1006.5 1027.2 1048.0 1068.8 193 T94 194 194 195 195 196 197 197 197 198 198 199 200 200 200 201 201 202 202 203 203 204 204 205 205 206 206 207 207 208 208 209 Aiv) 460.41 468.85 477.35 485.90 494.51 503-18 511.90 520.69 529-52 538.42 547-38 556.39 565-47 574.61 583-80 593-05 602.37 611.75 621.20 630.70 640.27 649.90 659-60 669.36 679.19 689.08 699.04 709.07 719.16 729.32 739-55 749-84 760.21 Diff. 844 850 855 861 867 872 879 896 901 908 914 919 925 932 938 945 950 957 963 970 976 983 989 996 1003 1009 1016 1023 1029 1037 1043 /{v) 0.43629 0.43809 0.43990 0.44172 0.44354 0.44538 0.44722 0.44908 C.45094 0.45282 0.45470 0.45659 0.45849 0.46040 0.46231 0.46424 0.46618 0.46812 0.47008 0.47205 0.47402 0.47601 0.47800 0.48001 0.48202 0.48404 0.48608 0.48812 0.49018 0.49225 0.49432 0.49641 0.49850 Diff. 180 181 182 182 184 184 186 186 188 189 190 191 191 T93 194 194 196 197 197 199 199 201 201 202 204 204 206 207 207 209 209 211 T(v) 7-95T 7-974 7-997 8.021 8.044 8.068 8.091 8.115 8.139 8.163 8.187 8.211 8.235 8.259 8.284 8.308 ^'333 8.357 8.382 8.407 8.432 8.457 8.482 8.507 8.533 8.558 8.584 8.610 8-635 8.661 8.687 8.713 8.739 Diff. TABLE I.-rCONTINUED. Siv) 798 797 796 795 794 793 792 791 790 789 788 787 786 785 784 783 782 781 780 779 778 777 776 775 774 773 772 771 770 769 768 767 766 1089.7 1 1 10. 7 1131.7 1152.7 1173.8 1195.0 1216.2 1237.5 1258.8 1280.3 1301.8 1323-4 1345-0 1366.6 1388.2 1409.8 1431-5 1453-3 I475-0 1496.8 1518.6 1540.4 1562.2- 1584.1 1606.0 1627.9 1649.9 1671.9 1693.9 1716.0 1738.0 1760. 1 1782.3 Diff. 210 210 210 211 212 212 213 213 215 215 216 216 216 216 216 217 218 217 218 218 218 218 219 219 219 220 220 220 221 220 221 222 222 A (7') 1770.64 1781.15 1791.72 1802.37 1813.10 1823.89 1834.76 1845.70 1856.71 1867.87 1879.08 1890.36 1901.70 1913.1i 1924.57 1936.10 1947.70 1959.36 1971.08 1982.87 1994.72 2006.64 2018.62 2030.68 2042.80 2054.98 2067.24 2079.56 2091.95 2104.41 2116.94 2129.54 2142.21 Diff. 051 057 065 073 079 087 094 lOI 116 121 128 134 141 146 153 160 166 172 179 185 192 198 j 206 i 212 I 1 2l8| 226 I 232 1 239 246 253 260 267 274 7(7.) I 0.50061 10.50273 I 0.50486 ! 0.50700 0-50915 0-51131 0.51348 0.51566 0-5 786 0.52008 0.52231 0.52454 0.52678 0.52904 0.53130 0.53357 0.53585 0.53813 0.54043 0.54273 0.54504 0.54736 0.54969 0.55203 0.55438 0.55674 0.55911 0.56148 0.56387 0.56626 0.56867 0.57108 0.57350 Diff. 212 213 214 215 216 217 218 220 222 223 223 224 226 226 227 228 228 230 230 231 232 233 234 235 236 237 237 239 I 239 241 241 242 244 T(v) 8.765 8.791 8.818 8.844 8.871 8.897 8.924 8.951 8.97.8 9.005 9.032 9.060 9.087 9.114 9.T42 9.170 9.197 9.225 ! 9-2531 9.281 ! 9-309 I 9-337 I 9365 i 9-394 i 9.422 j 9-450 9-479 9-507 9-536 9-565 9-593 9.622 9-651 TABLE I.— Continued. S{v) 765 764 763 762 761 760 759 758 757 756 755 754 753 752 751 750 749 748 747 746 745 744 743 742 741 740 739 738 737 736 735 734 733 1804.5 1826.7 1848.9 1871.1 1893.4 1915-7 1938.0 1960.4 2005.3 2027.7 2050.2 2072.8 2095-3 2117.9 2140.5 2163. 1 2185.8 2208.5 2231. 2 2253-9 2276.7 2299.6 2322.4 2345-3 2368.2 2391-1 2414.1 2437-1 2460.1 Diff. 222 222 222 223 223 223 224 224 225 224 225 226 225 226 226 226 227 227 227 227 228 229 228 229 229 229 230 230 230 231 2^1 2483.2 2506.31 231 2529.4' 232 A{v) Diff. 2154-95 2167.76 2180.64 2193-59 2206.62 2219.7 r 2232.88 2246.12 2259.44 2272.83 2286.30 2299.84 2313-45 2327.14 2340.91 2354-75 2368.67 2382.66 2396.74 2410.89 2425.12 2439-44 2453-83 2468.30 2482.86 2497-49 2512.21 2527.01 2541.89 2556.86 2571.91 2587.04 2602.25 281 288 295 303 309 3^7 324 332 339 347 354 361 369 377 384 392 399 408 415 423 432 439 447 456 463 472 /(v) 480 0.64271 488 497 505 513 521 530 0-57594 0.57838 0.58083 0.58330 0.58577 0-58825 0.59074 0.59324 0-59575 0.59827 0.60080 0.60334 0.60589 0.60845 0.61 103 0.61361 0.61620 0.61880 0.62142 0,62404 0.62667 0.62932 0.63198 0.63464 0.63732 0.64001 0.64542 0164814 0.65087 0.65361 0.65637 0.65913 Diff. 244 245 247 247 248 249 250 251 252 253 254 255 256 258 258 259 260 262 262 263 265 266 266 268 269 270 271 272 273 274 276 276 278 T{v) 9.680 9.709 9-738 9.767 9-797 9.826 9-855 9.885 5.914 9-944 9-973 10.003 10.033 10.063 10.093 10.123 10.153 10.184 10.214 10.244 10.275 10.306 10.336 10.367 10.398 10.429 10.460 10.491 10.522 10.554 10.585 10.616 10.648 Diff. 23 TABLE I.— Continued. S(v) 732 731 730 729 728 727 726 725 724 723 722 721 720 719 718 717 716 714 713. 712 711 710 709 708 707 706 705 704 703 702 701 700 2552.6 2575-8 2599.0 2622.3 2645.6 2668.9 2692.3 2715.6 2739.0 2762.5 2786.0 2809.5 2833.1 2856.7 2880.3 2903.9 2927,6 2951-3 2975-1 2998.9 3022.7 3046.5 3070.4 3094-3 3118.3 3142.3 3166.3 3190.3 3214.4 3238.5 3262.7 3286.9 3311.I Diff. 232 232 233 233 233 234 233 234 235 235 235 236 236 236 236 237 237 238 238 238 238 239 239 240 240 240 240 241 241 242 242 242 242 A{v) 2617.55 2632.94 2648.41 2663.97 2679.61 2695.34 2711.16 2727.07 2743-07 2759.16 2775-33 2791.60 2807.96 2824.41 2840.96 2857.60 2874.33 2891.15 2908.07 2925.08 2942.19 2959-39 2976.09 2994.09 3011.58 3029.17 3046.86 3064,66 3082.55 3100.54 3118.64 3136.84 3155-H Diff, 539 547 556 564 573 582 591 600 609 617 627 ; 636 I 645 6551 664 673 682 692 701 711 720 730 740 749 759 769 780 789 799 810 820 830 841 /{v) 0.66I9I 0.66470 0.66750 0.67031 0.67313 0.67596 0.67881 0,68167 0.68454 0.68742 0,69031 0.69322 0.69614 0.69907 0.70201 0,70496 0.70793 0,71091 0.71390 0.7I69I 0.71993 0.72296 0.72600 0.72905 0.73212 0.73520 0.73830 0.74I4I 0.74453 0.74766 0.75081 0.75397 0.75715 Diff, 279 280 281 282 283 285 286 287 289 291 292 293 294 295 297 298 299 301 302 303 304 305 307 308 310 311 312 313 315 316 318 319 T{v) 0.679 0.711 0.743 0.775 :o.8o7 0.839 0.871 0.903 0.936 :o.968 1. 00 1 1-033 1,066 1.099 1-132 1. 165 1. 198 1.231 1.264 1.297 1-330 1.364 1.398 1-432 1.465 1.499 1-533 1.567 1.60T 1.636 1,670 1,704 1-739 24 TABLE I.— Continued. S{v) 699 698 697 696 695 694 693 692 691 690 689 688 687 686 685 684 683 682 681 680 679 678 677 676 675 674 673 672 67. 670 669 668 667 3335-3 3359-6 3383-9 3408.3 3432.7 3457-1 3481.6 3506.1 3530.6 3555-2 3579-8 3604.4 3629.1 3653-8 3678.6 3703-4 3728.2 3753-1 3778.0 3802.9 3827.9 3852.9 3877-9 39030 3928.1 3953-3 3978.5 4003.7 4029.0 4054-3 4079.6 4105.0 4130.4 Diff. 243 243 244 244 244 245 245 245 246 246 246 247 247 248 248 248 249 249 249 250 250 250 25' 251 252 252 252 253 253 253 254 254 255 A (v) 3173-55 3192.06 3210.67 3229.39 3248.22 3267.15 3286.19 3305.33 3324.58 3343-95 3363-42 3383-00 3402.70 3422.50 3442.42 3462.45 3482.60 3502.86 3523-24 3543-73 3564.34 3585.07 3605.91 3626.88 3647.96 3669.17 3690.50 3711 94 373351 3755.21 3777.03 3798.98 3821.05 Diff. 1 85 I 1861 1872 1883 1893 1904 1914 1925 1937 1947 1958 1970 1980 1992 2003 2015 2026 2038 2049 ! 2061 I 2073 I 1 2084 2097 2108 2121 2133 2144 2157 2170 2182 2195 2207 2219 I{v) 0.76034 0.76354 0.76675 0.76998 0.77322 0.77648 0.77975 0.78304 0.78634 0.78966 0.79299 0.79633 0.79969 0.80306 0.80645 0.80985 0.81327 0.81670 0.82015 0.82362 0.82710 0.83059 0.83410 0.83762 O.84116 0.84472 0.84829 0.85188 0.85549 O.85911 0.86274 0.86639 0.87006 Diff. 320 321 3 3 324 326 327 329 330 332 333 334 336 337 339 340 342 343 345 347 348 349 351 352 354 356 357 359 361 362 363 365 367 369 T{v) 11.774 11.809 11.844 11.879 II. 914 11.949 11.984 12.020 12.055 12.091 12.126 12.162 12.198 12.234 12.270 12.306 12.342 12.379 12.415 12.452 12.489 12.526 12.563 12.600 12.637 12.675 12.712 12.750 12.787 12.825 12.863 12.901 12.939 25 TABLE I.— Continued. Siv) 666 665 664 663 662 661 660 659 658 657 656 655 654 653 652 65' 650 649 648 647 646 645 644 643 642 641 640 639 638 637 636 635 634 4155-9 4181.4 4206.9 4232.5 4258.1 4283.7 4309-4 4335.1 4360.9 4386.7 4412.6 4438.5 4464.4 4490.4 4516.4 4542.4 4568.5 4594-6 4620.8 4647.0 4673.2 4699.5 4725-9 4752.3 4778.7 4805.1 4831.6 4858.1 4884.7 4911.3 4938.0 4964.7 4991.4 Diff. 255 255 256 256 256 257 257 258 258 j 259 259 259 260 260 260 261 261 262 262 262 263 264. 264 264 264 265 265 266 266 267 267 267 268 A {7^) 3843-24 3865.57 3888.02 3910.60 3933-31 3956.16 3979-13 4002.24 4025.48 4048.86 4072.37 4096.01 4T19.79 4143-71 4167.77 Diff. 4340.12 4365-32 4390.67 4416.16 4441.81 4467.60 4493-55 4519.64 4545-89 4572.30 4598.86 4625.57 2233 2245 2258 2271 2285 2297 2311 2324 2338 2351 2364 2378 2392 2406 2419 4191.96 2434 4216.30 2448 4240.78 2462 4265.40 j 2476 4290.16 I 2491 4315.07 2505 2520 2535 2549 2565 2579 2595 2609 2625 2641 2656 2671 2687 /{v) 0-87375 0.87745 0.88II7 0.88490 0.88866 0.89243 0.89622 0.90002 0.90384 0.90768 0.9II53 0.9I54I 0.91930 0,92321 0.92715 0.931 10 0.93506 0.93904 0.94304 0.94706 0.95IIO 0.95516 0.95923 0.96333 0.96745 0.97158 0.97574 0.97991 0,98410 0.98831 0.99254 0.99680 1.00107 Diff. 370 372 373 376 377 379 380 382 384 385 388 389 391 394 395 396 398 400 402 404 406 407 410 412 413 416 417 419 421 423 426 427 429 T{v) 977 015 053 092 130 169 208 247 286 326 365 404 444 484 524 564 604 644 684 725 766 806 847 929 971 012 053 095 137 4 179 4.221 4-263 Diff. 26 TABLE I.— Continued. V S{zi) Diff. i A (7-) Diff. I{v) Diff. T(v) Diff. 633 632 63. 15018.2 i5C'45-o 15071.9 268 269 269 4652.44 4679.47 4706.65 2703 2718 ' 2735 1.00536 1.00967 1. 01401 431 434 436 14.305 14.348 14.390 43 42 43 6.'?o 629 628 15098.8 15125.8 15152-8 270 2701 270 i 4734.00 4761.51 4789.18 2751 2767 2784 1. 01837 1.02274 I.027I3 437 439 442 i 14.433 14.476 14.519 43 43 43 627 626 625 15179-8 15206.9 15234.0 271 271 272 4817.02 4845.02 4873.18 2800 2816 2833 I 03155 J. 03598 1.04044 443 446 448 14.562 14.605 14.648 43 43 44 624 623 622 15261.2 15288.4 15315-7 272 273 273 4901.51 4930.00 4958.67 2849 2S67 2883 1.04492 1.04943 1.05395 451 452 455 14.692 14.735 14.779 43 44 44 621 620 619 15343.0 15370.3 15397.7 273 274 274 4987.50 1 5016.51 5045-69 2901 2918 2935 1 1.05850 1.06307 1.06766 457 459 461 14.823 14.867 14.91 1 44 44 45 618 617 616 15425. 1 15452.6 15480.1 275 275 276 5075-04 5104.57 5134-27 2953 2970 2988 1.07227 1.07690 1. 08156 463 466 468 ,14.956 15.000 15.045 44 45 45 615 614 613 15507.7 15535.3 15563.0 276 277 277 5164.15 5194.21 5224.44 3006 3023 3042 1.08624 1.09095 1.09568 471 473 475 15.090 15-135 15.180 45 45 45 612 611 610 T5590.7 15618.4 15646.2 277 1 278 278 5254-86 i 5285.46 5316.24 3060 3078 3097 1 10043 1. 10520 I.IIOOO 477 48c 482 15-225 15.270 T5-316 45 46 45 609 608 607 1 15674.0 15701.9 15729.8 279 279 280 5347-21 5378.36 5409-71 31^5 3135 3153 1 I.II482 I.II966 1. 12452 484 486 489 15-361 15-407 15.453 46 46 46 606 605 604 15757-8 15785.8 15813-9 280 i 281 1 281 1 ! 5441.24 5472.95 5504.86 3171 3191 3210 I.I294I 1. 13433 1 1.13927 492 494 497 ■ 15-499 15-546 15.592 47 46 46 603 602 601 15842.0 15870. 1 15898.3 281 282 ' 283 1553696 i 556926 '5601.75 3230 3249 3268 1. 14424 1. 14923 1-15425 499 502 504 15.638 15.685 15.732 47 47 47 27 TABLE I.— Continued. V S{v) Diff. I 1 1 A (v) Diff. I{v) Diff. T{v) Diff. 600 599 598 15926.6 15954.9 15983-2 283 283 284 5634.43 I 3288 5667.31 J3309 5700.40 3329 1. 15929 1. 16435 1.16944 506 509 5'2 15.779 15.826 15.873 47 47 48 597 596 595 160IT.6 1 6040. 1 16068.6 285 285 285 1 5733-69 5767.18 5800.87 3349 3369 3389 1. 17456 1. 17970 1. 18487 5H 517 519 15.921 15.968 16.016 47 48 48 594 593 592 16097. 1 16125.7 16154-4 286 287 287 5834-76 5868.85 5903.16 3409 3431 3451 1. 19006 1. 19528 1.20053 522 525 527 16.064 16. 113 16. 16 I 49 48 48 591 590 589 16183.I 16211.8 16240.6 287 1 288 288 5937-67 5972.39 6007.32 3472 3493 3515 T. 20580 I.2IIIO 1. 21643 530 533; 535 , 16.209 16.258 16.307 49 49 49 588 587 586 16269.4 16298.3 16327.2 289 289 290 6042.47 6077.83 6113.41 3536 3558 3579 I. 22178 I. 22716 1.23257 538 541 544 16.356 16.405 16.454 49 49 50 585 584 583 16356.2 16385.2 16414.3 290 291 291 6149.20 6185.22 6221.46 3602 3624 3646 I. 23801 1.24348 1.24897 547 549 552 16.504 16.553 16.603 49 50 50 582 581 580 16443.4 16472.6 16501.8 292 292 293 6257.92 6294.61 6331-52 3669 3691 3714 1.25449 1.26004 1.26562 555 558 561 16.653 16.704 16.754 51 50 51 579 578 577 16531.1 16560.4 16589.8 293 294 294 6368.66 6406.01 6443-63 3735 3762 3783 1. 27123 1.27687 1.28253 564 566 570 16.805 16.855 16.906 50 51 52 576 575 574 16619.2 16648.7 16678.2 295 295 296 6481.46 65^9.52 6557.82 3806 3830 3854 1.28823 1.29396 I. 29971 573 575 579 16.958 17.009 17.060 51 51 52 573 572 571 16707.8 16737.4 16767. 1 296 297 298 6596.36 6635.14 6674.16 3878 3902 3926 1.30550 I.3II3' I.31716 581 585 588 17.112 17.164 i 17.216 52 52 52 570 569 568 16796.9 16826.7 16856.6 298 299 299 671342 6752.93 6792.68 3951 3975 4000 1.32304 1-32895 1-33489 591 594 597 17.268 17.320 17-373 52 52 28 TABLE I.— Continued. S{v) 567 566 565 564 563 562 561 560 559 558 557 556 555 554 553 552 551 550 549 548 547 546 545 544 543 542 541 540 539 538 537 536 535 6886.5 6916.4 6946.4 6976.5 7006.6 7036.8 7067.0 7097.3 7127.6 7158.0 7188.4 7218.9 7249.4 7280.0 7310.7 7341-4 7372.2 7403.0 7433-9 7464.8 7495-8 7526.8 7557.9 7589-1 7620.3 7651.6 7682.9 77H-3 7745-8 7777-3 7808.9 7840-5 7872.2 Diff. 299 300 301 301 302 302 303 303 304 304 305 305 306 307 307 308 308 309 309 310 310 311 312 312 313 313 314 315 315 316 316 317 317 Aiv) 6832.68 6872.93 6913-43 6954.18 6995.19 7036.46 7077.99 7119.78 7161.83 7204.15 7246.73 7289.58 7332.71 7376.11 7419.78 7463.74 7507-97 7552.48 7597.28 7642.36 7687.73 7733-39 7779-34 7825.58 7872.12 7918.96 7966.12 8013-55 8061.30 8109.36 8T57-73 8206.41 8255.41 Diff. 4025 4050 4075 4101 4127 4153 4179 4205 4232 4258 4285 43^3 4340 4367 4396 4423 4451 4480 4508 4537 4566 4595 4624 4654 4684 4716 4743 4775 4806 4837 4868 4900 4932 /{v) 34086 34686 35290 35897 36507 37120 37736 38356 38979 39606 40236 40869 41506 42146 42789 43436 44087 44741 45399 46060 46725 47394 48066 48742 49422 50106 50793 51484 52179 52878 53581 54287 54998 Diff. 600 604 607 610 613 616 620 623 627 630 637 640 643 647 651 654 658 661 665 669 672 676 680 684 687 691 695 699 703 706 711 715 Tiv) 7.425 7-478 7-531 7-584 7.638 7.691 7.745 7-799 7.853 7-908 7.962 8.017 8.072 8.127 8.183 8.238 8.294 8.350 8.406 8.462 8.519 8.576 8.633 8.690 8.747 8.805 8.921 8.979 9.038 9.096 9-155 9-215 Diff. 29 TABLE I.— Continued. S{v) 534 533 532 531 530 529 528 527 526 525 524 523 522 521. 520 519 518 517 516 5^5 514 513 512 511 510 509 508 507 506 505 504 503 502 7903.9 7935-7 7967.6 7999-5 8031.5 8063.5 8095.6 8127.8 8160.0 8192.3 8224.7 8257.1 8289.6 8322.1 8354-7 8387-4 8420.1 8452.9 8485.7 8518.6 8551-6 8584-7 8617.8 8651.0 8684.2 8717-5 8750-9 8784.3 8817.8 8851.4 8885.0 8918.7 8952.5 Diff 318 319 319 320 320 321 322 322 323 324 324 325 325 326 327 327 328 328 329 330 33^ 33'^ 332 332 333 334 334 335 33^ 33^ 337 33^ 338 A'iv) 8304.73 8354.36 8404.32 8454.61 8505.22 8556.16 8607.44 8659.06 8711.01 8763.30 8815.94 8868.92 8922.25 8975.93 9029.97 9084.36 9139.11 9194.23 9249.71 9305.56 9361.79 9418.39 9475-38 9532.74 9590.49 9648.62 9707.15 9766.06 9825.38 9885.09 9945.21 10005.74 10066.67 Diff. 4963 4996 5029 5061 5094 5128 5162 5195 5229 5264 5298 5333 5368 5404 5439 5475 5512 5548 5585 5623 5660 5699 5736 5775 5813 5853 5891 5932 5971 6012 6053 6093 6134 I{v) 55713 56431 57154 57881 58612 59347 60086 60830 61578 62330 63086 63847 64612 65381 66155 66933 67716 68504 69296 70092 70894 71700 72510 73326 74146 74971 75801 76636 77476 78321 79171 80026 80886 Diff. 718 723 727 731 735 739 744 748 752 756 761 765 769 774 778 783 788 792 796 802 806 810 816 820 825 830 835 840 845 850 855 860 865 T{v) 19.274 J9.334 19-394 ^9-454 19.514 19-574 19-635 19.696 19-757 19.819 19.881 19.943 20.005 20.067 20.130 20.193 20.256 20.319 20.383 20.447 20.511 20.575 20.640 20.705 20.770 20.835 20.901 20,967 21.033 21.099 21.166 21.233 21,300 30 TABLE I.— Continued. 501 500 499 498 497 496 495 494 493 492 491 490 489 488 487 486 485 484 483 482 481 480 479 478 477 476 475 474 473 472 471 470 469 ■S(v) 3 9020.2 9054.2 9088.2 9122.3 9156.4 9190.6 9224.9 9259-3 9293.8 9328.3 9362.9 9397.6 9432.3 9467.1 9502.0 9536.9 9572.0 9607.1 9642.2 9677-5 9712.8 9748.2 9783.6 9819. 1 9854.7 9890.4 9926.2 9962.0 9997-9 20033.9 20070.0 20106,2 Difif. 339 340 340 341 341 342 343 344 345 345 346 347 347 348 349 349 351 351 351 353 353 354 354 355 356 357 358 358 359 360 361 362 362 A{v) 0128.01 0189.78 0251.9 0314.5 0377.6 0441.0 0504.9 0569-3 0634.1 0699.3 0765.0 0831. 1 0897.6 0964.7 1032.2 1 100. 1 1168.6 1237-5 1307.0 1376.9 1447-2 1518.1 1589.4 1661.3 1733-7 1806.6 1880.0 1953-9 2028.4 2103.4 2178.9 2254.9 2331-5 Diff. 6177 6219 626 631 634 639 644 648 652 657 661 665 671 675 679 685 689 695 699 703 709 •713 719 724 729 734 739 745 750 755 760 766 771 /{v) .81751 .82622 .83498 •84379 -85265 -86157 •87054 -87957 .88865 .89778 .90697 .91622 •92552 -93488 .94430 •95378 .96332 .97292 .98258 .99230 2.00207 2.0II90 2.02180 2.03176 2.04179 2.05188 2,06203 2.07225 2.08253 2.09288 2.10329 2.II376 2.12430 Diff, 871 876 886 892 897 903 908 913 919 925 930 936 942 948 954 960 966 972 977 983 990 996 T003 1009 1015 1022 1028 1035 1041 1047 1054 1061 T{v) 21.367 21.435 21.503 21.572 21.641 21.710 21.779 21.848 2T.918 21.988 22.058 22.128 22.199 22.270 22.341 22.413 22.485 22.557 22,630 22,703 22.776 22.849 22.923 22.997 23.071 23.146 23.221 23.296 23.372 23.448 23-524 23.601 23.678 Diff. 31 TABLE I.— Continued, V S{v) Diff. A{v) Diff. /{v) Diff. T{v) Diff. 468 467 466 20142.4 20J78.7 20215.0 363 365 12408.6 '12486.3 12564.6 777 783 788 2.13491 2.14559 2.15635 ic68 i 1076 i 1082 23.755 23.833 23.911 78 78 78 465 464 463 20251.5 20288.0 20324.7 365 367 367 12643.4 12722.8 12802.7 794 799 805 2.16717 2.17806 2.18902 1089 1 1096 1 104 23.989 24.068 24.147 79 79 79 462 461 460 20361.4 20398.1 20435.0 367 369 369 12883.2 12964.3 13045-9 811 816 822 2.20006 2.21116 2.22233 i 1110 1117 1124 24.226 24.306 24.386 80 80 80 459 458 457 20471.9 20508.9 20546.0 370 371 371 13128. 1 13211.O 13294.4 829 834 841 2.23357 2.24489 2.25629 1132 1140 1147 24.466 24.547 - 24.628 81 81 82 456 455 454 20583.1 20620.4 20657.7 373 373 374 13378.5 134633 13548.6 848 853 859 2.26776 2.27931 2.29094 1155 1163 1171 24.710 24.792 24.874 82 82 82 453 452 451 20695.1 20732.6 20770.2 375 376 377 13634.5 13721.1 13808.3 866 872 878 2.30265 2.31443 2.32628 1178 1185 1193 24.956 25-039 25.122 ^3 83 84 450 449 448 20807.9 20845.6 20883.4 377 378 380 13896.1 13984.6 14073.7 885 891 898 2.33821 2.35022 2.36232 1201 1210 1218 25.206 1 25.290 25.374 84 84 85 447 446 445 20921.4 20959.4 20997.4 380 380 382 '4163.5 14254.0 14345. i 905 911 919 2.37450 2.38676 2.39911 1226 1235 1243 1 25.459 25.544 25.629 85 85 86 444 443 442 21035.6 21073.9 21112.2 385 14437.0 14529.5 14622.7 925 932 939 2.41154 2.42405 2.43665 1251 1260 1268 i 25.715 25.801 25.888 86 87 87 441 440 439 21150.7 21189.2 21227.8 385 386 387 14716.6 14811.2 14906.5 946 953 960 2.44933 2.46209 2.47494 1276 1285 1294 25.975 26.062 26.150 87 88 88 438 437 436 21266.5 213^5-3 2i34'4-2 388 389 389 15002.5 15099.3 15196.8 968 975 982' 2.48788 2.50091 2.51404 1303 1313 1322 26.238 26.327 ' 26.416 89 89 89 32 TABLE I.— Continued. S{v) 21383. 1 21422.2 21461.4 21500.6 21540.0 21579-4 21618.9 21658.5 21698.2 21738.0 21777.9 21817.8 21857.9 21898.1 21938.4 21978.7 22019.1 22059.6 Diff. 22100.2 22140,9 22 81.8 22222.7 22263.7 22304.8 22346.1 ^2387.4 22428.8 22470.4 225 12.0 22553-7 22595.6 22637.5 22679.6 391 392 392 394 394 395 396 397 398 399 399 401 402 403 403 404 405 406 407 409 409 410 411 413 413 414 416 416 417 419 419 421 422 A{v) 5295-0 5394.0 5493-7 5594-2 5695-4 5797-3 5900.0 6003.5 6107.9 6213. 1 6319.1 6425.9 6533-5 6641.9 6751.2 6972.2 7084.1 7196.8 7310.5 7425.0 7540.5 7656.8 7774.1 7892.2 8011.3 813^-3 8252.4 8374.4 8497.4 8621.4 8746.4 8872.3 Diff. 990 997 1005 012 019 027 035 044 052 060 068 076 084 093 T09 119 127 137 145 155 163 173 181 191 200 211 220 230 240 250 259 270 I{v) 2.52726 2.54057 2.55397 2.56746 2.58104 2.59471 2.60848 2.62235 2.63632 2.65039 2.66456 2.67883 2.69320 2.70767 2.72225 2.73692 2.75169 2.76658 2.78158 2.79668 2,81190 2,82723 2.84267 2.85822 2.87388 2.88965 2.90554 2.92155 2.93768 2.95393 2.97030 2.98679 3.00341 Diff. 33^ 340 349 i 358 367 377 387 397 407 417 427 437 447 458 467 477 489 500 510 522 533 544 555 566 577 589 601 613 625 637 649 662 674 T{v) 26.505 26.595 26.685 26.776 26.867 26.959 27.051 27.143 27.236 27.329 27.423 27.517 27.612 27.707 27.803 27-899 27-995 28.092 28.189 28.287 28.385 28.484 28.583 28.683 28.783 28.884 28.985 29.087 29.189 29.292 29-395- 29.499 29,603 33 TABLE I.— Continued. V Siv) Difif. A{v) Diff. I{v) Diff. Tiy) Diff. 402 401 400 22721.8 22764.0 22806.4 422 424 424 18999.3 19127.3 19256.2 1280 1289 1300 3.02015 I 3-03701 ' 3-05399 1686 1698 1710 29.708 29.813 29.919 105 106 106 34 TABLE II. For Spherical Projectiles. V S{v) Diff. A(v) Diff. 1 I{v) Diff. T{v) Diff. 2000 25 0.00 I i 0.00000 40 0.000 12 1990 1980 25 49 24 25 O.OI 0.02 I ! 2 00040 00080 40 41 0.012 0.025 13 12 1970 i960 1950 74 99 124 25 25 26 0.04 0.08 0.13 4 5 51 0.00121 00163 00205 42 42 43 0.037 0.050 0.063 13 13 13 1940 1930 1920 150 175 201 25 26 25 0.18 0.25 0.33 7J 8 1 9 0.00248 00292 00336 44 44 45 0.076 0.089 0.102 13 13 14 I9IO 1900 1890 226 252 278 26 26 26 0.42 0.53 0.65 13 0.00381 00427 00473 46 46 47 0.1 16 0.129 0.143 13 14 14 1880 1870 i860 304 357 26 27 26 0.78 0.92 1.07 14 15 17 0.00520 00568 00617 48 49 49 0.157 0.171 0.185 14 14 14 1850 1840 1830 383 409 436 26 27 27 1.24 1.43 1.63 19 20 21 0.00666 00716 00767 50 51 52 0.199 0.214 0.228 15 14 15 1820 181O 1800 463 490 517 27 27 28 1.84 2.07 2.31 23 24 26 0.00819 00872 00926 53 0.243 0.258 0.273 15 15 15 1790 1780 1770 545 572 600 27 28 28 2.57 2.84 3.14 27 30 31 0.00981 01036 01093 55 57 57 0.288 0.304 0.319 16 15 16 1760 1750 1740 628 656 684 28 28 28 3.45 3-78 4-13 35 37 0.01150 01209 01268 59 59 61 0.335 0.351 0.367 16 16 16 1730 1720 I7IO 712 741 769 29 28 29 4.50 4.89 5-30 39 41 43 0.01329 01390 01453 61 (>z 64 0.383 0.400 0.416 17 16 17 35 TABLE IL— Continued. V S{v) Diff. A{v) Diff; I{v) Diff. T{v) Diff. 1700 1690 1680 798 827 856 29 29 30 5-73 6.18 6.65 45 47 50 O.OI5I7 01582 01648 65 66 67 0.433 0.450 0.468 17 18 17 1670 1660 1650 886 915 945 1 29 30 30 7.15 7.67 8.21 52 54 56 O.OI7I5 01783 01853 68 70 71 0.485 0-503 0.521 18 18 t8 1640 1630 1620 975 1005 1036 30 3T 30 8.77 9-35 9-97 58 62 64 0.01924 01996 02070 72 74 75 0.539 0.558 0.576 19 18 19 I6I0 1600 1590 1066 1096 T127 30 31 31 10.61 11.27 11.96 66 69 72 0.02145 02222 02300 79 0.595 0.614 0.633 19 19 20 1580 1570 1560 1158 1189 1220 31 31 32 12.68 1344 14.22 76 78 82 0.02379 02460 02542 81 82 84 0.653 0.673 0.693 20 20 20 1550 1540 1530 1252 1284 1316 32 32 32 15-04 15.90 16.78 86 88 92 0.02626 02712 02799 86 87 89 0.713 0.734 0.755 21 21 21 1520 I5I0 1500 1348 1380 1413 32 17.70 18.65 19.63 95 98 100 0.02888 02979 03072 91 93 94 0.776 0.797 0.819 21 22 22 1490 1480 1470 1446 1479 1512 Z2> 7>Z 34 20.63 21.68 22.77 105 109 114 03166 03262 03360 - 96 98 ICI 0.841 0.863 0.885 22 22 23 1460 1450 1440 1546 1580 1614 34 34 34 23.91 i 25.10 26.34 ! 119 124 128 03461 03564 03669 103 105 107 0.908 0931 0.955 23 24 24 1430 T420 I4I0 1648 1682 1717 34 35 35 27.62 1 28.95 133 138 143 0.03776 03885 03997 109 112 114 0.979 T.003 1.028 24 25 25 1400 1390 1380 1752 1787 1823 35 35 31.76 33-25 34-79 149 154 160 0.041 1 1 04227 04346 116 119 122 1.053 1.079 1. 105 26 .6 26 36 TABLE II.— Continued. V S{v) Diff. A{v) Diff. I{v) Diff. T{v) Diff. 1370 1360 1350 1858 1894 1931 36 37 36 36.39 38.03 39-73 164 170 175 0.04468 04592 04719 124 127 129 1. 131 1. 158 1. 185 27 27 27 1340 1330 1320 1967 2004 2041 37 37 37 41.48 43-29 45-14 181 185 191 0.04848 04981 05II7 136 139 1. 212 1.239 1.267 27 28 27 I3I0 1300 1290 j 2078 2Tl6 2T54 38 38 47-05 49.01 51-04 196 203 212 0.05256 05398 05542 142 144 148 1.294 1.322 I-351 28 29 30 1280 1270 1260 2192 2231 2269 39 38 39 53-16 55-37 57.67 221 230 240 0.05690 05842 05998 152 156 160 1.38T 1.411 1.442 30 31 31 1250 1240 1230 2308 2348 1 2388 40 40 40 60.07 62.56 65.14 249 258 267 0.06158 06323 06492 165 169 174 1.473 1-505 1.538 32 33 1220 I2IO 1200 2428 2470 2512 42 42 22 67.81 70.59 73-54 278 295 156 0.06666 06846 07033 180 187 97 1.571 1.605 1.640 34 35 18 I I 90 2534 2556 2578 22 22 22 75-IO 76.70 78.32 t6o 162 165 0.07130 07229 07329 99 100 102 1.658 1.676 1.694 18 18 18 I 180 II75 1170 1 2600 2623 2646 '23 23 23 79-97 81.66 83-39 169 173 177 0.07431 07535 07641 104 106 108 1.712 I-731 I-751 19 20 19 I165 I 160 i'55 j i 2669 2692 2715 23 23 24 85.16 86.98 88.84 182 186 190 0.07749 07859 07972 no 113 115 1.770 1.790 1. 810 20 20 21 1150 1 145 1 140 2739 2763 2787 24 24 25 90.74 92.69 94.68 195 199 205 0.08087 08204 08324 117 120 122 1.831 1.852 1-873 21 21 22 1130 1125 2812 2837 2861 25 24 25 96.73 98.82 100.97 209 215 221 0.08446 08570 08697 124 127 130 1.895 1-917 1.940 22 23 23 37 TABLE II.— Continued. V Si^v) Diff. A{v) Diflf. I{v) Diff. T{v) Diff. II20 IITO 2886 2912 2938 26 26 26 103.18 J05-44 107.77 226 233 239 0.08827 08959 09094 132 '35 138 1.963 1.986 2.009 23 24 IIO5 IIOO 1095 2964 2991 3017 27 26 27 110.16 1 12,62 115-13 246 251 259 0.09232 09373 09516 141 '43 147 2.033 2.057 2.081 24 24 25 1090 1085 1080 3044 3071 3099 27 28 28 117.72 120.38 123.13 266 ■275 283 0.09663 09812 09965 149 153 156 2.106 2.132 2.158 26 26 26 1075 1070 1065 3127 3155 3184 28 29 29 125.96 128.87 131.87 291 300 308 0.10121 10280 10443 159 163 166 2.184 2.210 2.237 26 27 28 io6o 1055 1050 3213 3243 3273 30 30 30 134.95 138.12 141.38 317 326 338 0.10609 10-79 10952 170 173 177 2.265 2.293 2.321 28 28 29 1045 T040 1035 3364 30 31 31 144.76 148.22 151-77 346 355 364 0.11129 11310 1 1495 181 185 189 2.350 2.379 2.409 29 30 31 1030 1025 1020 3395 3427 3459 32 32 32 155-41 159-15 162.99 374 384 394 0.11684 II877 12074 193 197 202 2.440 2.471 2.502 31 31 32 1015 lOIO 1005 3491 3524 3557 Z2> 34 166.93 170.99 175-17 406 418 430 0.12276 12482 12693 206 211 215 2.534 2.566 2.599 32 1000 995 990 3591 3625 3660 34 35 35 179-47 183.90 188.46 443 456 470 0.12908 13128 13354 220 226 231 2.632 2.665 2.699 zz 34 ZS 985 980 975 3695 3731 3767 36 193.16 198.00 202.98 484 498 513 0.13585 13821 14062 236 241 246 2.734 2.770 2.806 36 37 970 965 960 3803 3840 3877 37 37 38 208.1 1 213.40 218.86 529 546 563 0.14308 14560 14818 252 258 264 2.843 2.881 2.920 38 39 39 38 TABLE II.— Continued. V S{v) Diff. A{v) Diff. I{v) Diff. T{v) Diff. 955 950 945 39^5 3953 3992 38 39 39 224.49 230.29 236.29 580 600 620 0.15082 15352 15628 270 276 283 2-959 2.999 3.040 40 41 42 940 935 930 4031 4070 41 10 39 40 41 242.49 248.86 255-43 637 657 676 O.I59II I620I 16498 290 297 304 3.082 3-125 3.168 43 43 44 925 920 915 4151 4192 4234 41 42 43 262.19 269.17 276.37 698 720 743 0.16802 17113 17432 311 319 327 3-212 3-257 3-303 45 46 47 910 905 900 4277 4320 4363 43 43 44 283.80 291.47 299.40 767 793 819 0.17759 18094 18437 335 343 352 3-350 3-397 3.445 47 48 49 895 890 885 4407 4451 4496 44 45 46 307-59 316.04 324-77 845 873 901 0.18789 I9I49 I95I8 360 369 378 3-494 3.544 3.595 50 51 52 880 875 870 4542 4589 4636 47 47 48 333-78 343-06 352.67 928 961 997 0.19896 20283 20680 387 397 407 3-647 3.700 3-754 53 54 55 865 860 855 4684 4732 4781 48 49 49 362.64 372.96 1032 1064 1099 0.21087 21505 21933 418 428 439 3-809 3.865 3.922 56 57 58 850 845 840 4830 4880 4931 50 5» 52 394.59 405-96 417.71 "37 1175 1216 0.22372 22823 23285 451 462 476 3-980 4.039 4.100 59 61 61 835 830 825 4983 5036 5089 53 53 54 429.87 442.45 455-47 1258 1302 1347 0.23761 24248 24746 487 498 511 4. 161 4.224 4.288 64 820 815 810 5143 5198 5253 55 55 56 468.94 482.89 497-33 1395 1444 1495 0.25257 25783 26323 526 540 553 4-354 4-421 4.489 67 68 70 805 800 795 5309 5366 5424 57 58 59 512.28 527-77 543-81 1549 1604 1661 0.26876 27444 28031 568 587 601 4-559 4.630 4.702 71 72 74 39 TABLE II.— Continued. V Siv) Diff. A{v) Diff. nv) Diff. T{v) Diff. 79Q 785 780 5483 5542 5602 59 60 61 560.42 577-64 595-48 1722 1784 1849 0.28632 29249 29883 617, 634 650; 4.776 4.852 4-929 76 77 79 775 770 765 5663 5725 5788 62 63 64 613-97 653.01 1916 1988 2062 0.30533 31203 31891 670 688 707 5.008 5.088 5-170 80 82 84 760 755 750 5852 5917 5983 65 66 67 673-63 695.01 717 19 2138 2218 2303 0.32-598 33325 34073 727 748 770 5254 5-340 5-427 86 87 90 745 740 735 6050 6118 6187 68 69 69 740.22 764.11 788.91 2389 248c 2574 0.34843 35634 36448 791 814 837 5-517 5-608 S-701 91 93 96 730 725 720 6256 6327 6399 71 72 73 814.65 841.38 869.14 2673 2776 2882 0.37285 38146 39033 861 887 912 5-797 5.894 5-994 97 100 102 715 710 705 6472 6546 6621 74 75 77 897.96 927.92 959.07 2996 3115 3238 0.39945 40885 41853 940 968 995 6.096 6.200 6.306 104 106 109 700 695 690 6698 6776 685s 78 79 80 991.45 1025.2 1060.2 3366 350 364 0.42848 43872 44926 1024 1054 1089 6.415 6.526 6.640 III ri4 116 685 680 675 6935 7016 7098 81 . 82 84 1196.6 1134.4 1173.8 378 394 409 0.46015 47143 48302 1128 1159 1192 6.756 6.875 6.997 1 119 122 125 670 665 660 7182 7267 7354 85 87 88 1214.7 1257.4 1301.8 427 444 463 0.49494 50722 51989 1228 1267 1307 7.122 7-249 7-380 127 131 134 655 650 645 7442 7531 7622 89 91 92 1348.1 1396.3 1446.5 482 502 523 0.53296 54645 56037 1349 1392 1436 7-5M 7-651 1 7.79T 137 140 143 640 635 630 7714 7808 7903 94 95 97 1498.8 1553.4 1610.2 546 568 592 0.57473 58955 60484 1482 1529 1579 7 934 ' 8.081 ' 8.231 147 150 154 40 TABLE II.— Continued. V Siv) Diff. A {v) Diff. I{v) Diff. T{v) Diff. 625 620 6.5 8000 8098 8198 98 100 lOI 1669.4 1731.2 1795-6 618 644 673 0.62063 63696 65386 1633 1690 1737 8.885 8-543 8.705 158 162 166 610 605 600 8299 8402 8507 103 1 105 107 1862.9 I933-I 2006.4 702 733 765 0.67123 68922 70781 1799 1859 1923 8.871 9.041 9.215 170 174 179 595 590 585 8614 8722 8833 108 III 1 12 2082.9 2162.9 2246.5 800 836 872 0.72704 74692 76747 1988 2055 2126 9.394 9.577 9-765 183 188 192 580 575 570 8945 9059 9175 114 116 118 2333-7 2424.8 2520.2 911 954 998 0.78873 81072 83348 2199 2276 2356 10.957 10.154 10.357 197 203 208 565 560 555 9293 9413 9535 120 122 124 2620.0 2724.3 2833.4 1043 1091 1142 0.85704 88144 90670 2440 2526 2617 10.565 10.778 10.997 213 219 225 550 545 540 9659 9785 9914 126 129 131 2947.6 3067.2 3192.4 1196 1252 1312 0.93287 95998 98808 2711 2810 2913 11.222 11-453 11.690 231 237 243 535 - 530 525 10045 10178 135 138 3323-6 3461.0 3605.0 1374 1440 1509 I.OI72I 1.04740 1.07873 3019 3247 11-933 12.183 12.440 250 257 264 520 515 510 10451 10591 10734 140 143 146 3755-9 3914.1 4080.1 1582 1660 1743 I.III20 1. 14486 I.I7981 ZZ(^(> 3495 3633 12.704 12.975 13-254 271 279 287 505 500 495 10880 11028 11179 .48 151 '53 4254-4 4437-3 4629.3 1829 1920 2017 I.21614 1.25393 T. 29312 3779 3919 4070 13-541 13.836 14.138 295 302 312 490 485 480 11332 11488 1 1648 156 160 162 4831.0 5042.8 5265-4 2118 2226 2340 1.33382 I.37614 1. 42013 4232 4399 4575 14-450 14.770 15.100 320 330 340 475 470 465 11810 11975 12143 168 172 5499-4 5745-5 6004.3 2461 2588 2724 1.46588 1. 51348 1. 56301 4760 4953 5157 15-440 15-790 16.150 350 360 370 41 TABLE II. —Continued. V S(v) Diff. A{v) ■ Diff. I{v) Diff. T{v) Diff. 460 455 450 12315 12490 12668 175 178 6276.7 6865.5 2868 3020 1. 61458 1.66826 ! 1. 72419 1 5368 5593 16.520 16.902 17.296 382 394 42 TABLE III. f) (^) Diff. Tan^ Diff. e (^) Diff. TanO Diff. o° oo' o 20 40 0.00000 00582 01164 i 582 582 582 0.00000 00582 01164 582 582 582 ' II 00 II 20 II 40 0.19560 20176 20794 616 618 621 0.19438 20042 20648 604 606 608 I 00 I 20 I 40 0.01746 02328 02910 582 582 583 0.01746 02328 02910 582 582 582 12 00 12 20 12 40 0.21415 22038 22663 623 625 627 0.21256 21864 22475 608 611 612 .2 00 2 20 2 40 0.03493 04076 04659 583 583 584 0.03492 04075 04658 583 583 583 13 00 13 20 13 40 0.23290 23920 24553 630 633 636 0.23087 23700 24316 613 616 617 3 00 3 20 3 40 0.05243 05827 06412 584 585 586 0.05241 05824 06408 583 584 585 14 00 14 20 14 40 0.25189 25827 26468 638 641 644 0.24933 25552 26172 619 620 623 4 00 4 20 4 40 0.06998 07585 08172 587 587 58S 0.06993 07578 08163 585 585 586 15 00 15 20 15 40 0.27112 27759 28409 647 650 654 0.26795 27419 28046 624 627 629 5 00 5 20 5 40 0.08760 09349 09939 589 590 591 0.08749 09335 09922 586 587 588 16 00 16 20 16 40 0.29063 29720 30380 657 660 663 0.28675 29305 29938 630 633 635 6 00 6 20 6 40 0.10530 II 122 11715 592 593 594 0.105 10 1 1099 11688 589 589 590 17 00 17. 20 17 40 0.31043 31710 32381 667 671 674 0.30573 31210 31850 637 640 642 7 00 7 20 7 40 0.12309 12905 13502 596 597 598 0.12278 12869 13461 591 592 593 18 00 18 20 18 40 0.33055 33733 34415 678 682 686 0.32492 33^36 33783 644 647 650 8 00 8 20 8 40 0.14100 14700 15301 600 601 603 0.14054 14648 1 15243 594 595 595 19 00 19 20 19 40 0.35101 35791 36486 690 695 699 0.34433 35085 35740 652 655 657 9 00 9 20 9 40 0.15904 16509 17116 605 607 608 0.15838 1 16435 1 17033 j 597 598 600 20 00 20 20 20 40 0.37185 37888 38596 703 708 713 0.36397 37057 37720 660 663 666 10 00 10 20 10 40 0.17724 18334 18946 610 612 614 0.17633 18233 18835 600 602 603 21 00 21 20 21 40 0.39309 40026 40748 717 722 728 0.38386 39055 39727 669 672 676 43 TABLE III.— Continued. . m Diff. Tan 6/ Diff. 6 (^) Diff. TanB Diff. 22° Oo' 22 20 2 2 40 0.41476 42208 42946 732 738 744 0.40403 41081 41763 678 682 684 33 33 00' 20 40 0.69253 70245 71248 992 roo3 1015 0.64941 65771 66608 830 837 843 23 00 23 20 23 40 0.43690 44439 45193 749 754 760 0.42447 43136 43828 689 692 695 34 34 34 00 20 40 0.72263 73290 74330 1027 1040 1052 0.67451 68301 69157 850 856 864 24 00 24 20 24 40 0-45953 46719 47491 766 772 778 0.44523 45222 45924 699 702 707 35 35 35 00 20 40 0.75382 76447 77525 1065 1078 1092 0.70021 70891 71769 870 878 885 25 00 25 20 25 40 0.48269 49054 49845 785 791 798 0.46631 47341 48055 710 714 718 36 36 36 00 20 40 0.78617 79723 80843 1 106 1 120 1 1 34 0.72654 73547 74447 893 900 908 26 00 26 20 26 40 0.50643 51448 52260 805 812 818 0.48773 49495 50222 722 727 731 37 37 37 00 20 40 0.81977 83126 8429 1 1 149 1 165 1T82 0-75355 76272 77196 917 924 933 27 00 27 20 27 40 0-53078 53904 54738 826 834 842 0.50953 51688 52427 735 739 744 38 38 3^ 00 20 40 0.85473 86670 87883 1197 1213 1231 0.78129 79070 80020 941 950 958 28 00 28 20 28 40 0.55580 56429 57286 849 857 865 0.53171 53920 54073 749 753 758 39 39 39 00 20 40 0.89114 90363 91629 1249 1266 1285 0.80978 81946 82923 968 977 ^987 29 00 29 20 29 40 0.58151 59025 59907 874 882 892 0.55431 56194 56962 763 768 773 40 40 40 00 20 40 0.92914 94217 95541 1303 1324 1343 0.83910 84906 859'2 996 1006 1017 30 00 30 20 30 40 0.60799 61699 62608 900 909 919 0.57735 58513 59297 778 784 789 41 41 41 00 20 40 0.96884 98247 99632 •363 1385 1407 0.86929 87955 88992 1026 1037 1048 31 00 31 20 31 40 0.63527 64455 65394 928 939 949 0.60086 60881 61681 795 800 806 42 42 42 00 20 40 1.01039 02468 03920 1429 1452 U75 0.90040 91099 92170 1059 1071 T082 32 00 32 20 32 40 0.66343 67302 68272 959 970 981 0.62487 63299 64117 812 818 824' 43 43 43 00 20 40 1.05395 06894 08418 1499 1524 1550 0.93252 94345 95451 1093 1 106 1118 44 TABLE III.— Continued. 44 oo 44 20 44 40 45 00 45 20 45 40 m 1.09968 I-II544 1.13148 1. 14779 1. 16439 1.18129 46 00 1. 19849 46 20 46 40 47 00 47 20 47 40 48 00 48 20 48 40 49 00 49 20 49 40 50 00 50 20 50 40 51 00 51 20 51 40 r. 21600 1.23384 1. 25201 1-27053 1.28940 1.30863 1.32823 1-34823 1.36863 1.38944 1. 41068 1.43236 1.45450 1. 47710 1. 50019 1-52379 I-54791 Diff. 1576 1604 1631 1660 1690 1720 1751 1784 1817, 1852 1887 1923 i960 2000 2040 2081 2124 2168 2214 2260 2309 2360 2412 2466 Tan S 0.96569 97700 98843 1. 00000 1.01170 1-02355 1-03553 1.04766 1.05994 1.07237 1.08496 1.09770 1.II06 1. 12369 1. 13694 1. 15037 r. 16398 1. 17777 1.19175 1-20593 1. 22031 1.23490 1.24969 1. 26471 Diff. 1131 1 143 1157 1170 1185 1213 1228 1243 1259 1274 129 1308 1325 1343 1361 T379 1398 1418 1438 1459 1479 1502 1523 52 00 52 20 52 40 53 00 53 20 53 40 54 00 54 20 54 40 55 00 55 20 55 40 56 00 56 20 56 40 57 00 57 20 57 40 58 00 58 20 58 40 59 00 59 20 59 40 60 00 (^) Diff. 1-57257 1.59779 1.62357 1.64995 1.67696 1.70460 1. 73291 1.76191 1. 79162 1.82207 1-85329 1.88530 1.91815 1.95186 1.98646 2.02199 2.05849 2.09600 2.13456 2.1742 2.21500 2.25697 2.3001 2.34468 2-39053 Tan^ Diff. 2522 2578 638 2701 2764 2831 2900 2971 3045 3122 3201 3285 3371 3460 3553 3650 3751 3856 3965 4079 4197 4321 4450 4585 4726 1.27994 1547 1. 29541 1569 1.31110 1594 1.32704 1619 1-34323 1645 1.35968 1670 1.37638 1.39336 1.41061 1.42815 1.44598 1.46411 1.48256 1.50133 1.52043 1.53986 1.55966 1.57981 1.60033 1. 62125 1.64256 1.66428 1.68643 1. 70901 1.73205 1698 1725 1754 1783 1813 1845 1877 1910 1943 1980 2015 2052 2092 2131 2172 2215 2258 2304 2351 45 THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO 50 CENTS ON THE FOURTH DAY AND TO $1.00 ON THE SEVENTH DAY OVERDUE. S"^' '.1 V2^ ^2M- i\oM ST 184S <'"?vB5lB Apr'59BBl V-V MAR 26 1989 OCT 3.) 1943 FEB 1 1944 S^ ^ea JUL 16 li-40 Ak^m^-^ ^■Ttj^ftl ^^ 2e3w.»'^ * "itV t ^t. 4 db'^ LD 21-100m-7,'40 (6936s) LIBRARY USE RETURN TO DESK FROM WHICH BORROWED LOAN DEPT. THIS BOOK IS DUE BEFORE CLOSING TIME ON LAST DATE STAMPED BELOW JilKf^^' i 4Rfi2 IBo'- JUN 61984 REC C1RMAY2 9 1984 SENT ON It I JUL 1 2 1 9 94 PNI U. C. BERKELEY LD 62A-20m-9,'63 (E709slO)9412A General Library University of California Berkeley