N^ THE GROUNDWORK OF PRACTICAL NAVAL GUNNERY THE GROUNDWORK OF PRACTICAL NAVAL GUNNERY A Study of the Principles and Practice of Exterior Ballistics, as Applied to Naval Gunnery And of the Computation and Use of Ballistic and Range Tables By PHILIP R. ALGER Professor of Mathematics, U. S. Navy Revised and Extended to Include the Formulas and Methods of Colonel James M. Ingalls, U. S. Army By the Officers on Duty in the Department of Ordnance and Gunnery United States Naval Academy 1914-15 ANMAPOLIS, MD. THE UNITED STATES NAVAL INSTITUTE 1915 ^^ COPTRIGHT, 1915, BY P. C. ALLEN Sec. and Tkeas. U. S. Naval Institute c^ Column 9 Column 3, Column 4. Column 5. Column 6. Column 8. Column 10. Column 11. Column 12. PREFACE. DEPAETMENT OF OKDNANCE AND GUNNERY. U. S. Naval Academy, Annapolis, Md., January 1, 1015. 1. Experience with the course of instruction of midshipmen in the science of exterior ballistics seemed to indicate that some modification of that course was neces- sar}', especially in view of the fact that the Bureau of Ordnance in its work uses the formulcB and methods devised by Colonel James ^I. Ingalls, U. S. Army, in the place of those given by Professor Philip K. Alger, U. S. Navy, in the text book prepared by him and used at the Naval Academy up to this date, for the computation of the data contained in the following columns of the range tables prepared by the Bureau of Ordnance and officially issued to the service: Angle of departure. Angle of fall. Time of flight. Striking velocity. Drift. Maximum ordinate. Change of range for variation of ± 50 foot-seconds initial velocity. Change of range for variation of ± Aw in weight of projectile. Change of range for variation of density of air of ± 10 per cent. 2. Exterior ballistics is taught at the Naval Academy in order that the midship- men, when they graduate and become officers, may be as familiar with the range tables, with the data contained in them, and with the methods by which this data is obtained and used ; in other words, with the use of the information which is furnished to aid them in using the gams successfully; as the time available for this instruction will permit. This being the case, everything has been omitted that does not bear directly upon the point assumed to be at issue, and no formulae have been retained that are not of use in connection with the practice of naval gunnery and the use of the range tables and of the gun afloat, or in the computation of the data contained in the range tables; except that certain preliminary formulae and discussions have necessarily been retained as vitally requisite for a thorough understanding of the later and mor^ practical parts of the work. An effort has been made to reduce the mathematical investigations to the lowest possible minimum consistent with a clear understanding of the practical portions of the work, in accordance with the views officially expressed by the Navy Department, by the Superintendent of the Naval Academy, and by the Academic Board ; but the subject is one that is almost purely mathematical, and which requires considerable preliminary mathematical work (as is the case with the science of navigation) before the practical features can be properly understood. The preliminary discussions of the trajectory have therefore been retained, but have been restricted as much as possible ; and, throughout, the work of revision has been carried out with the sole object in view of giving to the midshipmen, in the shortest possible time, a thorough practical knowledge of the underlying principles of naval gunnerv^ and of the computation and use of the range tables. 3. In brief, the advantages sought by this revision of the text book previously in use are: 339502 6 ' '• PREFACE (a) An arrangement that would appear more logical and consecutive to a mid- shipman taking up the study of the subject for the first time. (b) A more clear distinction between the methods and formula? that are purely educational and those that are actually used in practice. (c) The rendering more easily understood of quite a number of points in the older text book that seemed in the past to give great trouble to the midshipmen in their study of the subject. (d) The modification of the problems given in the text book to make them apply to modern United States Naval Ordnance. The problems in the older text book dealt largely with foreign ordnance, and exclusively with guns, projectiles, velocities, etc., that are now obsolete, or nearly so ; and, while many of the older problems have been retained as valuable examples of principles, a large number of problems dealing with present-day conditions and ordnance have been added. (e) An effort has been made to give a complete discussion of the practical use of the range tables, a subject but lightly touched upon in the older text book. In order to accomplish this a large number of officers, in the Atlantic fleet and elsewhere, were requested to contribute such knowledge as they might have on this subject, and the matter received from them has been incorporated in the chapter on this subject. The discussion of this point should therefore include all the most up-to-date practice in the use of the range tables. 4. In preparing this revision for the purpose indicated in the preceding para- graphs, the logical treatment of the subject seemed to indicate its division under two general heads, as follows : (A) The treatment of the trajectory as a plane curve; which, in turn, logically subdivides itself under two sub-heads, as follows : (a) General definitions, etc.; the trajectory in vacuum; the resistance of the air and the retardation due thereto ; the ballistic coefficient in its fullest form ; the equa- tion to the trajectory in air under certain specified and limited conditions ; and the approximate determination of the elements of the trajectory by the use of the above special equation. In other words, the features that are of educational rather than of practical value, but which are necessary to an understanding of the practical methods that are to follow. (b) The more exact and practical theories and formulae; that is, the ones that are generally used in practical work. This subdivision is not a rigid one, as it will be seen that some of the approximate formulae and methods are sufficiently accurate to permit of their use in practice, and they are so used ; but the general statement of the subdivision may be accepted as logical, with this one reservation. (B) A consideration of the variation of the actual trajectory from a plane curve, which treats of the influence upon the motion of the projectile of drift and wind, and of the effect upon the fall of the projectile relative to the target of motion of the gun and target. That is, having treated under the first division those computations that are not materially affected by the variations of the trajectory from a plane curve ; in the second division we treat of the effect of such deviations upon accuracy of fire. In other words, there are here to be discussed the steps taken to overcome the inaccu- racies in fire caused by the variation of the trajectory from a plane curve, in order to hit a moving target with a shot fired from a gun mounted on a moving platform, when there is a wind blowing. 5. Following these natural and logical divisions of the subject comes a' full discussion of the range tables, column by column, and of the methods of computing the data contained in them, and of using this data after it has been computed and tabulated. PEEFACE 7 6. Following this comes a consideration of the processes necessary to ensure that the guns of a ship shall be so sighted that the shot in a properly aimed salvo will fall well bunched. 7. Following this again, comes naturally a consideration of the inherent errors of guns, and of the accuracy and prol)al)ility of fire. 8. In accordance with the preceding statement of the natural and logical order in which the subject should be treated, this revised text book is therefore divided into six parts, as follows: PAET I. Chapters 1 to 5 Inclusive, general and approximate deductions, Preliminary definitions and discussion. The trajectory in vacuum. The resist- ance of the atmosphere to the passage of a projectile through it, and the retardation in the motion of the projectile resulting from this resistance. The ballistic coefiBcient. The equation to the trajectory in air when Mayevski's exponent is taken as having a value of 2; and a comparison between this equation and that to the corresponding trajectory in vacuum. The derivation from this special equation to the trajector}' in air of certain expressions for the approximate determination of the values of the elements of the trajectory. PAET II. Chapters 6 to 12 Inclusive. PRACTICAL methods. The computation and use of the ballistic tables, and of the time and space integrals. The differential equations giving the relations between the several ele- ments of the trajectory in air. Siacci's method. The time, space, altitude and inclination functions, and their computation and use. The ballistic formulae. Ballistic problems. PAET III. Chapters 13 to 15 Inclusive. THE variation OF THE TRAJECTORY FROM A PLANE CURVE. The variation of the trajectory from a plane curve; the forces that cause this variation ; and the consideration that it is necessary to give to it in the computations of exterior ballistics and in the use of the gun. Drift and the theor}^ of sights. The effect upon the travel of the projectile of wind and of motion of the gun; and the effect of motion of the target upon the fall of the projectile relative to the target. PAET IV. Chapters 16 to 17 Inclusive. RANGE tables ; THEIR COMPUTATION AND USE. The computation of the data contained in the range tables and the practical methods of using this data. 8 PREFACE PART V. Chapters 18 to 19 Inclusive, the calibration of single guns and of a ship's battery. The determination of the error of the setting of a sight for a given range ; the adjustment of the sight to make the shot fall at a given range ; and the sight adjust- ments necessary to make all the guns of a battery or ship shoot together. PART VI. Chapters 20 to 21 Inclusive. THE accuracy AND PROBABILITY OF GUNFIRE AND THE MEAN ERRORS OF GUNS. The errors and inaccuracies of guns. The probability of hitting under given conditions, and whether or not it would be wise to attempt to hit under these con- ditions. The number of shots probably necessary to give a desired number of hits under certain given conditions, and the bearing of this point upon the wisdom of attempting an attack under the given conditions, having in mind its effect upon the total amount of ammunition available. The probabilities governing the method of spotting salvos by maintaining a proper number of " shorts." 9. No claim is made to originality in any part of this revision; it is merely a compilation of what is thought to be the best and most modern practice from the works of two noted investigators of the subject, namely. Professor Philip R. Alger, U. S. Navy, and Colonel James M. Ingalls, TJ. S. Army. This revised work is based on Professor Alger's text book on Exterior Ballistics (edition of 1906), and the additions to it concerning the Ingalls methods are from the Handbook of Problems in Exterior Ballistics, Artillery Circular N, Series of 1893, Adjutant General's Office (edition of 1900), prepared by Colonel Ingalls. Further information has been taken from Bureau of Ordnance Pamphlet No. 500, on the Methods of Computing Range Tables. The two chapters on the calibration of guns were taken from a pamphlet on that subject written by Commander L. M. Nulton, U. S. Navy, for use in the instruc- tion of midshipmen, and Chapter 15 was furnished by officers on duty at the Naval Proving Ground at Indian Head. 10. To summarize, it may be said that the belief is held that the only reason for teaching the science of exterior ballistics to midshipmen and the only reason for expecting officers to possess a knowledge of its principles is in order that they may intelligently and successfully use the guns committed to their care. So much as is necessary for this purpose is therefore to be taught the undergraduate and no more ; and, as the information necessary for the scientific use of a gun is contained in the official range table for that gun, it may be said that this revision of the previous text book on the subject has been founded upon the question : " What is a range table, how is the information contained in it obtained, and how is it used ? " With a very few necessary and important exceptions such as the computations neces- sary in determining the marking of sights, the text of the book follows closely the question laid down above. 11. For the use of the midshipmen in connection with this revised text book, a reprint has been made of Table II for the desired initial velocities, from the Ballistic Tables computed by Major J. M. Ingalls, U. S. Army, Artillery Circular M, 1900; a reprint of the table from Bureau of Ordnance Pamphlet No. 500, for use in con- nection with Column 12 of the Range Tables; and a partial reprint of the Range PREFACE 9 Tables for Naval Guns issued by the Bureau of Ordnance; these being in addition to the tables previously available for midshipmen in the older text book. It is believed that these reprints will render available a range of practical problems far exceeding anything that has heretofore been possible for the instruction of midshipmen, and that by their use they will be graduated and commissioned with a far wider knowledge of the meaning and use of these tables, and of the underlying principles of naval gunnery than has ever been the case before. 12. The work of revision was done by the Head of Department, assisted by certain officers of the Department, and by criticisms and suggestions from numerous other officers, on duty elsewhere as weJl us at the Naval Academy.. Thanks are due to all those who helped in the work, and especially to Lieutenant (j. g.) C. T. Osburn, U. S. N., Lieutenant (j. g.) W. S. Farber, U. S. N"., and Lieutenant (j. g.) N". L, Nichols, U. S. N. Lieutenant Osbum carefully scrutinized the text, checked all sample problems worked in the text, and independently worked and checked the results of all the examples given at the ends of the chapters Lieutenants Farber and Nichols checked all the solutions given in the appendix, and Lieutenant Nichols prepared the drawing for all the figures. L. H. CHANDLER, Captain, U. S. Navy, Head of Department. CONTENTS. PEELIMINAEY. CHAPTER PAGE Introductory Order 5 Table of Symbols Employed 13 Table of Letters of Greek Alphabet Employed as Symbols 17 PART I. General and Approximate Deductions. Introduction to Part 1 19 1. Definitions and Introductory Explanations 21 2. The Equation to the Trajectory in a Non-Resisting Medium and the Theory of the Rigidity of the Trajectory in Vacuum 27 3. The Resistance of the Air, the Retardation Resulting Therefrom, and the Ballistic Coefficient 35 4. The Equation to the Trajectory in Air when Mayevski's Exponent is Equal to 2. 47 5. Approximate Determination of the Values of the Elements of the Trajectory in Air when Mayevski's Exponent is Equal to 2. The Danger Space and the Computation of the Data Contained in Column 7 of the Range Table 54 PART II. Practical Methods. Introduction to Part II 67 6. The Time and Space Integrals; the Computation of their Values for Different Velocities, and their Use in Approximate Solutions 69 7. The Differential Equations Giving the Relations Between the Several Elements of the General Trajectory in Air. Siacci's Method. The Fundamental Ballistic Formulae. The Computation of the Data Given in the Ballistic Tables and the Use of the Ballistic Tables 80 8. The Derivation and Use of Special Formulae for Finding the Angle of Departure, Angle of Pall, Time of Flight, and Striking Velocity for a Given Horizontal Range and Initial Velocity; that is, the Data Contained in Columns, 2, 3, 4 and 5 of the Range Tables. Ingalls' Methods 91 9. The Derivation and Use of Special Formulae for Finding the Coordinates of the Vertex and the Time of Flight to and the Remaining Velocity at the Vertex, for a Given Angle of Departure and Initial Velocity, which Includes the Data Contained in Column S of the Range Tables Ill 10. The Derivation and Use of Special Formula for Finding the Horizontal Range, Time of Flight, Angle of Fall, and Striking Velocity for a Given Angle of Departure and Initial Velocity 120 11. The Derivation and Use of Special Formulae for Finding the Angle of Elevation Necessary to Hit a Point Above or Below the Level of the Gun and at a Given Horizontal Distance from the Gun, and the Time of Flight to, and Remaining Velocity and Striking Angle at the Target; Given the Initial Velocity 125 12. The Effect Upon the Range of Variations in the other Ballistic Elements, which Includes the Data Contained in Columns 10, 11, 12 and 19 of the Range Tables 131 12 CONTENTS PART III. r,-rrAT,rn^T, The Vaeiation of the Tkajectory from a Plane Curve, „.^„ Introduction to Part III 143 13. Drift and the Theory of Sights, Including the Computation of the Data Con- tained in Column 6 of the Range Tables 145 14. The Effect of Wind Upon the Motion of the Projectile. The Effect of Motion of the Gun Upon the Motion of the Projectile. The Effect of Motion of the Target Upon the Motion of the Projectile Relative to the Target. The Effect Upon the Motion of the Projectile Relative to the Target of all Three Motions Combined. The Computation of the Data Contained in Columns 13, 14, 15, 16, 17 and 18 of the Range Tables 153 15. Determination of Jump. Experimental Ranging and the Reduction of Observed Ranges 166 PART IV. Range Tables; Their Computation and Use. Introduction to Part IV 1G9 16. The Computation of the Data Contained in the Range Tables in General, and the Computation of the Data Contained in Column 9 of the Range Tables. . 171 17. The Practical Use of the Range Tables 180 PART V. The Calibration of Single Guns and of a Ship's Battery. Introduction to Part V 213 18. The Calibration of a Single Gun 215 19. The Calibration of a Ship's Battery 226 PART VI. The Accuracy and Probability of Gunfire and the Mean Errors of Guns Introduction to Part VI 231 20. The Errors of Guns and the Mean Point of Impact. The Equation of Probability as Applied to Gunfire when the Mean Point of Impact is at the Center of the Target 233 21. The Probability of Hitting when the Mean Point of Impact is Not at the Center of the Target. The Mean Errors of Guns. The Effect Upon the Total Ammunition Supply of Efforts to Secure a Given Number of Hits Upon a Given Target Under Given Conditions. Spotting Salvos by Keeping a Certain Proportionate Number of Shots as " Shorts " 246 APPENDIX A. Forms to be Employed in the Solution of the Principal Examples Given in this Text Book 261 APPENDIX B. Description of the Farnsworth Gun Error Computer 315 APPENDIX C. The Use of Arbitrary Deflection Scales for Gun Sights •■■■ 326 Atmospheric Density Tables 331 TABLE OF SYMBOLS EIMPLOYED. 1. The symbols employed in this book are given in the following table. So far as possible they are those employed by the computers of the Bureau of Ordnance in their work of preparing range tables, etc. ; but a considerable number of additional symbols have been found necessary in a text book treatment of the subject. 2. In quite a number of cases it has been found necessary or advisable to use the same symbol to represent two or more different quantities ; but such quantities are, as a rule, widely diiferent from each other, and a reasonable amount of care will easily prevent any confusion from arising from this cause. 3. The symbols employed are : PERTAINING TO THE TRAJECTORY AS A PLANE CURVE. SYMBOL, R' X' R X w p j V V Vo Vh Vv U u (x, y) (J'o. 2/o) y t T to QUANTITY REPRESENTED UY IT. .Range in yards on an inclined plane. .Range in feet on an inclined plane. .Horizontal range in yards. .Horizontal range in feet. .Angle of departure. .Angle of fall. .Angle of elevation. .Angle of projection. .Angle of position. .Angle of jump. .Remaining velocity in foot-seconds at any point of the trajectory. .Initial velocity in foot-seconds. .Remaining velocity at the vertex in foot-seconds. .Remaining velocity at point of fall (or striking velocity) in foot-seconds. .Horizontal velocity at any point of the trajectory in foot-seconds. .Vertical velocity at any point of the trajectory in foot-seconds. .Pseudo velocity at any point of the trajectory in foot-seconds. .Pseudo velocity at the muzzle of the gun in foot-seconds; [7 = F. .Pseudo velocity at the vertex in foot-seconds. .Pseudo velocity at the point of fall in foot-seconds. . Coordinates of any point of the trajectory in feet. .Coordinates of the vertex in feet. .Ordinate of the vertex in feet; Y = y^- .Elapsed time of flight from the muzzle to any point of the trajectory in seconds. .Time of flight from muzzle to point of fall in seconds. .Time of flight from muzzle to vertex in seconds. .Angle of inclination of the tangent to the trajectory at any point to the horizontal. PERTAINING TO THE PROJECTILE. w. . . .Weight of the projectile in pounds. Aio. . . .Variation from standard in weight of projectile in pounds. d. . . .Diameter of the projectile in inches, c. . . .Coefficient of form of the projectile. C. .. .Ballistic coefficient. When written with numerical subscripts, as 1, 2, 3, etc., the several symbols represent successive approximate values of C as appearing in the computations. The same system of sub- scripts is used for a similar purpose with a number of other symbols. )3. . . .The Integration factor of the ballistic coefficient; normally /3 = 1. E . . . .Constant part of the ballistic coefficient for any given projectile, given by the formula K = w TABLE OF SYMBOLS EMPLOYED PERTAINING TO ATMOSPHERIC CONDITIONS. Sj. . . . Standard density of the atmosphere, in work taken as unity. 5 . . . . Density of the atmosphere at the time of firing, and subsequently repre- senting the ratio -^ = ,- . 5i 1 /. . . .Altitude factor of the ballistic coefficient. PERTAINING TO WIND AND SPEEDS. W . . . .Real wind; force in feet per second. (S. . . .Angle between wind ahd line of fire. Wx- . . .Component of W in line of fire in feet per second. Wiox. . . .Wind component in feet per second of 12 knots in line of fire. Wz Component of W perpendicular to line of fire in feet per second. Wi2«. . . .Wind component in feet per second of 12 knots perpendicular to the line of fire. X. . . .Range in feet without considering wind. X' . . . .Range in feet considering wind. T . . . .Initial velocity in foot-seconds without considering wind, y . . . . Initial velocity in foot-seconds considering wind. (^. . . .Angle of departure without considering wind. 0'. . . .Angle of departure considering wind. T. . . .Time of fiight in seconds without considering wind. T' . . . .Time of flight in seconds considering wind. AXir- . . .Variation in range in feet due to Wj- AXj,,r- ■• .Variation in range in feet due to a wind component of 12 knots in the line of fire. ARffr. . . .Variation in range in yards due to Wj. ARiiW- ■ ■ .Variation in range in yards due to a wind component of 12 knots in the line of fire. 7.... Angle between the trajectories relative to the air and relative to the ground. Dw ■ ■ .Deflection in yards due to W^. Dy.w ■ ■ .Deflection in yards due to a wind component of 12 knots perpendicular to the line of fire. G. . . .Motion of gun in feet per second. Gx- ■ ■ .Component of the motion of the gun in the line of fire in feet per second. G,2a;. . . .Motion of gun in line of fire in feet per second for a component of motion of gun in that line of 12 knots. Gz- ■ ■ .Component of the motion of the gun perpendicular to the line of fire in feet per second. Gi22. . . .Motion of gun perpendicular to line of fire in feet per second for a com- ponent of motion of gun of 12 knots in the same direction. AXq. .. .Variation in range in feet resulting from G^. AXyQ. .. .Variation in range in feet due to a motion of the gun of 12 knots in the line of fire. Ai?o. . . .Variation in range in yards resulting from Gx- Aliv.G- ■■ .Variation in range in yards due to a motion of the gun of 12 knots in the line of fire. Do. .. .Deflection in yards due to G-. Z)i„f;. .. .Deflection in yards due to the motion of the gun of 12 knots perpen- dicular to the line of flre. T. . . .Motion of target in feet per second. Tj. .. .Component of the motion of the target in the line of flre in feet per second. Tjoj. .... Motion of target in line of fire in feet per second for a component of motion of target in that line of 12 knots. T-. .. .Component of the motion of the target perpendicular to the line of fire in feet per second. Ti2« .... Motion of target perpendicular to line of fire in feet per second for a component of motion of target of 12 knots in the same direction. TABLE OF SYMBOLS EMPLOYED 15 AJr- • • .Variation in range in feet resulting from Tj-. AXi2T. ■ ■ .Variation in range in feet due to a motion of the target of 12 knots in tlie line of fire. ARt- ■ ■ .Variation in range in yards resulting from Tx- ARi2T- ■ ■ .Variation in range in yards due to a motion of the target of 12 knots in the line of fire. Dt- ■■ .Deflection in yards due to Tg. Di2T- •■ .Deflection in yards due to a motion of the target of 12 knots perpen- dicular to the line of fire, a. . . .Angle of real wind with the course of the ship, a' . . . . Angle of apparent wind with the course of the ship. W . . . .Velocity of the real wind in knots per hour. W". . . .Velocity of the apparent wind in knots per hour. PERTAINING TO THE THEORY OF PROBABILITY. X. . . .Axis of; axis of coordinates lying along range, for points over or short of the target. Y. . . .Axis of; axis of coordinates in vertical plane through target for points above or below the center of the target. Z....Axis of; axis of coordinates in vertical plane through target for points to right or left of the center of target. (^1, Vi), etc. . . .Coordinates of points of impact in vertical plane through target. ^z. . . . Summation of 2;,, z^, etc. 2y. . . .Summation of 1/1, y-,, etc. «... .Number of shot. yz. . . .Mean deviation along axis of Z, that is, above or below. yy. . . .Mean deviation along axis of Y, that is, to right or left. yx- ■ ■ .Mean deviation along axis of X, that is, in range. P. . . .Probability that the deviation of a single shot will be numerically less than the given quantity a. — . .. .Argument for probability table. 7 PERTAINING TO VARIATIONS IN THE BALLISTIC ELEMENTS. AX. . . .Variation in the range in feet. AR. . . .Variation in the range in yards. A(sin 2^) . . . .Variation in the sine of twice the angle of departure. ArA- ■ • .Quantity appearing in Table II of the Ballistic Tables in the Ar column pertaining to " A." With figures before the subscript Y it shows the amount of variation in V for which used. (Be careful not to con- fuse this symbol with AV or dV.) SY . . . .Variation in the initial velocity. (Be careful not to confuse this symbol with A,-,i or AV.) AV. . . .Difference between V for two successive tables in Table II. (Be careful not to confuse this symbol with Ay^ or dV.) AY j<,. .. .Variation in the initial velocity in foot-seconds due to variation in the weight of the projectile in pounds. Figures before the to show the amount of variation in that quantity in pounds. AX;r. .. .Variation in range in feet due to a variation in V in foot-seconds. Figures before the Y show the amount of variation in that quantity in foot-seconds. Ai^r- ■• -Variation in range in yards due to a variation in Y in foot-seconds. Figures before the Y show the amount of variation in that quantity in foot-seconds. AC . . .Variation in the ballistic coefl^cient in percentage. AXq. . . .Variation in range in feet due to a variation in C in percentage. Figures before the C show the amount of variation in that quantity in per- centage. Ai?c'- ■• -Variation in range in yards due to a variation in C in percentage. Figures before the C show the amount of variation in that quantity in percentage. 16 TABLE OF SYMBOLS EMPLOYED A5. . . .Variation in S in percentage. Ai'g Variation in range in feet due to a variation in 5 in percentage. Figures before the 5 sliow the amount of variation in that quantity in per- centage. ARg .... Variation in range in yards due to a variation in 5 in percentage. Figures before the 5 show the amount of variation in that quantity in percentage. Aw. . . .Variation in w in pounds. AXy, Variation in range in feet due to a variation in w in pounds. Figures before the w show the amount of variation in that quantity in pounds. AX'w That part of AXw in feet which is due to the reduction in initial velocity resulting from Aio. AX"w That part of AXy^ in feet which is due to Am; directly. Ai?,„ Variation in range in yards due to a variation in w in pounds. Figures before the w show the amount of variation in that quantity in pounds. AR\, That part of AR^ in yards which is due to the reduction in initial velocity resulting from A%o. AR"u, That part of A72,„ in yards which is due to Aw directly. H Change in height of point of impact on a vertical screen through the target, in feet, due to a change of AR in R in yards. Figures as sub- scripts to the H show the change in R necessary to give that par- ticular value of H. MATHEMATICAL AND MISCELLANEOUS. g Acceleration due to gravity in foot-seconds per second; g = 32.2. dx. . . .Differential increment in x. dy... .Differential increment in y. ds. . . .Differential increment along the curve, that is, in s. dv . . . .Differential increment in v. dt. . . .Differential increment in t. du. . . .Differential increment in u. a. . . .Mayevski's exponent. A Mayevski's coefficient. R. . . .Total air resistance in pounds. Rf... .Total air resistance in pounds under firing conditions. iSs--- -Total air resistance in pounds under standard conditions, p. . . .Radius of curvature of the trajectory at any point in feet. fc. . . .The value of -tt , in which A is Mayevski's constant and C is the ballistic coefficient. e. . . .The base of the Naperian system of logarithms; e = 2.7183. n. . . .The ratio between the range in vacuum and the range in air for the same angle of departure. Ty. . . .Value of the time integral in seconds for remaining velocity v. Tv ■■ .Value of the time integral in seconds for initial velocity Y. Sv ■ ■ .Value of the space integral in feet for remaining velocity v. *SV. . . .Value of the space integral in feet for initial velocity V. Tji. . . .Value of the time function in seconds for pseudo velocity u. Tv ■ ■ .Value of time function in seconds for initial velocity V. 8u- • • .Value of space function in feet for pseudo velocity u. Sy ■■ .Value of space function in feet for initial velocity V. Au- • ■ .Value of altitude function for pseudo velocity u. Ar- ■■ .Value of altitude function for initial velocity V. lu. . . .Value of inclination function for pseudo velocity u. Iy_ .. .Value of inclination function for initial velocity V. Su ... .Value of space function in feet for pseudo velocity u^. Su . . . .Value of space function in feet for pseudo velocity m„. Tu ... .Value of time function in seconds for pseudo velocity u . TABLE OF SYMBOLS EMPLOYED 17 /u„. AS. AT. AA. AI. z. a. h, a', t' . . A, B. A', T'.. . A ' and B = -r A x. ■ n n D' D I D d h E K K' n S a 7 (a. a') (b, 6') (c, c') (.d,d') . .Value of time function in seconds for pseudo velocity Uo. . .Value of altitude function for pseudo velocity u . . .Value of altitude function for pseudo velocity u„. . .Value of inclination function for pseudo velocity u . . .Value of inclination function for pseudo velocity Uo. . . Difference between two values of the space function. . . Difference between two values of the time function. . . Difference between two values of the altitude function. . . Difference between two values of the inclination function. . . General expression for value of argument in Column 1 of Table II of the Ballistic Tables, for any point of the trajectory; 2; = -^ . . .Special expression for value of argument in Column 1 of Table II of the Ballistic Tables, for the whole trajectory; Z = -^. . General values of Ingalls' secondary functions. i Special value of Ingalls' secondary functions for whole trajectory. . .Angle of departure for a horizontal distance x. . .A ratio used in computing the drift; in which k is the radius of gyration of the projectile about its longitudinal axis, and R is the radius of the projectile. . .A special ratio used in computing the drift. . . The twist of the rifling, used in computing the drift. . .Ingalls' secondary function for drift. . .Drift in yards. . .Sight radius in inches. . .Deflection in yards (used with R in yards in deflection computations). . . Distance in inches which the sliding leaf is set over to compensate for the, deflection D in yards. . . Permanent sight-bar angle. . . Sight bar height in inches. . .Penetration of armor in inches. . .Constant used in computing penetration of armor. . .Constant used in computing penetration of armor. . . Height of target in feet. . .Danger space in general. • |-.\ngles for plotting fall of shot in calibration practice. Coordinates of point of fall of shot for plotting in calibration practice. LETTERS OF THE GREEK ALPHABET USED AS SYMBOLS. Letter. Pronunciation. .Alpha. .Beta. 7. . . .Gamma. A or S. . . .Delta. . .Epsilon. Letter. Pronunciation. Lett€ r. Pronunciation. e. . ..Theta. S or a. . . . Sigma. X.. . .Lambda. <*. ...Phi. fj.. . ..Mu. ^- ...Psi. p.. ..Pi. ..Rho. u , . . .Omega. PAET I. GENERAL AND APPROXIMATE DEDUCTIONS. INTRODUCTION TO PART I. Part I of this text book includes the preliminary definitions and discussions, from which we pass to the derivation of the equation to the trajectory in a non- resisting medium, and thence to the principle of the rigidity of the trajectory in vacuum. As the next step comes the discussion of the resistance of the atmosphere to the passage of a projectile through it, and of the retardation in the motion of the pro- jectile resulting from this resistance; and following this, a consideration of the ballistic coefficient in all its forms ; and following this again, a statement of Mayevski's general and special formula for retardation, that is, of the general formula, and of the numerical results of experiments to determine the resistance and retardation. We are then prepared to take up the derivation of the equation to the trajectory in air, having determined the forces that act upon a projectile in flight. We will find that it is only possible to determine such an expression for certain given velocities, but having thus determined it, we will be able to get a comparison between the trajectories in vacuum and in air under similar conditions (for the special conditions under which we were able to derive the equation to the trajectory in air). We can also derive certain formulae for determining approximate values of the elements of the trajectory in air, by the use of the equation above described. This completes the preliminary, educational and approximate consideration of the trajectory as a plane curve, and enables us to pass on, in Part II, to the more practical methods actually employed by artillerists in making such computations as are necessary for the successful use of guns. It must be noted, however, that some of the approximate methods given in Part I give results that are sufficiently accurate to render them available for general use, and that these results are actually so used in practice. CHAPTEE 1. DEFINITIONS AND INTRODUCTORY EXPLANATIONS. Symbols Introduced. B' . . . . Eange in yards on an inclined plane. X' . . . . Range in feet on an inclined plane. 11. . . . Horizontal range in yards. X . . . . Horizontal range in feet. • (^. . . .Angle of departure. tu . . . . Angle of fall. p. . . . Angle of position. ;'.... Angle of jump. i// . . . . Angle of elevation. i//' . . . . Angle of projection. V . . . . Initial velocity in foot-seconds. V. . . .Eemaining velocity at any point in the trajectory in foot-seconds, t'o . . . . Eemaining velocity at the vertex in foot-seconds. Vcj. . . .Eemaining velocity at the point of fall, or striking velocity, in foot- seconds. Vh .... Horizontal velocity at any point of the trajectory in foot-seconds. Vf. . . .Vertical velocity at any point of the trajectory in foot-seconds. u. . . .Pseudo velocity at any point of the trajectory in foot-seconds. U . . . . Pseudo velocity at the muzzle of the gun in foot-seconds; U= V, Vq. . . .Pseudo velocity at the vertex in foot-seconds. Uu. . . . Pseudo velocity at the point of fall in foot-seconds. 1. Ballistics is the science of the motion of projectiles, and is divided into two Definitions, branches ; namely, interior ballistics and exterior ballistics. 2. Interior ballistics is that branch of the science which treats of the motion of the projectile while in the gun and of the phenomena which cause and attend this motion. 3. Exterior ballistics is that branch of the science which treats of the motion of the projectile after it leaves the gun. The investigations to be conducted under exterior ballistics therefore begin at the instant when the projectile leaves the muzzle of the ffuu. 22 EXTERIOE BALLISTICS 4. There are certain definitions connected with the travel of the projectile after it leaves the gun, and certain symbols which are used to represent the quantities covered by these definitions. These definitions and symbols will now be given. Figure 1. Elements of Tbajectort. BOII:^ = Ang]e of Departure. fl^=: Point of Fall, Horizontal Range. DnO = w = Ang\e of Fall. Oi/ =3 X= Horizontal Range. Oi/ ^Line of Position. JI/Off:=p = Angle of Position. D'ME=:w' — Ansle of Fall. OJ/ = X' = Range. 50if :=!/''=: Angle of Projection. BOA^j:^ Angle of jump. AOM = \p = Angle of Elevation.

is the angle of departure ; H the point of fall on the horizontal plane; DHO = w is the angle of fall; and OH = X (or E) is the horizontal range. If the target be at M, then OM is the line of position; MOH = p is the angle of position; D'ME = o/ is the angle of fall; and OM = X' (or R') is the range. BOM = xl/' is the angle of projection, which coincides with the angle of departure, , when the angle of position, p, is zero. OA represents the position of the axis of the bore at the instant before firing, and OB its position at the instant the projectile leaves the muzzle, the angle between them, BOA=i, being the a;\?le of jump. The angle AOM = xp is the angle of elevation. It will be seen that and V cos — .-.t/o^^ ■> . (^) ^ 2 y ^ cos- (f> and (2) is the equation to the trajectory in vacuum, which trajectory, from the form of its equation, is evidently a parabola with a vertical axis. 36. From the above equation various expressions may be derived from which we can readily determine the values of the different elements of the curve. Angle of 37. Differentiatnig (2) and putting tan (9 for ^, we get inclination. ^^ tan 6 = tan d) — --„ -2— -— (3) V'^ cos^ (j> which gives the inclination of the curve to the horizontal at any point. Horizontal ' 38. Putting ^ = in (2), wc find two values of x, the first zero, and the second range, -r^o • -p j, sm /i(p consequently the horizontal range, OH, is given by 9 -^^ V^sm2 .^. This shows that, for a given initial velocity, the range increases with the angle of V- departure up to ^ = 45°, when it reaches its maximum value of — ; and that the same range is given by either of two angles of departure, one as much greater than 45° as the other is less than 45°. Variations 39. From (3) we see that, as x increases, d decreases from its initial value of <^, in angle of q^ inclination. ^^^\^i\ j^ becomes zero when tan 4>= ,.„ ^ „ , : that is. when V^ cos^ ^ _ 72 cos^ _ 7^ cos^ 4* sin "^ _ 7^ sin 4 cos 4 ^~ g ~ gcoscj> g 7^ sin 2; and that for x = X, 6= —(f>. That is to 7^ cos- say, the highest point or vertex, S, of the curve is midway of the range, and the angle of fall, w, is equal to the angle of departure, 4. * Sin 2(p = 2 sin ^ cos . GENERAL AND APPROXIMATE DEDUCTIONS 29 40. The maximum ordinate is obtained by putting x=j= ^^ in (3), ^^"J^^t'e^*^^ whence we have for the coordinates (x^, y^) of the vertex _ V- sin 24> _X _ Y^ sin- (^ _ X tan /kx ""^ 2g 3" y^- 2g 4 ^""^ This value of y^ is directly given from the fact that the vertical component of the initial velocity is V sin <^, since the height to which a body will rise in vacuum is equal to the square of the vertical velocity divided by 2g. 41. The vertical and horizontal components of the velocity at any time being Remaining respectively -^^ = T' sin (f> — gt and -^ =V cos (f>, we have for the remaining velocity at any point (x, y) = V'- - 2gtV sin <^ + g't- = V- - 2g (vt sin

duration of the trajectory, or time of flight, T, is given by rp_ X _ / 2Z tan .r^\ Vcos~ y g ^ ' The second of these values is obtained directly from the consideration that if gravity did not act the projectile would rise to the height (X tan ^) while moving X hori- zontally, and so the distance which it falls from the tangent to the curve at the origin under the action of gravity during the time of flight T is given by X tan = yT'' 43. To determine the range, X', on an inclined plane, let (x^, y^) be the coordi- Range on nates of the target M, in Figure 2, p being the angle of position, and i/^' = <^ — p the angle of projection. Then from (2) we get 27- 9 sin cos p — cps (f) sin p _ sin( — ^) tan o= ^ =tan — tan p — cos <^ cos p cos (f) cos p Therefore x,= ^-^ X '^^i-P)^0' = ^ T^ x s^^fcos(f + p) g cos p g cos p Whence, since X' — x^ sec p, we have X'=^^' X sinf cos(f + ;?) /g) g cos^ p * Sin' tt> + cos' = 1. 30 EXTEEIOR BALLISTICS dtpfrture ^^- "^^ determine the angle of departure necessary to give a given range on an on an inclined plane, we have incline. ^ ' But x^ = X' COS p and y^ = X' sin p; therefore X' sin p = X' cos p tan cA- ?^^^^ SK^cos-t^ or 2X'V'^ cos p sin <^ cos <^ — 2X'y^ sin p cos^ — gX'^ cos- p = But sin ^ cos 0=-J sin 3 and cos- ^ = |(l + cos 2^), therefore X'y^(sin 2^ cos p — cos 2^ sin p) =gX'^ cos^ p + Z'F^ sin p whence sin(2(/) — /)) = ^^ cos^ p + sin p (9) 45. If p were zero, as the angle of departure was \p', we would have for the hori- zontal range from (8), X= • sin i/^' eosxp', and so the ratio of the range on an inclined plane to the horizontal range, for the same angle of projection, would be ^= ^^^^f'^f) =secp{l-tsind,'tsinp) (10) X cos ij/ cos^ p The value of the second member of (10) is very nearly unity so long as ij/' and p are small angles, being, for example, 0.9992 for i/^' = 3°, p = 5° ; and it therefore follows that, with small angles of projection and position, we may consider the range as independent of the angle of position. Figure 3 Theory of 46. The assumption in the last line of the preceding paragraph is called the of the assumption of the rigidity of the trajectory, and evidently consists in supposing that the gun, trajectory, and line of position (a chord of the trajectory) may be turned through a vertical angle, as illustrated in Figure 3, without any change of form. The trajectory is assumed to be rigid in practice when the same sight graduations are used GENEEAL AND APPEOXIMATE DEDUCTIONS 31 in firing a gun at objects at different heights, or when the gun itself is at different heights relative to the target. 47. As an example of the mathematical work relative to the trajectory in vacuum, and to show the proper logarithmic forms for such work, suppose we have given, in vacuum, an angle of elevation (i/') of 7° 50', an angle of jump (;) of +10', an initial velocity of 2600 f. s., and an angle of position (p) of zero; and desire to compute the horizontal range, time of flight, striking velocity, angle of fall, coordi- nates of vertex, and time to and remaining velocity at the vertex. (Use Table VI where convenient.) X Z tan ^o=-2-; i\=vco^<^ F = 2600 2 log 6.82994 colog 6.58503 — 10 log 3.41497 ^ = 8° 00' sec 0.60425 tan 9.14780—10. ..cos 9.99575 — 10 20 = 16° 00'.. .. sin 9.44034 — 10 Sr=i32.2 colog 8.49214—10 4 colog 9.39794—10 JCi= 57865.5'... log 4.76242 log 4.76242 log 4.76242 7' = 22.475 log 1.35170 3/0 = 2033.1 log 3.30816 v^ = 2574.65 log 3.41072 Eesults. 7? = 19388.5 yards. x^ = 9644.25 yards. r = 22.475 seconds. ^o = 3033.1 feet. (0 = 8° 00'. ^0 = 11.2375 seconds. i'^ = 2600f. s. -^o = 2574.65 f. s. 48. And, again, suppose the angle of position (p) to be 30°, the angle of pro- jection (i{/') to be 2° 25' 45", and the initial velocity 2900 f. s. ; and it is desired to find the range on the incline and the time of flight, both in vacuum. X' = ^^ sin ^' cos (^' + /^) . CO _^ . therefore, x=X' cos p, and g cos^ p X t=^ , whence T = ^' ^^^ f , where cf> = il^' + p 7 = 2900 2 log 6.92480 colog 6.53760-10 p = 30° 00' 00" sec 0.06247 2 sec 0.12494 cos 9.93753-10 f = 2° 25' 45" sin 8.62721-10 z=32° 25' 45" cos 9.92637-10. ... sec 0.07363 ^ = 32.2 log 1.50786. . . .colog 8.49214-10 2 log 0.30103 Z' = 24916.5 log 4.39649 log 4.39649 r = 8.8155 log 0.94525 Eesults. i2' = 8305.5 yards. T = 8.8155 seconds. 32 EXTERIOE BALLISTICS EXAMPLES. 1. If the initial velocity and angle of departure are as given in the first two columns of the following table, compute the horizontal and vertical components of the velocity at the point of origin, in vacuum. Give results obtained by both the use of logarithms and by the use of the traverse tables without logarithms. DATA. ANSWERS. Problem. Initial velocity. f. 3. Angle of departure. By logs. By traverse tables. Vh- f. s. f. s. Vh- f. s. f. s. 1 1000 1100 1250 1400 1500 1750 2000 2400 2600 2900 2° 00' 00" 3 15 42 4 25 16 5 10 25 10 12 14 7 30 00 5 00 00 9 21 15 12 37 54 17 24 24 999 1098 1246 1394 1476 1735 1992 2368 2537 2767 35 63 96 126 266 228 174 390 569 868 999 1098 1246 1394 1476 1735 1992 2368 2537 2767 35 2 63 3 96 4 126 5 265 6 229 7 174 8 390 9 569 10 868 Note. — It will be seen from the above that the traverse tables give the results correctly to the nearest foot-second, which is all that is required in ordinary work. 2. The data being as given in the first three columns of the following table, find the results, in vacuum, required by the other columns. DATA. ANSWERS. Problem. Initial velocity, f. s. Angle of eleva- tion. Angle of jump. Angle of depar- ture. Horizon- tal range. Yds. Time of flight. Sees. Angle of fall. Striking velocity. f. s. 1 1000 1100 12.'^0 1400 1.500 1750 2000 2400 2600 2900 5° 27' 4 32 3 33 2 15 7 22 8 12 12 37 7 50 3 07 16 35 + 7' + 3 — 3 — 5 + 6 — 7 — 10 + 3 + 5 5° 34' 4 35 3 30 2 10 7 28 8 12 12 30 7 40 3 10 16 40 1999 1995 1971 1533 6002 8951 17500 15767 7719 47840 6.03 5.46 4.74 3.29 12.11 15.50 26.89 19.89 8.92 51.66 5° 34' 4 35 3 30 2 10 7 28 8 12 12 30 7 40 3 10 16 40 1000 2 1100 3 1250 4 1400 5 1500 6 1750 7 2000 8 2400 9 2600 J 10 2900 GENERAL AND APPROXIMATE DEDUCTIONS 33 3. Given the initial velocities and angles of departure in the table below, com- pute the coordinates of the vertex, and the time to and remaining velocity at the vertex, in vacuum. . DATA. ANSWERS. Problem. Initial velocitv. f. s.' Angle of departure. Yds. Feet. to. Sees. f. s. 1 1000 1100 1250 1400 1500 1750 2000 2400 2600 2900 5° 34' 4 35 3 30 2 10 7 28 8 12 12 30 7 40 3 10 16 40 999 998 986 767 3001 4475 8750 7884 3860 23920 146 120 90 44 590 967 2910 1592 320 10742 3.01 2.73 2.37 1.64 6.05 7.75 13.44 9.94 4.46 25.83 995 2 1096 3 1248 4 1399 5 1487 6 1732 7 1953 8 2379 9 2596 10 2778 4. A body is projected in vacuum with r=1000 f. s., and an angle of departure of 30°. Where is it after 3 seconds ? Where after 10 seconds ? Ansivers. 3 seconds. a;=:1500V3 feet. t/ = 1355 feet. 10 seconds. a;=5000V3 feet. tj = 3390 feet. 5. A body is projected in vacuum from the top of a tower 200 feet high, with a velocity of 50 f. s., and an angle of departure of 60°. Find the range on the hori- zontal plane through the foot of the tower, and the time of flight. Answers. Range = 128 feet. Time = 5.13 seconds. 6. What is the angle of departure in vacuum in order that the horizontal range may be: (a) Equal to the maximum ordinate of the trajectory; and (b), equal to three times the maximum ordinate ? Answers, (a) 75° 57' 51". (b) 53° 07' 48". ^ 7. Compute the initial velocity and angle of departure in vacuum in order that the projectile may be 100 feet high at a horizontal distance from the gun of a quarter of a mile, and may have a horizontal range of one mile. Answers. For 1 mile = 5280 feet. 5° 46' 05". 922 f . s. For 1 mile = 6080 feet. 5° 00' 07". 1062 f. s. 8. The angle of position is 45°, the angle of projection is 1° 16' 31", and the initial velocity is 1500 f. s. Compute the range on the incline and the time of flight, in vacuum. Answers. i?'=: 1433 yards. 7 = 2.93 seconds. 9. What must be the angle of projection in vacuum for an initial velocity of 400 f. s. in order that the range may be 2500 yards on a plane that descends at an angle of 30° ? Answer. Angle of projection. 34° 35' 56" or 85° 24' 04". 10. A body is projected in vacuum with an angle of departure of 60°, and an initial velocity of 150 f. s. Compute the coordinates of its position and its remaining velocity after 5 seconds ; also the direction of its motion. Atiswers. a- = 375feet. 7/ = 247feet. ^=(-)22° 31'25". i; = 81f. s. 34 EXTERIOE BALLISTICS 11. A 12" mortar shell weighing 610 pounds, fired with an initial velocity of 591 f. s., and an angle of departure of 73°, gave an observed horizontal range in air of 1939 yards, and a time of flight of 36 seconds. What would the range and time of flight have been in vacuum ? Ansivers. 5 = 2022 yards. r = 35.10 seconds. 12. The measured range in air of a 12" shell of 850 pounds weight, fired with 2800 f. s. initial velocity, and an angle of departure of 7° 32', was 11,900 yards, and the time of flight was 19.5 seconds. What would the range and time of flight have been in vacuum ? Ansivers. i2=: 21,097 yards. r = 22.8 seconds. CHAPTEE 3. THE RESISTANCE OF THE AIR, THE RETARDATION RESULTING THERE- FROM, AND THE BALLISTIC COEFFICIENT. w. d. a. A. R. Rf. Rs' 8,. 8. c. C. /• /?■ Y. dv. dt. K. New Symbols Introduced. . . Weight of projectile in pounds. . . Diameter of projectile in inches. . . Mayevski's exponent, . . Mayevski's constant. . . Total air resistance in pounds. . .Total air resistance under firing conditions in pounds. . . Total air resistance under standard conditions in pounds. . . Standard density of air, taken as unity. . . Density of air at time of firing, and subsequently representing the ,.8 8 ratio Y" — 'i" • . , Coefficient of form of the projectile. . . Ballistic coefficient. , . Altitude factor. . . Integration factor. . .Maximum ordinate, or ordinate of the vertex, in feet; Y = yQ. . . Differential increment in v. . . Difi^erential increment in t. . . Constant part of ballistic coefficient for a given projectile; K=~-j^, 49. The first investigations of the resistance offered by the air to and the resultant retardation in the travel of the projectile were in the nature of practical experiments conducted from time to time by a number of persons, and indeed, although later mathematical investigations have led to a fuller understanding of the subject, the formulas still in use for the determination of atmospheric resistance and tlie retardation resulting from it are the results of these experiments ; in other words, they are partly empirical, and not purely mathematical. The list of men who con- ducted these experiments includes many names prominent in the records of scientific research, such as Tartaglia, Galileo, Newton, Bernouilli, Eobins, Count Eumford, Dr. Hutton, Wheatstone, Bashforth, Mayevski and Zaboudski. The results of Bash- forth's experiments, as expressed in formulas by Mayevski, and modified and extended by Zaboudski, form the basis upon which calculations for resistance and retardation still rest. 50. As it is manifestly simpler to determine experimentally the retardation pro- duced in the flight of a projectile than it is to attempt to measure the atmospheric pressure opposing its motion, the experiments have naturally taken that direction. To measure the retardation, a gun is given a very slight elevation, and the velocity of the projectile is measured at two points sufficiently far apart to make the two velocities (v^ and v^) appreciably different, and yet near enough together to ensure, as closely as possible, that the resistance of the air does not change from one point of measurement to the other. The two pairs of screens used for measuring the velocities should be at about the same level, so that the effect of gravity may be neglected. Measure- ment of retardation, 36 EXTEEIOR BALLISTICS Having determined these two velocities, it is evident that the retardation of the pro- jectile while traveling the distance, x, between the two points of measurement is Vj^ — v^. Let w he the weight of the projectile, R the resistance of the air in pounds (total resistance). Then, since the work done by the resistance must equal the loss of energy of the projectile, we have whence. K=^(V-«.') (11) where R is taken to be the total resistance of the air in pounds which corresponds to the mean velocity — ^ — ~ . As an example of the use of the above formula, suppose we have a 12" pro- jectile weighing 870 pounds, fired through two pairs of screens 300 feet apart, and the measured velocity at the first pair was 2819 f. s. and at the second pair was 2757 f. s. These measured velocities are the mean velocities for the spaces traversed between the two screens of each pair, that is, we may assume that each velocity is the velocity at the point midway between the two screens of its own pair. The distance given as 300 feet is the distance between the midway points of each pair of screens. Also, for determining the value of iVj^ — V2^), we know that v-^^ — v^^^ (v^ + Vo) (I'j — tjg), and the work becomes: v^ + V2 = 557G log 3.74632 v^-V2 = 62 log 1.79239 w = 870 log 2.93952 a:=300 log 2.47712 colog 7.52288-10 2^ = 64.4 log 1.80889 colog 8.19111-10 5 = 15567.5 pounds log 4.19222 in which R is the resistance for the mean of the two measured velocities, that is, for ^1 + ^2-2788 18. Expert- 51. As the result of many such measurements with different projectiles and results, different velocities, it has been shown that the resistance of the air is proportional to : 1. The cross-sectional area of the projectile; or, what is the same thing, the square of its diameter, which is the caliber. 2. The density of the air; or, what is the same thing, the weight of a cubic foot of the air. 3. A power of the velocity, of which the exponent varies with the velocity, but may be considered as a constant within certain limits of velocity. 4. A coefficient which varies with the velocity, with the form of the projectile, and with the assumed value of the exponent; but which may be considered as a con- stant for any given projectile between the same limits of velocity for which the exponent is considered as a constant. Mayevski's 52. In accordance with these four experimentally determined laws, we may write a general formula expressing the retardation of a projectile caused by the atmospheric resistance to its flight ; which is Mayevski's formula. It would be : Retardation = f^- = -A ^^ i'« ( 12) dt w in which a is Mayevski's exponent, A is Mayevski's constant coefficient, c is the formula, GENEEAL AND APPROXIMATE DEDUCTIONS 37 coefficient of form of the projectile, d the diameter of the projectile, w the weight of the projectile, and v the velocity. The acceleration, which is negative in this case, is of course represented by - ^ . 53. The quantity S in the above equation represents the ratio of the density of half-saturated air for the temperature of the air and barometric height at the time of firing to the density of half-saturated air for 15° C. (59° F.) and 750 mm. (39.5275") barometric height. The values of 8 for different readings of the barometer (in inches) and thermometer (in degrees Fahrenheit) may be found in Table III of the Ballistic Tables. 54. In the above expression, c is the coefficient of form of the projectile. It will be readily understood that if certain results are obtained with a projectile of a given shape, a change in the shape of the projectile will change the results. There- fore the factor c is introduced, and values for it for different projectiles are deter- mined experimentally, as explained later. 55. Mayevski adopted as standard the form of projectile in most common use at the time he conducted his experiments, which was one about three calibers in length, with an ogival head the radius to the curve of which ogive was two calibers, and for that projectile called the value of c unity. He also used a temperature of 59° F. and a barometric height of 29.5275" as standard, thus reducing the value of 8 to unity also. His general expression then becomes dv m By determining velocities experimentally as explained in paragraph 50, he proceeded, on this formula as a basis, to derive specific laws for finding the retardation at difl^erent velocities. 56. As the result of these experiments he derived the following expressions : w (13) V between 3600 f. s. and 2600 f. s. dv . d- ~dr~ ^^ w ^1.55 log li = 7.60905- -10 V between 2600 f. s. and 1800 f. s. dv . d'^ dt w ^1.7 log ^2 = 7.09620- -10 V between 1800 f. s. and 1370 f. s. dv _ A d- dt ~ ^ w ^2 log ^3 = 6.11926- -10 V between 1370 f. s. and 1230 f. s. dv . d- dt ~ ^* w ^3 log .1, = 2.98090- -10 V between 1230 f. s. and 970 f. s. dv ^ d- dt ~ ' w j;5 log 15 = 6.80187- -20 V between 970 f. s and 790 f. s. dv . d^ dt ~ « IV ^3 log .46 = 2.77344- -10 V between 790 f . s . and f . s. dv _ A d- dt ~ ^ ' w y2 log 1, = 5.66989 - -10 (14) 38 EXTERIOR BALLISTICS 57. From the above expressions, by using the appropriate one, the retardation for any velocity in foot-seconds may be calculated for the standard projectile and standard condition of atmosphere as adopted by Mayevski ; and thence for any other projectile or atmospheric condition by applying the proper multipliers. Of course the total resistance of the air in pounds (R) may be found by multiplying the mass of the projectile by the retardation, so we have R = A — v<'X~=A-^^~v^ (15) w g g 58. For instance, given a 6" shell weighing 105 pounds, traveling with a velocity of 2500 f. s. ; to find the resistance and retardation under standard atmospheric conditions, provided it be a standard shell. dv A d^ a -D A d"" „ -^rr = —A — V" R = A — v" at w g For 2500 f. s., Mayevski's constants are a=1.7 and log A = 7.09620 -10. t' = 2500 log 3.39794 loglog 0.53121 a = 1.7 log 0.23045 t'"= log 5.77640 ...loglog 0.76166 A= log 7.09620-10 ^' = 36 loo- 1.55630 ^(fV log 4.42890 log 4.42890 w = 105 log 2.02119 g = 32.2 log 1.50786 -— =-255.69 f. s log 2.40771 i2 = 833.75 pounds log 2.92104 In the ease of the experimental firing, suppose the above shell gave measured velocities of 2525 f . s. and 2475 f . s. at two points 488.88 feet apart, to find the resistance : ^= Yfx ("^'-^"> = Jgx^"^ + '^^ ("^-"^) Vj + V2 = 5000 log 3.69897 Vj-t;, = 50 log 1.69897 w = 105 log 2.02119 2^ = 64.4 log 1.80889 colog 8.19111-10 a; = 488.88 log: 2.68920 colog 7.31080-10 22 = 833.75 pounds log 2.92104 59. These results being only for standard conditions, we must introduce another factor if we desire results for any other conditions. This factor is known as the ballistic coefficient, and is denoted by C. It is a most important quantity to which great attention must be paid and which we must strive to thoroughly understand, for it enters constantly into nearly every problem in exterior ballistics. It represents the combination of the different elements already explained as well as some other elements which will now be discussed. 60. Introducing the ballistic coefficient, equation (12) becomes: — =-—«;« a6) dt C ^ ' In other words, the values resulting from the use of the specific formulae given in (14) must be divided by C for the individual case and conditions in order to get results for any other than standard conditions. GENERAL AND APPROXIMATE DEDUCTIONS 39 61. The value of C in its most complete form is given by the expression : Ballistic coefficient. in which r- iii^ (17) «' = the weight of the projectile in pounds. d = the diameter of the projeetile in inches (caliber of the gun). c = the coefficient of form of the projectile. / = the altitude factor. ;8 = the integration factor. 8 = the density of the air at the time of firing. Si = the adopted standard density of the air. 62. Considering these factors in detail : IV and d are standard characteristics of the projectile, and need no special remark. c is a quantity representing the form of the projectile. It may readily be con- ceived that for a projectile more sharply pointed than the standard its value would be less than unity and that the shell would suffer less retardation than the standard projectile (for the modern 12", long-pointed shell, for instance, c = O.Gl), whereas for a blunter shell, its value would exceed unity (for a flat-headed projectile the value of c is probably greater than 3) . It has been found that the value of c depends to some extent upon the smoothness and the length of the projectile, but that it depends primarily upon the shape of the head. Apparently the form of the head near its junction with the cylindrical body is also a most important' factor in deter- mining the resistance of the air to the flight of the projectile. The U. S. Navy method of determining the value of c for any projectile by experimental firing will be explained later. §1 represents the adopted standard density of the air, which has been taken as the density of half-saturated air for 15° C. (59° F.) and 750 mm. (29.5275") barometric height. 8 represents the density of half-saturated air for the given tem- perature and barometric height at the time of firing. Assuming that 8^ = 1 for the standard condition, Table III of the Ballistic Tables has been computed for the values of -s— for different readings of the thermometer in degrees Fahrenheit and ^1 8 8 8 barometer in inches; and as -^^then becomes ^r-, when this table is used, -«— may be f>i 1 Ol replaced by 8 in the formulae. Hereafter the symbol 8 will be used to represent this TW ratio, and (17) then becomes C' = ^^— ,-, in which 8 is taken from Table III. / is a factor which enters in cases in which we can no longer assume that the density of the air is the same at all points of the trajectory, owing to the fact that the path of the projectile rises to a considerable height above the level of the gun, or passes to a considerable vertical distance below the level of the gun, as when firing from an elevated battery down to the water. In such cases, /, which is known as the altitude factor, must enter into the computation of the value of the ballistic coefficient. / is the ratio of the density of the air at the gun to the mean density of the air through which the projectile actually passes. The mean height of the projectile during flight is ordinarily taken as two-thirds the height of the vertex of the trajectory, which would be exact were the trajectory in air a true parabola. This is therefore ordinarily practically correct, and the mean density of the air is taken to be the same as the density at a height of |1'. The value of / for any height in feet will be found in Table V of the Ballistic Tables. Weight and diameter. Coefficient of form. Density factor. Altitude factor. 40 EXTERIOR BALLISTICS For firing when gun and target are in approximately the same horizontal plane, correction for altitude is ordinarily a needless refinement for trajectories for which the time of flight does not exceed about 12 seconds. In computing range table data, the correction for altitude is generally started when such correction would produce a variation in the angle of departure of about one minute in arc. In all problems in which the vertical distance of the point aimed at above or below the horizontal plane of the gun is such that the rigidity of the trajectory cannot be Figure taken for granted, the ballistic coefficient should be corrected for altitude; and an examination of Figure 4 will show that the mean height of the trajectory from gun to target is less or greater than two-thirds the height of the target above the hori- zontal plane of the gun according as the target is nearer to or further from the gun than the vertex of the complete trajectory. Thus, if the target be at A^, the mean height of the arc OA^ is less than f^/^, and approaches -~- more and more as A-^ is nearer and nearer 0; while if the target be at Ao the mean height of the arc OA2 is greater than ^y^, being about equal to 2/2 when the abscissa of A, is fX. If, therefore, all possible refinements are to be introduced into the calculations, the relative positions of target and vertex must be determined before fixing the value of /. Generally speak- ing, however, it will be sufficiently accurate to give / the value corresponding to . .Y-Y. FlGUEE 5. The two preceding paragraphs explain how to determine the value of / for either : (1) A horizontal trajectory with a maximum ordinate sufficiently great to make it necessary to correct for altitude; or (2) in the case in which there is a material difference in height between the gun and the target. For the third possible case, a very likely one in naval operations, that of long-range firing at an elevated target, it is apparent that we have here both of the conditions calling for the use of the factor / as previously discussed. In the light of what has been said, an approximate rule that would probably not lead to material error in most cases, is to take the value of / GENERAL AND APPROXIMATE DEDUCTIONS 41 from the table for a height equal to two-thirds the maximum ordinate or two-thirds the height of the target, whichever of the two gives the greater height. The rule given in the preceding subparagraph is of course only approximate, and a reference to Figure 5 will show at once that a closer approximation to the mean height of travel would really be ^a + ^Y sec p, and that the value of / would then be taken from the table for the height determined by the above expression (not for two- thirds of it) . It is believed, however, that the first rule given is sufficiently accurate for all ordinary cases; although special consideration should be given to this point in all cases involving peculiar conditions. ^ is a quantity known as the integration factor, and will be explained later. For integration the present it may be assumed to be equal to unity, and it will therefore disappear from the formula for the value of the ballistic coefficient for all our practical purposes. 63. C, as given in its fullest form in paragraph 61, is sometimes known as the reduced ballistic coefficient, the form used by Mayevski, C = — ,^ , being called the ballistic coefficient. The expression for the value of this ballistic coefficient should always be remembered in its fullest form, however, and the different factors entering into it allowed to drop out by becoming unity as the conditions of actual firing approach the standard conditions. 64. Suppose we desire to find the value of the ballistic coefficient for the 12" gun, w = 870, c = 0.61, for 30.1-1" barometer and 24.5° F., when the highest point of the trajectory is 3333 feet. The formula is C = -^,^. We could work it out directly from this formula, but where investigations are to be carried out in regard to any one particular gun and projectile, it is convenient to work out the combined value of the constant factors for that projectile, that is, for iv, c and dr, and having once determined this, thereafter for the given gun and projectile we have only to apply 8 and / to this constant to get the value of the ballistic coefficient under the given conditions. Expressed mathematically this is : K=^ C=J-K cd- 8 and for the above problem the work becomes : w = 8r0 log 2.93953 c = .61 log 9.78533-10 colog 0.21467 d^ = 14:-i loir 2.15836 colog 7.84164-10 Z = 9.9045 log 0.99583 Now, from Table III, for 30.14" and 24.5°, we find 8= 1.0947. And, from Table V, for a height of =2222 feet, we find /= 1.059. Hence, for our special case, o we have A'= log 0.99583 /= 1.059 log 0.02490 8 = 1.0947 log 0.03929 coloir 9.96071-10 (7 = 9.5816 log 0.98144 65. Referring to the second part of paragraph 58, we see that we found experi- mentally that a certain resistance existed at the time of firing to the passage of a certain projectile through the air. Let us now suppose that the coefficient of form of the projectile used was c=0.61, and that the barometer stood at 30.14", and the thermometer at 24.5° F. at the time of firing. What would be, from this experimental 42 EXTEEIOR BALLISTICS firing, the resistance to a standard projectile under standard atmospheric conditions? The air on firing, being more dense than standard, the resistance would be less under standard conditions, that is, Rs = -„- Rf. The projectile used being more tapering than o the standard, would pass more easily through the air, and the resistance to the standard projectile would be more than that measured, that is, jRg= — Rfj and com- bining, Rs=j-Rf Rf = 833.7b log 2.92104 8=1.0947 log 0.03929 colog 9.96071-10 c = .61 log 9.78533-10 colog 0.21467 i2, = 1248.6 pounds log 3.09642 66. Suppose that we have given that, for a 12" gun: w = 870; c = 0.61; two measured velocities at points 920 feet apart were 2840 f . s, and 2810 f . s. ; to deter- mine the resistance, and then discuss the difl:erence between the results obtained by actual firing and by the use of Mayevski's formula. For simplicity in computation, consider the atmospheric conditions as standard. By Mayevski's formula R = A^^ v''; Mean velocity = 2825 f. s.; a = 1.55; log A = 7.60905 -10 i; = 2825 log 3.45102 loglog 0.53794 a=1.55 log 0.19033 v«= log 5.34900 loglog 0.72827 c=.61 log 9.78533-10 A= log 7.60905-10 d- = lU log 2.15836 g = 32.2 log 1.50786 colog 8.49214-10 B = 2476.7 pounds log 3.39388 By actual firing : R= ^''_ {v^--v^-)= -^(v^ + v.) (v^-v^) t;, + r, = 5650 log 3.75205 t;i-V2 = 30 log 1.47712 w = 870 log 2.93952 2g = 64:A log 1.80889 colog 8.19111-10 a; = 920 log 2.96379 colog 7.03621-10 72 = 2488.9 pounds log 3.39601 If by our experimental firing we find as above that the resistance is 2488.9 pounds for the given projectile, when moving with a velocity of 2825 f . s., and assuming that at this velocity the resistance varies as the 1.55th power of the velocity, what would be the value of Mayevski's constant A in this case? E^A — V therefore A = -,?— R g cd-V^ 72 = 2488.9 log 3.39601 v^ (from preceding problem) . . .log 5.34900 colog 4.65100 — 10 ^ = 32.2 log 1.50786 c=0.61 log 9.78533-10 colog 0.21467 ^2 = 144 log 2.15836 colog 7.84164-10 A = .0040849 log 7.61118-10 GENERAL AND APPROXIMATE DEDUCTIONS 43 But Mayevski gives for A for this velocity a value of A = .0040649, of which log A = 7.60905 -10. This difference simply means that Mayevski's value is the mean of a large num- ber of values obtained by experimental firings, such as the one worked out above ; and that the value of A found above is the one resulting from a single firing only. The difference between them therefore simply represents the difference between a mean value resulting from many experiments and one of the many individual values that go to make up such a mean. Note. — See Chapter 15 for a more full discussion of the coefficient of form, the subject being there treated according to the most modern practical methods now employed in the U. S. Navy. In this later consideration a different method of determining the values of the coefficient of form and ballistic coefficient is employed, and in fact a somewhat different conception of what the coefficient of form really is is adopted. A comprehension of the methods of this present chapter is however very necessary in understanding the explana- tions contained in Chapter 15. EXAMPLES. 1. Compute the value of K ic=^ J-^ = "{^-^j i^ the ballistic coefficient in the cases given in the following table ; giving the values of log K and colog E. Do not use Table VI of the Ballistic Tables in these computations; as this question calls for the computation of the data contained in that table. DATA. ANSWERS. Problem. Gun. ^ Cd^' d. In. If. Lbs. c. V. f. s. K. log K. colog K. A 3 3 4 5 5 6 6 ? 7 8 10 10 12 13 13 14 14 13 13 33 50 50 105 105 105 165 165 260 510 510 870 1130 1130 1400 1400 1.00 l.no 0.67 1 . 00 0.61 0.61 1 . 00 0.61 1.00 0.61 0.61 1.00 0.61 0.61 1.00 0.74 0.70 0.70 1150 2700 2900 3150 3150 2600 2S00 2800 2700 2700 2750 2700 2700 2900 2000 2000 2000 2000 1.4444 1.4444 3.0783 2.0000 3.2787 4.7814 2.9166 4.7814 3.3673 5.5201 6.6598 5.1000 8.3605 9.9045 6.6863 9 . 0355 10.2040 10.2040 0.1.5970 0.15970 . 48832 0.30103 0.51570 0.67956 0.46189 0.67956 0.52728 0.74195 0.82346 0.70757 0.92224 0.99583 0.82519 0.95596 1.00877 1.00877 9.84030— 10 B 9.84030— 10 c 9.51168 — 10 D E F G H I 9.69897 — 10 9.48430 — 10 9.32044—10 9.53511—10 9.32044 — 10 9.47272—10 J 9.25805— 10 K 9.17654—10 L 9.29243— 10 M 9.07776—10 N 0.00417 — 10 9.17481 — 10 P 9.04404 — 10 Q 8.99123 — 10 R 8.99123—10 , 44 EXTERIOR BALLISTICS 2. Using the values of K found in example 1 preceding, determine the values of 8, /, and log C for the conditions given in the following table. Correct for maxi- mum ordinate or for height of target according to rule, but consider every trajectory whose time of flight is greater than five seconds as requiring correction for altitude. DATA. ANSWERS. Gun. Diff. in Atmos- f3 R'nge. Yds. Time of flight. Max. ord. Value of ht. of gun and phere. 5. f- lU log (7. o d. If. T. Sees. Feet. log Z. target. Bar. Ther. f^. In. Lbs. f.s. Feet. In. °F. A.. . 3 13 1.00 1150 2600 8.25 277 0.15970 300 .30.33 24.7 1.1011 1.0050 0.12004 B... 3 13 1.00 2700 4400 9.25 366 0.15970 150 30.13 17.5 1.1113 1.0063 0.11660 C, . 4 33 0.67 2900 3900 5.10 105 0.48832 200 29.92 15.7 1.1080 1.0037 0.44538 T) , . 5 50 1.00 3150 4300 6.18 154 0.30103 225 29.83 12.4 1.1135 1.0040 0.25606 F„ F, 50 0.61 3150 4300 5.19 108 0.51570 90 29.57 29.3 1.0644 1.0022 0.48956 F.. , f) 105 0.61 2G00 14800 31.56 4215 0.67956 1200 29.45 33.8 1.0502 1.0753 0.68982 G 6 105 1.00 2800 4000 5.56 124 0.46489 350 29.37 39.41 1.0352 1.0060,0.4.5247 H , 6 105 0.61 2800 3700 4.. 57 85 0.67956 200 29.07 43.2 1.0170 1.0037 0.67384 T 7 165 1.00 2700 7000 11.76 563 0.52728 None 28.95 48.7 1.0009 1.0095 0.53100 J 7 165 0.61 2700 7400 10.61 455 0.74195 175 28.83 50.3 0.9936 1.00810.74824 K 8 ?m 0.61 2750 8300 11.49 532 0.82346 450 28.73 52.8 0.9852 1. 0091 !o. 83387 T„ 10 510 1.00 2700 10100 16.57 1116 0.70757 500 28.. 58 69.3 0.9475 1.0193 0.73929 IM 10 510 0.61 2700 11000 15.69 997 0.92224 1100 28.47 95.7 0.8936 1.019010.97927 N }?. 870 0.61 2900 23500 37.61 5758 0.99583 1500 28.36 97.4 0.8867 1.1072 1.09228 0, 13 11.30 1.00 2000 10400 21.53 1889 0.82519 700 28.27 99.8 0.8790 1.0328 0.89522 P , 13 1130 0.74,2000 11300 22.28 2005 0.95596 508 28.21 74.8 0.9243 1.0351 1.00513 Q. 14 1400 0.70 2000 14100 28.36 3264 1.00877 800 28.20 71.3 0.9310 1.058311.06443 K... 14 1400 0.70 2600 14400 21.83 1925 1.00877 700 28.71 84.6 0.9225 1.0335 1.05811 3. Given the measured velocities of a projectile at two points, as determined by experimental firing, as given in the following table, determine the resistance of the air at the mean velocity between the two points of measurement. If the atmospheric conditions at the time of firing were as given, what would be the corresponding resistance under standard atmospheric conditions? DATA. ANSWERS. DATA. ANSWERS. Problem. Projectile. Dist. between points of measure- ment. Yds. Measured velocities at. Rf. Lbs. Atmosphere. d. In. w. Lbs. Bar. In. Ther. °F. Its- First point. f. s. Second point. f.s. Lbs. 1 3 5 6 7 12 13 14 6 13 60 105 165 870 1130 1400 7b 80 90 95 100 105 110 125 200 2650 2250 2550 2680 2870 1910 2540 1951 2600 2200 2500 2580 2800 1880 2460 1874 220.78 767.76 1444.5 4492.2 17022.0 6045.5 23188.0 533.55 28.00 29.00 30.00 30.50 31.00 .30.00 29.00 29.53 50 60 70 80 90 i 59 228.55 2 783.43 3 1454.7 4 4539.8 5 17281.0 6 5257.0 7..... 22021.0 533.55 GENERAL AND APPROXIMATE DEDUCTIONS 45 4. Under the conditions given in the following table, compute the total atmos- pheric, resistance to the passage of the projectile, and the resultant retardation in foot-seconds. DATA. ANSWERS. Problem. Projectile. Velocity, f. s. ' Atmosphere. Retardation, f.s. Resistance. Lba. d. In. to. Lbs. c. Bar. In. Ther. °F. 1 3 5 6 7 12 13 14 14 3 13 60 105 165 870 1130 1400 1400 13 1.00 1.00 0.61 0.61 0.61 0.95 0.70 0.70 0.93 2300 1500 1300 2700 2850 1100 850 2500 650 30.00 31.00 29.00 28.50 29.45 30.15 28 . 67 29.33 30.40 20 55 47 82 90 95 64 75 80 493.39 130.. 53 44.23 141.16 86.92 13.74 3.43 70.20 12.54 199 19 2 243.23 144 "^4 3 4 723.. 33 2348.40 482.27 149 '^7 5. 6 7 8 3059 90 9 5.06 5. What is the resistance of the air to a baseball of 3" diameter, weighing 8 ounces, moving at 100 f. s.; supposing the resistance of a sphere to be 1.25 times that of a standard ogival ; and what would be its retardation ? Answers. Resistance, 0.16337 pound. Retardation, 10.531 foot-seconds. 6. Given the data in the following tables, compute the value of the constant A in Mayevski's formula, for each individual case. DATA. ANSWERS. Problem. Projectile. 5. Velocity, f. s. Resist- ance. Lbs. Retar- dation. f. s. Value of a. Value d. In. tv. Lbs. c. of A. 1 6 6 6 12 14 14 70 70 70 870 1400 1400 1.00 1.00 1.00 0.61 0.70 0.70 1.0000 1.0200 1 . 0200 0.9354 0.9610 0.9606 1912.5 1818.0 1859.5 2850.0 850.0 2500.0 533.6 491.8 1248.6 3052. 4 86! 92' 3.4328 1.70 1.70 1.70 1.55 3.00 1.70 0.001259 9 0.00124023 3 . 0030296 4 0.0040649 5 0.000000059353 6 0.001248 7. A 6" projectile, weight 70 pounds, is fired through screens, and the velocities measured at two points 200 yards apart are 1951 f. s. and 1874 f. s. What was the mean resistance of the air? Answer. 533.57 pounds. 46 EXTERIOE BALLISTICS 8. A 6" ogival-headed projectile, weight 70 pounds, is fired through screens 150 yards apart, and its velocities at the first and at the second pairs of screens are 1846 f. s. and 1790 f. s., respectively. A 6" flat-headed projectile of the same weight is fired through the same screens, and gives velocities of 1929 f. s. and 1790 f. s., respectively. What was the resistance of each projectile ? If the first was a standard projectile, what was the coefficient of form of the second ? Ansivers. First, R= 491.82 pounds. Second, E = 1248.65 pounds. Coefficient of form of second = 2.5388. 9. A 12" projectile, weight 850 pounds, gave measured velocities of 1979 f. s. and 1956 f . s. at points 500 feet apart. What was the mean resistance of the air ? If the density of the air at the time of firing was 1.02 times the standard density, what would be the resistance in a standard atmosphere ? Ansivers. Rf = 2389.0 pounds. J?« = 2342.2 pounds. 10. Determine the resistance of the air to and the consequent retardation of a standard 3" projectile, weight 13 pounds, when moving: (1) at 2800 f. s.; (2) at 2000 f. s. Answers. (!) Eesistance = 250.34 pounds. Eetardation=: 620.09 f. s. (2) Eesistance = 142.67 pounds. Eetardation = 353.38 f . s. 11. Determine the resistance of the air and the consequent retardation in the following cases. (Standard atmosphere ; and c= 1.00 in each case.) DATA. ANSWERS. Problem. Projectile. Velocity. f. 8. Resistance. Lbs. Retardation. d. In. w. Lbs. f. s. 1 4 4 6 G 8 8 10 10 12 12 12.5 12.5 33 33 100 100 250 250 500 500 850 850 802.5 1000.0 2800 2000 2800 2000 2800 2000 2800 2000 2800 2000 1400 1400 445.1 253.6 1001.4 570.7 1780.2 1014.5 2781.5 1585.2 4005.4 2282.7 1251.6 1251.6 434.3 2 247.5 3 322.4 4 183.8 229.3 6 130.7 7 179.1 8 102.1 9 151.7 10 86.5 11 50.2 12 40.3 CHAPTEE 4. THE EQUATION TO THE TRAJECTORY IN AIR WHEN MAYEVSKI'S EXPONENT IS EQUAL TO 2. New Symbols Introduced. p. . . .Radius of curvature, in feet, of the trajectory at any point. A k. . . .The ratio -^, where A is Mayevski's constant, and C is the ballistic coefficient. £. . . .The base of the Naperian system of logarithms; £ = 3.7183. n. . . . The ratio of the range in vacuum to the range in air for the same angle of departure. 67. Mayevski's equations, as given in (14) and (16), show that, strictly speak- ing, A and a can only be regarded as constants in the general expression for the retardation caused by the resistance of the air dt C ^ ^ within certain limited ranges of the value of the velocity, v. As will shortly be seen, in the attempt to derive the equation to the trajectory in air, it will be possible to succeed in cases where the value of a is an integer, that is, for cases within the range of the formulse in (14) where the value of v lies between 1800 f. s. and 1370 f. s. (a = 2), or within the lower limits where a = 3, 5 or 2 again; and, unfortunately, these limits do not include the initial velocities for modern high-powered guns. This is true, as will be seen, because the derivation of the required equation involves the integration of certain expressions in which the value of a must appear as an exponent, and it is impossible to make such integrations except when a is a whole number. The equation which we will derive will therefore only be correct for the limited range of values of v for which a = 2. Could all the integrations be performed, for the decimal as well as for the integral values of a, a series of equations to the trajectory for the different limits of the initial velocity, could be derived and tabu- lated in the same way in which Mayevski's expressions for retardation shown in (14) were tabulated, but as it is, for the reason given, such equations cannot be derived for the initial velocities that are used at the present day; namely, from 1800 f. s. up, so some other method of obtaining solutions must be found. This is done by the use of certain differential equations, as will be explained later. 68. Meanwhile it is of value to follow through the derivation of the equation to the trajectory in air when a = 2, both for educational purposes and in order to make a comparison with the equation to the same curve in vacuum. We must remember, however, that this equation will only be correct for initial velocities for which a=2 in (14), and that for all other initial velocities results obtained by its use will be only approximate. 69. Assuming that the axis of the projectile coincides with the tangent to its Forces path at every point, which is very nearly the case with modern rifled guns, the resultant action of the resistance of the air will likewise coincide with the axis, and the trajectory will be the same as if the mass of the projectile were concentrated at its center of gravity and moved under the action of two forces only, one the constant acting'. 48 EXTERIOR BALLISTICS vertical force of gravity, w, and the other the variable resistance of the air, — X -yrv^, acting in the tangent. Figure 6 represents the trajectory, which is, of course, a plane curve, under the foregoing suppositions, and Figure 6(a) represents the two forces acting upon the projectile at any point, the resistance of the air being denoted by — /, in which / is the retardation, -^, which we are now taking as proportional to v^ dt Figure 6 70. Taking vertical and horizontal axes at the point of departure, 0, let V be the initial velocity, cf) the angle of departure, v the velocity at any point whose coordinates are (x, y), and Vn the horizontal component of the velocity at that point. A p is the radius of curvature of the curve at that point. Then, letting lc= -T^in equa- dv tion (18), we can put -,-7- = — A;i'-, and, since w has no horizontal component, the acceleration parallel to the axis of X is given by d-x dr- = — kv- cos 6 but d^x df dt * V cos d = Vh, and v= -^j ; whence (19) may be written dt dvj, dt — kvi, ds IF' dvn _ = — kds (19) (20) (21) and integrating (21) between corresponding limits of Vh and s we get I-'cos t)ft=7cos<^€-^'» (22) lOffef ge ^h = —ks n Fcos d> loge =ks Next resolving along the normal, since the acceleration towards the center of curva- ture IS given by the expression — , p being the radius of curvature at that point, P we have — g cos 6 (23) GENERAL AND APPEOXIMATE DEDUCTIONS 4.9 ds But v = Vh sec 9, and p= — j^, whence (23) may be written do Vh^ sec- cW=—g cos 6 ds= —gd.v (as dx = ds cos 6) ; sec- 6 d6= — ^ -^ (24) gdx Now substituting in (2-i) the value of vi, given in (22), we get sec- ede=- Yr^^—^ €-'''dx (25) V^ cos^ 71. In the case of the flat trajectory, in which the angle of departure does not exceed -t° or 5°, the difference between the values of s and x is so small that it may be practically disregarded, and x may be substituted for s in (25), giving, after integration * tan^ tan ^ = tan 0- — ^/— ^ (e^*^-l) (26) 2kV- cos^ But c-^-^j when expanded by j\laclaurin's theorem, equals f 1 + 21CX + 2k-x- + ^k^x^ + so that e"^-^ - 1 = 2 A-.r ( 1 + kx + § fc^T- + ) whence, substituting in (26) and writing ^ for tan 6, we have or, integrating between corresponding limits of x and y, y = xtancf>- gy^^^y^ (l + t^-'^ + ^^-'-^'+ • • • •) (^7) But the greatest value of kx is always a small fraction in any trajectory flat enough to justify the substitution of x for s which has already been made; hence we may neglect the terms beyond k'x- in the expansion, and write for the equation to the trajectory in air when a = 2 Equation to trajectory in r.2 ^ = .r tan c^- ^f^ (l + ^kx + ik^x^) (28) ^t'"' * The integration in paragrapli 71 is as follows: From (25) From calculus we know that ( sec- d dO = tan 6 + d, C^ being the constant of integration. From calculus we know that fe" cf?/ = e* + C'j, C, being the constant of integration. Now let y = 2kx and the above becomes whence f 1 f „, , .„, 1 , „ The integration between the limits given above therefore becomes (tan . + C.) - (tan ^ + C.)=- -,.^L p^ ,.- + c,) - (^ .' + C.)] tan ^ z. tan <^ - y^^^^ X ^ (e=*- l) t The expansion in paragraph 71 is: From either algebra or calculus we have that «^ = 1 + 2/ + if- + ll^ + ||- +• • • -etc. whence, if we let y ^ 2kx we have Ak-x' , %k^x^ , 16k*x' e^fc- = 1 -f- 2kx + -2~ + 3^^ + 4 X 3 X 2 + • • • ■^^''- or e-^x = 1 ^ 2A-J- (1 -\- kx + ^ k^'x' + |A: V + etc.) 50 EXTERIOR BALLISTICS The ratio n 72. Comparing this equation with (2), it will be seen that its first two terms represent the trajectory in vacuum, and that it only differs from the latter by having other terms, subtractive like the second, and containing higher powers of k and x. 73. The value of k in the equation to the trajectory in air (28) just deduced is, as already stated, — ^ Assumptions made. c= Scd- where A is the experimentally determined coefficient, and is the ballistic coefficient. As a matter of general interest, it may be stated that, for the value of .4 = 0.0001316, the value assigned to A by Mayevski when a = 2, the value of k for our naval guns from the 6-pounder up to the 13" gun, varies from about 0.00011006 to about 0.00002022, for the standard projectile and standard density of the air. 74. If we put y = in equation (28), we get for values of x^ one equal to zero, denoting the origin, and another, the range X, given by V- sin 2(^ Z(l + pX + iFZ^) = 9 (29) But the second member of (29) is the range in vacuum for the same initial velocity, V, and the same angle of departure, (f>, so we see that the expression l + ikX + WX^= Z(invacuum) ■^ -^ Z(mair) This ratio will be found to play an important part in many ballistic problems, and will hereafter be designated by the letter n. Hence we have for the range in air the expression -^^71sin2^ (30) gn (31) or, if it be desired to find the value of n for a given range, _ 72 sin 2 75. Since k is a very small fraction, the value of ti, which is evidently unity for X = 0. increases slowly with X, and for moderate values of X is only slightly greater than unity. These deductions follow from the form of the equation l + ^kX + ik'X^ = n 76. It is Avell to summarize here that the following suppositions have been made in deriving the equation to the trajectory in air when a = 2, and these suppositions must be held to be correct in all consideration of this equation ; and the equation is inaccurate to whatever degree results from the lack of correctness of any one or more of these assumptions : 1. That a=2, and that the corresponding value of A is correct. 2. That the axis of the projectile coincides with the tangent to the trajectory at every point, and that the resistance of the air will therefore act along the same tangent. 3. That the curve is so flat that we may consider dx=ds without material error. •4. That kx is so small in value that any term involving powers higher than k'x^ may be neglected. 77. The following examples show the form for work under the formulae derived in this chapter. It must be remembered, be it again said, that these formulas are derived on the assumptions given in the preceding paragraph, and results obtained by their use are therefore only approximately correct for the usual present-day initial velocities. GENERAL AND APPROXIMATE DEDUCTIONS 51 For a 6" gun, given that the initial velocity is 2600 f. s., and that an angle of departure of 4° 14' 30" gives a range of 7000 yards, to compute the value of the ratio between the ranges in vacuum and in air for that angle of departure; that is, the value of n. _ V- sin 2^ 7 = 2600 log 3.41497 2 log 6.82994 2(^ = 8° 29' 00" sin 9.16886-10 g = 32.2 log 1.50786 colog 8.49214-10 Z = 21000 log 4.32222 colog 5.67778-10 n = 1.4748 log 0.16872 78. Given that the angle of departure for the 12" gun of 2900 f. s. initial velocity (^ = 870 pounds; c = 0.61) for a range of 10,000 yards is 4° 13' 12", compute the approximate value of the ordinate at a distance of 2000 yards from the gun, and compare it with the ordinate in vacuum at the same point for the same angle of departure. In vacuum y = x tan S — 7:^7^^ — ^— •^ 2V^ cos^ (^ T • , . nqx^ 1 V^ sin 2d> In air y = x ia.xi. dt — ~^^ — 5— where n=: =^— £- 2 y^ cos^ ^ gX TForfc in Vacuum. a;=6000 log 3.77815 2 log 7.55630 <^ = 4° 13' 12" tan 8.86797-10. . sec 0.00118 2 sec 0.00236 ^ = 32.2 log 1.50786 2 log 0.30103 colog 9.69897-10 7 = 2900 ..log 3.46240 2 log 6.92480.. 2 colog 3.07520-10 442.710 lo, may be used over a considerable range of values of (f>, since it increases slowly with mcreases of range, provided the trajectory be reasonably fiat. We shall still denote by n the ratio of the range in vacuum to the range in air, which is now given by Approximate 80. To determine the approximate horizontal range, put y = in (32), and "'^'range. solve for X. An X factor will divide out, so one value of x is zero, for the origin, as was to be expected, and the remaining equation is It is at once apparent that this is an awkward equation for logarithmic work, and furthermore not very accurate for work with five place logarithmic tables owing to the decimal value of Jc. Angle of fall. 81. Differentiating (32) we get 4^ = tan ^ = tan <^ - -^^^-^ ( 1 + hx) (34) ax V- cos- ^ But the angle of fall, w, is the negative of the value of at the point of fall, where x = X, hence tan w = — tan d> + ^^^^ — .^ — (1 + TcX) V- COS" <^ , . 2 y ^ sin (f> COS ^ V ,1 • 1 ■.. and since -=7iX, this may be written 9 . 4- , , 2tan H ^ ( 1 + A:A ) = tan [2- -j (35) GENEEAL AND APPROXIMATE DEDUCTIONS 55 From (35) we see that the angle of fall is always greater than the angle of departure, but can never reach double the latter.* dr 82. Eeturning to equation (32) and writing—,'- for Vh, and x for s, we have Time of = T' cos but from l + -p.Y = n we get that 1 + ^ = -^^^ , therefore y^3n_fi^ Z^ (36) 4 K cos ^ * This follows from the form of the expression, for from paragraph 75 we know that 7^ = 1 + gfcX + Jfc-X'', from which we see that n is unity when .Y = and increases very slowly with X, k being a very small decimal. Therefore is always less than unity and 2 is always greater than unity; and the angle of fall must therefore always be greater than the angle of departure. Also as n must always be greater than unity for any real range, then — must always be a positive real number, and therefore the value of 2 must always be less than 2; therefore the angle of fall can never become twice as great as the angle of departure. f The integration in paragraph 82 is as follows: From integral calculus Uvdu^eV + C C being the constant of integration. Now let y = A:.T and we have L^^dX = -^ t^^d (kX) = -j- (t^^) + C Therefore e*^di=Fcos0 dt Jo Jo becomes (^+ o] — U^ + A = (V cos 4> X T + C,) — (V cos 4> X + C,) Ci being the constant of integration in the second term. The above becomes — ^ — — YT cos (p t The expansion of e*-r following the integration is as follows: From either calculus or algebra we know that .vz=l + 7/ + -||- + -^ + ....etc. and substituting kX for y, and neglecting the higher powers than the square, we have e''^=l + kX + ^ whence ^^j _ 1 = fcx + -^ whence X (l + ~] = VT cos cp 66 EXTEEIOE BALLISTICS nX As the range in vacuum for the same values of V and (^ would be nX, then Fees ^ is the time of flight in vacuum, and so we see that the time of flight in air is always less than it would be in vacuum, approaching three-fourths the latter's value as a minimum.* For flat trajectories, cos may, of course, be taken as unity. Remaining 83. Equation (32), with the substitution of x for s, gives the value of the hori- velocity. i \ / 7 . . zontal component of the velocity at any point m the trajectory, and smce the strikmg velocity is the horizontal velocity at the point of fall multiplied by sec w, we have V cos d> '^ f.'^-^ cos w whence, putting for t^^ the first two terms of its expansion, and calling ? equal to unity. The striking velocity, therefore, is always less than the initial velocity, being reduced to -^ when n = 3, a value which it seldom reaches.f Coordinates 84. Since the trajectory is horizontal at its highest point, we obtain the abscissa of vertex. . {xq) of the vertex by putting 6 = in (34), thus getting ? =XQ{l-\-'kxQ), and since "^ =nX, this may be written Xq{1-\-1cXq) = -^r-, a quadratic equa- 9 '^ tion, the solution of which gives iCo = ~ 97/" — ^ which, since A;Z = |(n — l), may be written if we divide both numerator and denominator of the second term of (38) by w we get _ Vl + 3n(n-l)-l ^ .33. '''- 3(/i-l) ^ . ^'^^^ / 1 +3-A-J- _ n" n n 3- A n from which we see that when n is infinity, the value of x^ becomes— ^ =0.58-3r, * We know that n = 1 + ifcX + Jfc^'Z^ therefore n increases slowly with the range and is always greater than unity. Therefore 3n is always greater than 3 and 3» + 1 is always greater than 4 but less than 4n. Therefore — 7^- is always less than unity, and as 4w Fcos0 is the time of flight in vacuum, the time of flight in air, which is T = ."'" X y- — must be always less than ^ r ; that is, the time of flight in air is always less than it would be KCOS^ in vacuum. We also see that ^^ ^T^ + ;i^ =^r + T- , and therefore that if n becomes in 4n. 4m 4 4» infinitely great then p"— becomes -j-, which is evidently the maximum possible value of ^7^ ; and we therefore see than the time of flight in air cannot be less than three- 4?t fourths of the time of flight in vacuum. f As in the note to paragraph 82, n is necessarily greater than unity, therefore 3n — 1 2 Is necessarily greater than 2, and „ ■, is therefore necessarily less than unity; there- fore the final velocity is always less than the initial velocity. GENEEAL AND APPEOXIMATE DEDUCTIONS 57 from which we see that the abscissa of the vertex is always greater than half the range but never reaches 0,58X,* The ordinate of the vertex may be obtained by substituting the value of Xq obtained from (38) in the equation to the trajectory (32), but an equally accurate and much sim^Dler determination is given by !/.= 2|-" (39) This assumes the height of the vertex to be that from which a body would fall freely (in vacuum) in half the time of flight. Actually the vertical velocity of the pro- jectile is reduced by air resistance, but since the time the projectile takes to describe the descending branch of the trajectory is somewhat greater than half the time of flight, the air resistance is approximately allowed for by equation (39). 85. As an example of the use of the foregoing formula, we will compute the various elements of the 500-yard trajectory of the old pattern United States magazine rifle, for which the initial velocity is 2000 f. s., and the angle of departure for the given range is. 0° 31' 35". After finding the value of n, we may, as the angles are small, use ^ and w in place of their tangents, and the formulae then are y2sin2(^ [^ 1\^ rp 3n + l ^ Z . ^, _ 2 ^. _ Vl + 3/K/^- l)-l Y- ,/ - 91^ 3(n-l) ^^' ^"~ 8 7 = 2000 log 3.30103 2 log 6.60206 2 = l° 03' 10" sin 8.26418-10 (/ = 32.2 log 1.50786 colog 8.49314-10 Z = 1500 log 3.17609 colog 6.82391-10 n = 1.5216 log 0.18229 — = 0.6572 colog 9.81771-10 n 2 — ^ = 1.3428 log 0.12801 3?; = 4.5648 3?z.-M = 5.5648 log 0.74545 3^-1 = 3.5648 ..log 0.55204 colog 9.44796-10 <^ = 31.6' log 1.49969. sec 0.00002 4 log 0.60206 colog 9.39794-10 Z=1500 log 3.17609 7 = 2000 log 3.30103 colog 6.69897-10. log 3.30103 2 log 0.30103 :42.43' lo2 1.62770 r= 1.0434 lo£T 0.01847 i'a, = 1122 loff 3.05002 * Let us suppose that g . ^. ="9' • Solving this quadratic for n, we find that under these conditions » = 1; and therefore if n be greater than unity the value of the left-hand member above must be greater than I; therefore the value of Xo must always be greater than -^ for a trajectory in air; and, as x^ = O.SSi when n is infinity, we also see that the value of x^ can never reach 0.58Z. 58 EXTEEIOR BALLISTICS n = 1.5216 3n = 4.5648 ..log 0.65942 w-l = 0.5316 ..log 9.71734-10 3(n-l) =1.5648 log 0.19446 colog 9.80554-10 3n(w-l) =2.3810 ..log 0.37676 l + 3n(n-l) =3.3810 ..log 0.52905 Vl + 3n(n-l) =1.8388.^ log 0.26453 Vl + 3n(n-l) -1 = 0.8388 log 9.92366-10 X = 1500 log 3.17609 a:o = 804.05 log 2.90529 T = 1.0434 . .log 0.01847 2 log 0.03694 ^ = 32.2 log 1.50786 8 \os 0.90309 coloff 9.09691-10 ^0=^4.3824 log 0.64171 a) = 0° 42' 24". a-o = 368.02 yards. T= 1.0434 seconds. ?/o = 4.3824 feet. i?,,= 1122f. s. 86. As another example, take the ease of the 6" gun with 7" = 2400 f. s., and an angle of departure of 2°, for which the range is 3100 yards. n= g^ ; tana> = tan<^(^2--j, T- -^ ^ V^o^ ' ^-"Sr^^^ y = 2400 log 3.38021 2 log 6.76042 2 = 4° sin 8.84358-10 ^ = 32.2 log 1.50786 colog 8.49214-10 Z = 9300 log 3.96848 colog 6.03152-10 ri = 1.3417 log 0.12766 — = 0.7453 colog 9.87234 — 10 n 2- -=1.2547 log 0.09854 n 3n = 4.0251 3n + l = 5.0251 log 0.70115 3/1-1 = 3.0251 ..log 0.48074 colog 9.51926-10 cj) = 2° tan 8.54308-10. . sec 0.00026 4 log 0.60206 colog 9.39794-10 Z = 9300 log 3.96848 7 = 2400 log 3.38021 colog 6.62979-10. . log 3.38021 2 ." log 0.30103 a, = 2° 30' 38" tan 8.64162-10 r = 4.871 log 0.68762 t;, = 1586.7 log 3.20050 w = 2° 30' 38". r= 4.871 seconds. r„= 1586.7 f. s. GENERAL AND APPROXIMATE DEDUCTIONS 59 87. If the value of n be known, A; may be found from n = 1 + '^kX, and then by substituting the proper value of x in the equation to the trajectory (32), the corre- sponding value of y, the ordinate of the trajectory at a distance x from the gun may be computed ; and this was the way in which the last problems in the examples under the last chapter were worked. If, however, we know the angles of departure corre- sponding to various ranges (which data is contained in the range table for the gun) the approximate value of y for any value of x, for the trajectory for a given range, may be more readily found as follows. Referring to Figure 7, let (x', y') be the coordinates of the point M on the trajectory for which = \p' + p. Then, by the principle of the rigidity of the trajectory, if the angle of departure were i/'', the horizontal range would equal OM, or what is practically the same thing, x'. Conse- quently, if we taken from the range table the angle of departure for a range x', and subtract it from the angle of departure for the given trajectory, the result will be the angle /;. Then y —x tan [). Figure 7. 88. By the term " danger space " is meant an interval of space, between the point of fall and the gun, such that the target will be hit if situated at any point in that space. In other words, it is the distance from the point of fall through which a target of the given height can be moved directly towards the gun and still have the projectile pass through the target. Therefore, within the range for which the maximum ordinate of the trajectory does not exceed the height of the target, the danger space is equal to the range, and such range is known as the " danger range.'' Referring to Figure 8, AH = S is the danger space for a target of height AB — h, in the case of the trajectory OBH. It will be seen from Figure 8 that, when the value of li is very small in comparison with the range, the danger space is given with sufficient accuracy by the formula *S = /icotw (40) Danger space. 60 EXTERIOR BALLISTICS This assumes that the tangent at the point of fall is identical with the curve from H to B, whereas it really passes above B, so that the result given by (10) is somewhat too small. — X. Figure 8. 89. A more accurate formula for the danger space is deduced as follows : Calling the very small angle AOB^^fj), we have, very nearly, tan A<^= ^ and tanw=-|p, whence, equating the two values of h derived from the preceding expressions, li = X tan A<^ = S tan w, h or, as A<^ is very small, A'A<;f> = ^ tan w, or, since A= -^ X — o hX x-s = ^tan whence S = h cot w ( ^^^ — ^ ]=h cot< ^+x + x^ + - whence, neglecting the higher powers of the fraction -^r , and putting for S in the expression its approximate value of h cot w S^ll cot 0) (l+^") (41) 90. As a further example of the use of the equations derived in this chapter in determining the approximate values of the quantities concerned, we have the follow- ing: Given that, for the 6" gun (m; = 105, c = 0.61), the initial velocity is 2600 f. s., and that the angle of departure for a range of 5500 yards is 3° 02' 24"; to find the approximate values of the angle of fall, time of flight and striking velocity. n= ~^ ; tan a) = tan d>[2 ; 7= — -^ — X ~ ; i\= -^ gX ' ^\ nj' 4 7cos<^ '" 3n-l y = 2600 log 3.41497 2 log 6.82994 2*^ = 6° 04' 48" -. .. sin 9.02491-10 5^ = 32.2 log 1.50786 colog 8.49214-10 Z=16500 log 4.21748 colog 5.78252-10 n = 1.3474 log 0.12951 = 0.74215 colog 9.87049-10 GENERAL AND APPROXIMATE DEDUCTIONS 61 2- ^- = 1.25785 log 0.09963 n 3w = 4.0432 3n + l = 5.0423 log 0.70263 3w-l = 3.0423 ..log 0.48319 colog 9.51681-10 = 3° 02' 24" tan 8.72516-10. . sec 0.00061 4 log 0.60206 colog 9.39794-10 Z = 16500 log 4.31748 7 = 2600 log 3.41497 colog 6.58503-10. . log 3.41497 3 log 0.30103 :3°49'19" tan 8.82479-10 r = 8.011 log 0.90368 r^ = 1709.3 log 3.23281 (0 = 3° 49' 19". r = 8.011 seconds. v^ = 1709.3 f. s. To find the approximate co-ordinates of the vertex for the trajectory given above. _ yi + -dn{n- l)-ly. _gT^ "' 3(n.-l) ' ^'~~S~ n = 1.3474 3« = 4.0422 . .log 0.60662 ^-1 = 0.3474 ..log 9.54083-10 3(n-l) =1.0423 log 0.01795 colog 9.98205-10 3n(n-l) =1.4043 ..log 0.14745 l + 3n(/i-l) =2.4043 ..log 0.38099 Vl + 3w(n-l) =1.5506.^ log 0.19050 VH-3n(n-l) -1 = 0.5506 log 9.74084-10 X = 16500 \os 4.21748 a-o = 9085 log 3.95832 T = 8.011 . . .log 0.90368 2 log 1.80736 ^ = 32.2 log 1.50786 8 log 0.90309 colog 9.09691-10 7/0 = 258.30 log 3.41313 a;o = 3028.3 yards. 2/0 = 358.30 feet. For the conditions given in the preceding problem, to find the danger space for a target 20 feet high. There are two formulae possible, of which the longer is the more exact, and is 'the one used in computing the values of the danger space for a 20-foot target given in Column 7 of the range tables. It should be used whenever exactness is required. We will compute by both and compare the results. ;S = /icotw *S' = /ieotwfl+ — Y~] 62 EXTERIOR BALLISTICS n = 20 log 1.30103 (0 = 3° 49' 19" cot 1.17519 S = li cot CO = 299.38 log 2.47623 log 2.47622 log 2.47622 Z= 16500 log 4.21748 ^^^^=0.0181 log 8.25875-10 A^^ +1 = 1.0181 log 0.00779 ^ = 304.8 log 2.48402 By approximate formula 99.795 yards. By more exact formula 101.600 yards. The variation in the above more exact result from the value given in the range table is due to the fact that the value of the angle of fall used above is only approximate, standard Throughout this book, in working sample problems showing the computation of problem. ^^^^ ^^^^ ^^^ ^^^ range tables, the work will be done in each case for what will be known as the " standard problem " of the book. This will be for a range of 10,000 yards, for the 12" gun for which 7 = 2900 f. s., w = 870 pounds, and c = 0.61. This is the gun for which the range table is Bureau of Ordnance Pamphlet No. 298; which table is given in full in the edition of the Range and Ballistic Tables, printed for the use of midshipmen in connection with this text book. For this gun and range, we know, by methods that will be explained later, that the angle of fall, w, is 5° 21' 10"; therefore, to determine the danger space for a target 20 feet high, at the given range, the work for getting the data in Column 7 of the Range Table is as follows : S = h cot.{l+^-^l^) As we desire our result in yards, however, we may reduce all units of measurement in the formula to yards, and the expression then becomes A=6.6667 log 0.82391 log 0.82391 3 (0 = 5° 21' 10" cot 1.02827 cot 1.02827 R = 10000 log 4.00000 colog 6.00000 - 10 -^ cot w o R h , -— cot ( = 0.0071 log 7.85218-10 1+ - ^ „ =1.0071 log 0.00307 6'2o = 'M.655 yards log 1.85525 91. We can now make a comparison between the trajectory in vacuum and that in air for the same initial velocity and angle of departure. Figure 9 represents on the same scale the trajectories in air and in vacuum of a 12" projectile weighing 870 pounds, c = 0.61, fired with an initial velocity of 2900 f. s., at an angle of departure of 4° 13.2' ; the range in vacuum for this angle of departure being 38315.3 feet (12771.7 yards), and in air of standard density, being 30,000 feet or 10,000 yards. In the figure the ordinates of both curves are exaggerated ten times as compared with the abscissae, in order that the curve may be seen. GENERAL AND APPROXIMATE DEDUCTIONS 63 If gravity did not act, the projectile would move in the tangent to the curve 0Q^Q2, and in traveling the horizontal distance x = OA, would rise to the height -4 ' ?3 -'- Figure 9. Comparison between Trajectory in Vacuum and that in Air for same

- J^ 2 V^ cos^ <^ We have already computed the value of x tan ^ as above and found it to be 1475.7 feet. Therefore, computing the second term of the right-hand member of the above equation, we have g = S2:2 log 1.50786 a;=-. 20000 log 4.30103 Slog 8.60206 2 log 0.30103 colog 9.69897-10 «^ = 2900 2 colog 3.07520-10 <^ = 4° 13.2' 2 sec 0.00236 ^^;^,-^ = 769.93 feet log 2.88645 a; tan <^ = 1475.7 feet AP, = ,j, = ,t.n,-^-^S^,-^ = 705.8 feet 64 EXTERIOE BALLISTICS When the resistance of the air also acts, retarding the motion of the projectile, it takes longer for it to move OA horizontally, and so gravity has longer to act, and it has fallen the further distance and the ordinate of the trajectory in air is AP' = y^ = x tan <^- ^^-^^ n+ijcx + ^Jc'x-) ■^^ ^ 2f- cos- ^ V ' 3 I a / or ?/2 = a;tan«^-^f , . , 72 gij^ 2^ m which n = .^^-^ 9^ gnx'' cos- ^ 7 = 2900 , '3 log 6.92480 2 cos- <}> so at a:= 30,000 feet, we solve for P^P" a; = 30000 log 4.47712 2 log 8.95424 S' = >S,ooo-^26oo = 5106.1 -2967.1 = 2139 feet. If the ballistic coefficient were not unity, then we W'ould have 7 = 0.939 X C and 5 = 2139 X C. The variation from unity of the ballistic coefficient may, of course, be caused by a variation in any of its factors, such as coefficient of form, density of atmosphere, etc. 98. By means of equations (51) and (52), if we have given the ballistic coeffi- cient, one of the velocities, and any one of the other quantities, we can find the remain- ing quantities. 99. Suppose that a 6" projectile weighs 105 pounds; that its coefficient of form is 0.61 ; and that it has an initial velocity of 2562 f . s. ; and that we desire to know its velocity after it has traveled 3 seconds, and also how far it will travel in that time ; when the barometer stands at 30.00" and the thermometer at 40° F. C-g^; T=C(T,-T,J, or T, = ^+T,^; S = C{S,-S,.J iv = 10d log 2.02119 8 = 1.056 log 0.02366 colog 9.97634-10 c = 0.61 log 9.78533-10. .colog 0.21467 fZ- = 36 W 1.55630 eolos: 8.44370-10 C= log 0.65590 colog 9.34410-10 r = 3 loff 0.47712 -^ = 0.66255 log 9.82122-10 r„ = 1.01840 From Table I. r,^ = 1.68095 whence. ^2 = 2122.2 f.s. From Table I. Also S'„,- f ; T = C{T,.-T,:) bed' ^ w = S70 log 3.93952 S=z0.997 log 9.99870-10. .colog 0.00130 c = 0.61 log 9.78533-10. .colog 0.21467 d- = U4: log 2.15836 colog 7.84164-10 C= log 0.99713 colog 9.00287-10 ^ = 12000 log 4.07918 -— = 1208.0 log 3.08205 -S',, =3227.5 From Table I. 5'„, =2019.5 hence t7^ = 2900 f. s. From Table I. Also T,^ = 1.072 From Table I. r,,^ = 0.625 From Table I. 2^^, _ 2^^,^-0.447 log 9.65031-10 C= log 0.99713 T = 4A4:0G log 0.64744 Therefore the initial velocity was 2900 f. s., and the elapsed time was 4.4406 seconds. 101. Again, suppose the projectile given in the preceding paragraph started with an initial velocity of 2900 f. s., and traveled for 3 seconds, under atmospheric conditions as given ; how far did it go in that time ? C = as before. T = C{ T,, - T,, ) . hence T,„ = -^ + T,^ ; S = C{ S,^_ - S,^) C = as in preceding paragraph colog 9.00287 — 10 T = 3 log 0.47712 -^ = 0.302 .■ log 9.47999 - 10 C '' T,^ =0.625 From Table I. r,^ = 0.927 hence v, = 2635 f. s. From Table I. Also ^,, = 2853.6 From Table I. *S,^ = 2019.4 From Table I. S^^-S,^= 834.2 log 2.92127 C= log 0.99713 *S' = 8287 log 3.91840 Therefore the space traversed was 2763.3 yards. 102. The foregoing methods are of course only strictly applicable to such parts of the trajectory as may without material error be considered as straight lines, since in deducing (51) and (52) we have entirely neglected the effect of gravitation. They will give sufficiently accurate results when applied to any arc of a trajectory if the length of the arc be not materially greater than the length of its chord, and if the latter's inclination to the horizontal be not greater than 10° to 15°; but they are principally applied to the entire trajectories of guns fired with angles of departure not exceeding 3° or 4°, giving the striking velocity and time of flight for ranges as great as 5000 yards, in the case of medium and large guns of higb initial velocity, with as much accuracy as is obtainable by any other method of computation. It will PEACTICAL METHODS 75 be seen later that these formulre may be correctly employed in dealing with the pseudo velocity, and that the fornmlaj then become generally serviceable. 103. By means of the formula^ derived in this chapter we may see how to determine by experimental firing the value of the coefficient of form of any given projectile. To do this the projectile may be fired through two pairs of screens at a known distance apart, and the velocity measured at each pair of screens. Considering the first pair of screens, which gave a measured velocity of v^, we know that this measured velocity is the mean velocity for the distance between the two screens of the first pair, that is, it is the actual velocity at a point half way between the two screens of the first pair; and similarly for the second pair of screens. Therefore the points of measurement giving the distance traversed while the velocity is being reduced from i^^ to v^ are the two points half way between the two screens of each pair, respectively. It will also be understood that this distance between the two pairs of screens must be great enough to furnish a material reduction in the velocity, but not great enough to violate the assumption on which we have been working; namely, that the force of gravity does not afi^ect the flight of the projectile while traversing the distance under consideration. Also, to avoid introducing errors resulting from the action of gravity, the two pairs of screens should be in the same horizontal plane. We then have the formulae : /in S = CiSv, — Sv^), or, as 0=.-^—^ > for s^^ch firing S = ^—.— {S V-, — S V,) , from which from which we can solve for the value of c from the observed velocities, by the use of Table I of the Ballistic Tables. 104. As an example of the above, a Krupp 11.024" gun fired a projectile weigh- ing 7G0.4 pounds through two pairs of screens 328.1 feet and 6561.7 feet, respectively, from the gun, giving measured velocities of 1694.6 f. s. and 1483.3 f. s., respectively, at the pairs of screens. The value of 8 at the time of firing being 1.013, find the value of c for the projectile used. Using formula (53), we have ,«^,.^ = 7389.5 From Table I. aSi., = 6377.3 From Table I. ^,^-.9,, =1012.2 log 3.00527 w = 760.4 log 2.88104 8 = 1.013 log 0.00561 colog 9.99439-10 f? = 11.024 log 1.04234 2 log 2.0SJ68 2 colog 7.91532-10 /S = 6233.6 loi? 3.79474 coloS = 150 log 2.17609 ■— = 25.4 log 1.40503 5... =4616.5 From Table L >S'i,^ = 4591.1 whence t;i = 2134.5 f.s. From Table I. That is, the initial velocity in this case is 2134.5 foot-seconds. PEACTICAL METHODS 77 EXAMPLES. Note. — The answers to these problems, being obtained by the use of formulas and methods discussed in this chapter, all depend for their accuracy upon the correctness of the assumption that in every case the effect of gravity upon the flight of the projectile is negligible for the portion of the trajectory involved. They are, therefore, only accurate within the limits imposed by this assumption. 1. Given the data in the first five columns of tlie following table and two velocities ; or one velocity and either S and T, compute the data in the other two of the last four columns. Projecti e. Atmos phere. V,. f.s. t'o. f.'s. 8. Feet. T. Sees. Problem. d. In. to. Lbs. c. Bar. In. Then °F. A 3 3 4 5 5 6 6 6 7 1 8 10 10 12 13 13 14 14 13 13 33 50 50 105 105 105 165 165 260 510 510 870 1130 1130 1400 1400 1.00 1.00 0.67 1.00 0.61 0.61 1.00 0.61 1.00 0.61 0.61 1.00 0.61 0.61 1.00 0.74 0.70 0.70 28.00 29.00 30.00 31.00 30.00 29.00 28.00 28.25 29.. 50 30.75 31.00 30.00 29.00 28.00 29.00 29.53 30.00 29.00 5 10 20 30 40 50 00 70 80 90 95 100 85 75 59 52 45 11.50 2700 2900 31.50 3150 2600 2800 2800 2700 2700 2750 2700 2700 2900 2000 2000 2000 2600 935 2300 2552 2003 2561 2013 2247 2474 2313 2133 2.541 2114 2316 2154 1833 1756 1900 2343 3978.0 1763.3 3024.2 6720.9 5549.4 9779.2 5581.0 5289 . 8 4471.5 10799.0 4562.4 10942.0 11920.0 27530.0 4746.7 9042.8 3947.9 8767.7 3.8810 B 0.7078 C 1.1110 D 2.6819 E 1 . 9556 F 4.2756 G 2.22.53 H 2.0087 I 1.7897 J 4.4980 K 1.7241 L 4.5808 M 4.7642 N 11.0280 2.4774 P 4.8340 2.0289 R 3.5527 2. Given the data contained in the first seven columns of the following table, compute the value of the coefficient of form of the projectile in each case. DATA. ANSWERS. Problem. Projectile. Value of 5. Distance of pairs of screens from gun.* Measured veloci- ties at. Value of c. d. In. Lbs. First pair. Feet. Second pair. Yds. First pair. f.s. Second pair, f.s. A 3 3 4 5 5 6 6 6 7 13 13 33 50 50 105 105 105 165 1.0.57 0.989 l.Ul 1.062 0.899 0.950 1.107 1.009 0.937 75 150 150 200 200 200 250 250 250 500 1000 1500 1500 1200 1200 i)00 800 900 1100 2650 2875 3130 31.30 2550 2750 2760 2650 1017 2090 2390 2399 2830 2348 2461 2620 2448 1.00570 B 1.00120 C... D 0.672.50 1.01260 E 0.59322 F 0.62045 G 1.01390 H 0.60165 I 0.97983 * The distance given in these columns is in each case the distance from the muzzle of the gun to a point midway between the two screens composing the pair. EXTERIOR BALLISTICS 3. Given the data for actual firing contained in the first six columns of the following table, compute the initial velocity of the gun in each case. DATA. ANSWEPvS. Projectile Di.stance of screens from gun. Elapsed Problem. Value of 5 time between Initial velocity. d. In. First Second screens. f.s. Lbs. c. screen. screen. Sees. Feet. Feet. J 7 165 0.61 150 350 0.925 0.074 2716 K 8 260 0.61 200 400 1.021 0.074 2717 L 10 510 1.00 250 450 0.899 0.074 2722 M 10 510 0.61 2.50 450 1.125 0.075 2681 X 12 870 0.61 250 450 0.937 0.071 2828 13 13 14 14 1130 1130 1400 1400 1.00 0.74 0.70 0.70 250 250 250 250 550 550 600 600 1.015 0.954 1.115 0.913 0.153 0.149 0.176 0.136 1976 P 2024 2001 R 2585 4. The initial velocity of a 3" 13-pound projectile, c = 1.00, is 2800 f. s. Atmos- pheric conditions being standard, what are the elapsed times until its velocity is reduced to 3600 f. s., 2400 f. s. and 2200 f. s.? Answers. 0.341, 0.729 and 1.176 seconds. 5. The initial velocity of a 3" 13-pound projectile, c = 1.00, is 2800 f. s. Atmos- pheric conditions being standard, what are the spaces traversed while the velocity is being reduced to 2600 f . s., 2400 f . s. and 2200 f . s. ? Answers. 922, 1890 and 2917 feet. 6. A 12" projectile weighing 850 pounds, c=1.00, has an initial velocity of 2400 f . s. What is the remaining velocity at 1000, at 2000 and at 3000 yards range ; first in an atmosphere of standard density ; and, second, when the thermometer is at 82° F., and the barometer at 29.20"? Ansivers. 2256, 2118, 1986 f.s. 2264, 2133, 2008 f.s. 7. A 12" projectile weighing 850 pounds, c=1.00, has an initial velocity of 2400 f. s. What is the time of flight for 1000, for 2000 and for 3000 yards range; when the temperature is 82° F., and the barometer is at 29.20"? Answers. 1.283, 2.647, 4.092 seconds. 8. How long does it take a 6" projectile weighing 100 pounds, having an initial velocity of 2000 f. s., to travel 1000, 2000 and 3000 yards, and what is the remaining velocity at each range; the temperature being 44° F., and the barometer at 30.15"? Answers. 1.614, 3.492, 5.668 seconds. . 1725, 1485, 1282 f. s. 9. A Krupp 5.91" gun, projectile weighing 112.2 pounds, having an initial velocity of 1667.6 f. s., in experimental firing under standard atmospheric conditions, gave as a mean of ten shots, measured velocities of 1656 f. s. at 164 feet from the gun, and 1358 f. s. at 4921 feet from the gun. Compute the velocities at those ranges and from a comparison between the observed and computed ranges, determine the coefficient of form of the projectile used. Answers. Computed velocities . ... 1656. . . . 1363 f. s. Observed velocities .... 1656 .... 1358 f . s. Value of c 1.00 PRACTICAL METHODS 79 10. A Knipp 5.91" gun, projectile weighing 112.2 pounds, having an initial velocity of 1763.6 f. s., in experimental firing under standard atmospheric conditions, gave measured velocities of 1740.7 f. s. at 328 feet from the gun, and 1369 f. s. at 6562 feet from the gun. Compute the velocities at those ranges, and from a com- parison between the observed and the computed ranges, determine the coefficient of form of the projectile used. Amwers. Computed velocities. .1740 1348 f. s. Observed velocities.. .1740.7. . . 1369 f. s. Value of c 1.00 11. A Krupp 12" gun, projectile weighing 1001 pounds, having an initial velocity of 1721.7 f . s., in experimental firing under standard atmospheric conditions, gave measured velocities of 1711 f. s. at 328 feet from the gun; 1692 f. s. at 984 feet from the gun, and 1518 f. s. at 6563 feet from the gun. Compute the velocities at those ranges, and from a comparison between the observed and computed ranges, determine the coefficient of form of the projectile used. Answers. Computed velocities. .1711. . . .1690. . . .1521 f . s. Observed velocities.. .1711. ... 1692. .. .1518 f . s. Value of c 1.00 12. A 4.72" gun, firing a projectile weighing 45 pounds, c=1.00, in experi- mental firing, when the thermometer was at 65° F. and the barometer at 30.43", gave a measured velocity of 2204 f. s. at a point 175 feet from the muzzle. What was the initial velocity? Answer. 2228 f. s. 13. A Krupp 11.024" gun, projectile weighing 760.4 pounds, in experimental firing, under atmospheric conditi-ons when 8 = 1.013, gave measured velocities of 1746 f. s. and 1529 f. s. at points 328 and 6562 feet from the gun, respectively. Com})ute the value of the coefficient of form of the projectile used. Anstver. c = 0.9992. 14. A flat-headed, 6" projectile, weighing 70 pounds, when fired through two pairs of screens 150 feet apart, gave measured velocities at those two points of 1881 f. s. and 1835 f. s. Atmospheric conditions being standard at the time of firing, compute the value of the coefficient of form of the projectile. Answer. c=2.46. 15. A Krupp 9.45" gun, projectile weighing 474 pounds, in experimental firing, under atmospheric conditions when 8 = 1.06, gave measured velocities of 1719 f. s. and 1460 f. s. at points 328 feet and 6562 feet, respectively, from the gun. Compute the value of the coefficient of form of the projectile used. Answer. c = 0.9969. CHAPTER 7. THE DIFFERENTIAL EaiTATIONS GIVING THE RELATIONS BETWEEN THE SEVERAL ELEMENTS OF THE GENERAL TRAJECTORY IN AIR. SIACCI'S METHOD. THE FUNDAMENTAL BALLISTIC FORMULA. THE COMPU- ^TATION OF THE DATA GIVEN IN THE BALLISTIC TABLES, AND THE USE OF THE BALLISTIC TABLES. New Symbols Introduced. u. . . . Pseudo velocity at any point of the trajectory in foot-seconds. du. . . . Differential increment in u. Su- • • • Value of space function in feet for pseudo velocity u. Sy. . . . Value of space function in feet for initial velocity V. Tu. " ' Value of time function in seconds for pseudo velocity u. Tr. . . ' Value of time function in seconds for initial velocity V. Au. . • .Value of altitude function for pseudo velocity u. Ay. ■ . .Value of altitude function for initial velocity V. In- ■ ■ • Value of inclination function for pseudo velocity u. Iv. . ■ ■ Value of inclination function for initial velocity V. 109. From the hypothesis already made in paragraph 69 that the resultant atmospheric resistance acts in the line of the projectile's axis, which itself coincides with the tangent to the trajectory at every point, it follows that the trajectory is a plane curve. Por, if a vertical plane be passed through the gun and through any point of the 'trajectory, that plane will contain the only two forces acting upon the projectile while it is at that point, namely, gravity and the resistance of the air; and so their resultant will lie in that plane also, and there will be no force tending to draw the projectile from that plane, and so the next consecutive point of the curve must lie in the same plane also ; and so on to the end. 2t Figure 11. Derivation 110. Figure 11 represents a portion of the trajectory with the two forces acting of the differ- ^^^^ ^^^ projcctile, namely, its weight, lo, acting vertically downward, and the ential equa- tions. resistance of the air, R=-Xj^v'^, acting along the tangent to the curve in a direc- PEACTICAL METHODS 81 f tion opposite to that in which the projectile is moving. Kinetic equilihrium results from the balancina; of these two forces bv the inertia forces — ■ X -rr actinsr in the g dt tangent, and — X — acting in the normal, p being the radius of curvature of the curve at the point under consideration.* Eesolving forces along the normal to the traiectorv, the inertia force — X — , commonlv called the centrifugal force, must g 9 ^ balance the resolved part of w along the same line, whence we have — X — =w cos 6. or — =gcosd (54) P In other words, the acceleration towards the center of curvature is the resolved part of g in that direction. But the radius of curvature p= — -jr- , whence v-dO= —g COS 6 ds= —gdx, or gdx——v-d9 (55) doc 111. Dividing each side of (55) by dt and putting Vh for -jj, we have qvh= —v^ —J- , whence qdt= = — ;: , ^ dt ^ Vft i; cos ^ whence gdt= —v sec 6 dd (56) 112. By putting cot 6 dy for dx in (55) we get g cot 6 dy= —v-d6 or gdy= —v- tan 6 dO (57) 113. By putting cos 6 ds = dx in (55) we get g cos 6 ds= —v'dO or gds= —v- sec 6 dO (58) 114. Now resolving horizontally, since the horizontal component of the atmos- pheric resistance, — X -T7 f " cos 6, is the only force which acts to produce horizontal 9 ^ acceleration, -^r-- = —f^ , which in this case is negative acceleration, we have ' dt~ dt ' "^ dVk - d(vcose) __A^ f, dt Jt C ' '°' ^ but from (56) we know that gdt= —v sec dO, therefore the above expression becomes gd(v cos 6) =~v^<'^^^de (59) 115. Grouping the expressions derived above together, we have The diflfer- ential equa- gd(v cos 6) = -^ v^'^^^'dO (60) '"°°'- gdx=-v~de (61) gdt=-v sec Odd (62) gdy=-v- tan Ode (63) gds=-v- sec Odd (64) and these equations, (60) to (64) inclusive, are the differential equations giving the relations between the several elements of the trajectory; and, could (60) be integrated, thus giving a finite relation between v and 6, one of those variables could * See any standard work on the subject for the derivation of this expression. 6 EXTERIOR BALLISTICS Pseudo velocity. be eliminated from equations (61) to (64) inclusive, and then values of x, y, t and s could all be obtained either exactly or by quadrature. (The " quadrature " method need not be taken up in this elementary treatise on the subject.) 116. Since we cannot integrate (60), it is necessary to resort to methods of approximation, and a method devised by Major F. Siacci, of the Italian Army, has been generally adopted by artillerists because of its comparative simplicity and readiness of application. It is to be noted that in all the preceding chapters dis- cussing the trajectory in air and deriving mathematical expressions in regard to it, we have dealt with approximate methods only, and we now see that we are again, and for our final and most approved methods, driven to fall back upon another approxi- mate system. However, this one, Siacci's method, has been found to be accurate within all necessary limits for ordinary ballistic problems, and for our purposes we may consider it as exact, in contradistinction to the more approximate methods that we have hitherto considered. It should not be forgotten, however, that the method is not literally exact, and that it might be possible, under unusual and peculiar con- ditions, that results might vary appreciably from those actually existent in practice. It is not ordinarily necessary to consider this point, but should some unusual and peculiar problem present itself, it would be necessary to consider whether the con- ditions were such as to introduce material inaccuracies into results obtained by the ordinary methods. 117. Taking rectangular axes in the plane of fire; origin at the gun; X, as always, horizontal, and positive in the direction of projection; Y, as ahvays, vertical, and positive upward; let V, in Figure 12, be the initial velocity, and the angle of departure, and let v be the remaining velocity at any point, P, of the trajectory. Then, resolving v vertically and also parallel to its original direction, and designating by u the component parallel to the initial velocity, we have u = v cosd sec (l> (65) JC Figure 13. 118. The quantity u, which is represented in Figure 12, is known as the " pseudo velocity," and its use in the solution of practical ballistic problems is due to Major Siacci, and is the essence of his method. It will readily be seen that u=V at the origin, where 6 = cf>, and also at another point in the descending branch of the tra- jectory where 6=—4>. At the vertex, where 6 = 0, the pseudo velocity differs most PKACTICAL METHODS 83 from the true velocity, its value there being ii= v sec , but if be small the differ- ence is very small, and for flat trajectories {(f> not greater than about 5°), u may be considered as equal to v throughout the entire trajectory without material error. 119. Substituting u cos <^ for v cos 6 and ^^ for v in (60), we get siacci's cos u method. gau- ^ X ^^g«,,i,^ u a(f whence ^. - X -^ = -^^ sec^ 6 d6 (66) A w<«+i) cos<"-^^^ ^ ' Now the value of a definite integral remains unchanged when for any variable under the integral sign is substituted a constant which is the mean value of that variable-^ between the limits of integration. Thus for the variable quantity * ^ ^, ,,1 we may substitute a constant, p, thereby enabling {QG) to be integrated without introducing any error, provided we assign to y8 its proper value. It may be said here that, for all practical purposes, in the use of the methods of exterior ballistics in connection with the ordinary problems of naval gunnery, /8 may always be considered as equal to unity without exceeding the limits of accuracy within which we are otherwise able to work. In certain special classes of firing this value of ^ = 1 cannot be used without introducing material error, the principal such classes being mortar and high-angle firing; but as these are not methods within the province of ordinary naval gunnery, the value of ^=1 will be adopted in the formulae throughout the rest of this book, and the factor ^ will be allowed to disappear from the formulae, as, when it is equal to unity, it does not affect the results obtained by their use. (For a further discus- sion of the value of /?, see Alger's Text Book on Exterior Ballistics, Edition of 1906, page 58, et seq., and other standard works on the subject.) We know that COS" <^ = cos<«--+2'<^ = cos<»-2)<^ cos- <^ and by substituting ^ C0s'""-' • COS'^d) , C0S*^ cosc-^'^ ^ ^ whence, by substitution in (6G) and expressing the direct integration, we get NovHf wecall — ^- rT^ny —^u, then (67) becomes, after integration, tan 2 COS" (j> / \. / 120. Now let us put ^^l^^f for -y in (61), and we will have COS (I dr. = — u- cos- ^ sec- 6 (16 or, substituting for sec- 6 dd its value from {QQ) whence |%/.r = - C £ -1- X ^^ (69) * The constant factor cos"-2i0 is taken out with the variable factor ^— j- for the cos"* — ' 6 reason that their product, the mean value of which we call /3, differs very little from unity. 84 EXTERIOE BALLISTICS whence x = CiSu-Sr) (70) 1 f du in which S,, stands for — . A JM 121. Proceeding in exactly the same way with (62), we get gdt= -u cos cf> sec^ e de= — X 4- X ^ [' ., C [^ 1 ^ du or \ dt= \ -J- X -—■ Jo cos<^ iv A u'^ or t=:-^{Tu-Ty) • (71) cos<^ in which T„ stands for t- — -r- A J w* 122. Now, multiplying (68) by the differential of (70), and putting dij for tan Q dx, we have dxj — tan <^ dx — - ^--- (7« — ly) X ^- X 2cos-(^ ' '> '- A m(«-^' and by putting — -,- f/w -^mX -7^1^ —Au, this becomes, after integrating. C 2/-a;tan<^=— 5— [A„-Af-/f('S'«-/Sf)] -^ 2 cos^ ^ ^ -" and, tinally, dividing by (70), we get l^=un.t-^-^(i^-iv) (re) X 2cos <^\Om — Of / Note that in this substitution that Au means the A function of u, otherwise known as the altitude function, this A having of course nothing in common with the constant A on the other side of the equation. The bauistic 123. The formulas given iji equations (65), (68), (70), (71) and (73) are formula. j,j-^q^^,j^ ^g ^j^g ballistic equations, and are the ones on which are based all the principal problems in exterior ballistics. They are here repeated, grouped, for convenience, as follows : x=C(Su-Sv) (73) -^ = tan - — ^ fl"^ -ly) (74) tan ^ = tan -— ^ (7„-2f) (75) 2 cos- <^ t=-^{Tu-Ty) (76) cos ^ v = ucos (l> secO (77) 124. These ballistic formulae express the values of (a) The two coordinates of any point of the trajectory ; (b) The tangent of the angle of inclination of the curve to the horizontal at any point of the trajectory : (c) The time of flight to any point of the trajectory; and (d) The remaining velocity at any point in the trajectory directly as functions of 1. A new variable, u, known as the pseudo velocity, and already defined; 2. The ballistic coefficient; 3. The angle of departure; and 4. The initial velocity. PRACTICAL METHODS 85 The formula could therefore be used in solving problems by working out the values for each case represented by the symbols S, A, I and T, with their appropriate subscript letters in each case; but, in order to facilitate the process, the values of these integrals corresponding to all necessary velocities have been worked out and made available in the columns of Table I of the Ballistic Tables. For all velocities between 3600 f. s. and 500 f. s., using the velocities as argiiments in the left-hand column of the table, headed u, we may find the corresponding values of the desired integrals, under the appropriate columns, headed, respectively, Su, Au, h and Tu. 1 f" du f 125. As shown bv (71), the value of the definite integral j- — — when The time '' ^ A. ]y Vr \ function. multiplied liv —^1 measures the time of flight from a point where the pseudo ^ • cos / velocity is V to one where it is u. We have represented this definite integral by Tu — Tv, and we require a table from which its value may be taken for any given values of V and u. But, as explained in Chapter 6, the values of Tv{Tu) in Table I were calculated by the use of the formula ^ _ _1_ f« _dv_ '-^ J 3600 '' and the integral given above is the same as this, except that u, a velocity, is sub- stituted for V, a velocity; so that the results for the same velocity, whether real or pseudo, would be the same, and what we formerly called Tv is the same thing that we now designate Tu — T^^^q. Consequently, we have always that Tu^ — Tu=Tv„—Tv^, and the tabulated time functions may be used indiscriminately for either real or pseudo velocities, provided the proper quantity be taken as an argument. 1 f" du 126. In the same way, in (70), the value of the integral — -j- -j^p^] , which The space we have represented by Su — Sr, measures (when multiplied by C) the horizontal space traversed while the pseudo velocity changes from Y to u, and, as the values of Su in Table I, as explained in Chapter 6, were calculated by the use of the formula cf If'' dv o,,= — ^ .'3600 '^ and the integral given above is the same as this, except that u, a velocity, has been substituted for v, a velocity, so that the results for the same velocity, whether pseudo or real, would be the same, and what we formerly called Sv is the same thing that we now designate Su — Ss^qq. Consequently, we now have 8u^ — Su^ = Sv^ — Sv^, and the tabulated values of the space function, Su, may be used indiscriminately for either real or pseudo velocities, provided the proper quantity be taken as an argument. 127. As shown by (68), the value of the definite integral f- [ when The incUna A JyU'- \ tion functio C \ multiplied bv r ^ — measures the change in the tangent of the inclination of the '■ -^ 2 COS^ 4>J or, trajectory from the point where pseudo velocity is V to the point where it is u. We have represented this definite integral by lu — Iv, and we require a table from which its value, for any given values of V and u may be taken. To supply this, the values of '^"-have been computed for values of u from 3600 f. s. to 500 f. s., and -19_ A. Joo u' placed in Table I under the heading 7„, with u as an argument in the left-hand column. Then, as in the case of the time and space functions, we always have Iu^ — Iu^ = Iv^ — Iv^, and these values may be used for either true or pseudo velocities indiscriminately, provided the proper argument be used. It would have been equally 86 EXTERIOR BALLISTICS well to have tabulated the values of __2^r« du ,(a+ '—, making IggQQ = instead of ■^3600 = -03138, as is the case with the integration performed as indicated for a lower limit of infinity. This would have made the series of values of the inclination func- tion, like those of the space, time and altitude functions, begin at the imagined origin where w=3600 f. s., but as we deal entirely with differences, the point of origin is immaterial. 128. The equations from which 7„ are computed are obtained by substituting successive values of A and a in T = _ ll { ^^^ " A Jii<«+i' and integrating, the first integration being between infinity and u (u from 3600 f. s. to 2600 f. s.) ; the second between 2600 and u {u from 2600 to 1800) ; the third between 1800 and u (u between 1800 and 1370); and so on; the results being as follows : u between 3600 f. s. and 2600 f. s. J _ [4.00897] ^"" " u'-'' u between 2600 f. s. and 1800 f. s. j^^ [4.48170] ^Q QQ,,, u between 1800 f. s. and 1370 f. s. h= [5-38806] +001776 ir V between 1370 f. s. and 1230 f. s. /«= [8-35032] +0 03033 u between 1230 f. s. and 970 f. s. 1^= [14-30751] +0.10912 u between 970 f. s. and 790 f. s. ^ [8.55778] _o.o5028 (78)^ u between 790 f. s. and f. s. /„= [5-83743] _o.4i960 Note. — The above formulse give numerical values correct to five places only. In computing the tables the numerical values were carried out correctly to seven or more places. The altitude 129. As explained in paragraph 122, Au — Ay stands for the value of the definite function. A du 77^' we 1 f" du integral t \ ^u (a-i) > °^' substituting for /„ its value of — A Jy W * have . 2g_ [« du ^' A^-a jyu^"""-^^ * The numbers enclosed in brackets are the logarithms of the constants and not the constants themselves. PRACTICAL METHODS 87 and in order that the value of this integral may be found for any given values of 2g f" (hi V and u, the values of — have been computed for values of u from 3600 f. s. to 500 f. s., and will be found in Table I under the heading Au, with values of u as arguments in the left-hand column. Then, as in the case of the space, time and inclination functions, we have always Au^—Au^ = Av^—Av^, and the tabulated values of the altitude function may be used for either real or pseudo velocities indis- criminately, provided the proper argument be used. 130. The equations from which the values of Au are computed are obtained by substituting the successive values of A and a in the expression A -- J5L A-a f du and integrating, exactly as was done in Chapter 6 in finding the values of Tv, except that the values of lu to be substituted in the integral expression for the value of Au — Av must include the constants of integration whose values are given in (78). The results are : u between 3 GOO f. s. and 2600 f. s. u between 2600 f. s. and 1800 f. s. Au= LL^-JJll] _ [1.08179]m''-3- 39.264 u between 1800 f. s. and 1370 f. s. 4„=iMC3^^-[2.49233]logii+ 1052.0 u between 1370 f. s. and 1230 f. s. ^^^[1M6^+ [5.80321],,,, 3^ u between 1230 f. s. and 970 f. s. [26.60254] [11.75890] u between 970 f. s. and 790 f. s. ^__^ [^508228] _ i^Mnil ^gj^^j w* u u between 790 f. s. and f. s. A,= IM^J^] +[4.31515]logw-68192.0 Note. — The above formulae give numerical values correct to five places only. In computing the tables the numerical values were carried out correctly to seven or more places. 131. In the preceding work in this chapter we have followed the methods given by Professor Alger in his most excellent book on Exterior Ballistics, which are the generally accepted methods, and have derived certain ballistic formulae as given in (79) * The numbers enclosed in brackets are the logarithms of the constants and not the constants themselves. Table I. 88 EXTEEIOE BALLISTICS equations (73) to (77) inclusive. We have also shown how the values of the space (Su), time (Tu), inclination (/«) and altitude (Au) functions for varying values of the real or pseudo velocities have been computed and made readily available in Table I of the Ballistic Tables. Professor Alger accepts the results already obtained as being sufficient for all practical purposes, and uses these equations in the form in which we already have them (rearranged to suit each special problem) in the solution of ballistic problems. Colonel Ingalls, however, proceeded still further with the reduction of these formulae, and by most noteworthy mathematical work succeeded in getting resulting expressions that vastly reduce the labor of the computer below that involved in the use of the formulse as they stand above. These reductions are somewhat involved, but, when once carried through, so simplify the solutions of problems and reduce the labor connected therewith, that Ingalls' methods have become generally accepted for work of this nature. Their acceptance and use involved the computation of another extensive table. Table II of the Ballistic Tables, but with this table and Ingalls' formulas, the work of the computer is reduced to a minimum. Ingalls' methods and formula may be more appropriately considered in the solution of certain special problems, and the study of them is therefore deferred to the next chapter. Vse of 132. As an example of the use of Table I, let us suppose that it is desired to take from it the values of the four functions corresponding to a velocity, either real or pseudo, of 2727 f. s. For determining the value of Su, we have: For ?i = 2720, g^ = 2517.7+ ^^''t^ ^^ =2517.7 + 41.2 = 2558.9 and similarly .4„ = 97.94+ ^•^^^^^ =97.94 + 1.98 = 99.92 I„ = .04791+ -QQQ'^S X 13 ^,04791 + .00036 = .04827 r„ = .802+ ^^^|^^ = .802 + .015 = .817 133. And, similarly, suppose that we want to find the value of the real or pseudo velocity (which one it is depends upon the formula in use) corresponding to a value of /„ = . 05767. The nearest tabular value to this is /«=. 05775, corresponding to m=2430 f. s., from which, by interpolation, w(ort;)=2430+ ^^^=2430 + 2.2 = 2432.2 f. s. 134. If we had desired to find the value of Au corresponding to the value of 7„ = . 05767 given above, we could find u as just described, and then find the value of Au thus A„ = 151.46- '■'''^^■^ =151.46-0.43 = 151.03 PEACTICAL METHODS 89 or we could proceed without findino; the value of u, thus .4„ = 151.46-^^^ X ^ = 151.46 -0.43 = 1.51.03 This latter method will frequently be found necessary in the use of Table IT, and it should be practiced until it can be done quickly and accurately. 135. As an example of the problems that may be solved by the methods indicated in this chapter, let us suppose we have a 3" gun, standard projectile weighing 13 pounds, with an initial velocity of 2800 f. s., and that the angle of departure for a horizontal range of 3000 yards is 1° 53'; and that we desire to determine the pseudo velocity and the horizontal distance traversed at the moment when the projectile has been in flight 4 seconds; atmospheric conditions being standard. C=~; t= -^ (T,-Tv) ^vhence Tu=^-^ +Ty; x = OiSu-Sr) Cl- cos (p L iv = rs log 1.11394 d- = d log 0.95424 colog 9.04576-10 C= log 0.15970 colog 9.84030-10 ^ = 4 : log 0.60206 =l° 53' cos 9.99977-10 ^-—^ = 2.768 log 0.44213 n-^ 0.734 From Table I. T„ = 3.502 whence w=1411 f. s. From Table I. Su = 7769.2 From Table I. »Sy = 2329.1 From Table I. 6'u-;SV = 5440.1 log 3.73561 C= loff 0.15970 a;= 7858 log 3.89531 Pseudo velocity 1411 f. s. Distance traveled 2619 yards. EXAMPLES. 1. In the 4000-yard trajectory of a 3" gun (7 = 2800 f. s.), = 3° 10'. What is the pseudo velocity at a point horizontally distant from the gun 2000 yards, and what is the time of flight to that point? Anstvers. 1671 f. s. ; 2.791 seconds. 6. The 4000-yard trajectory of a 6" gun (7 = 2400 f. s., w = 100 pounds) has tf> = 2° 52'. What is the pseudo velocity at a point horizontally distant from the gun 2000 yards, and what is the time of flight to that point? Answers. 1829 f. s.; 2.856 seconds. 7. In the 4000-yard trajectory of a 5" gun (7 = 2550 f. s., w = 50 jwunds), ^ = 3° 01'. What is the pseudo velocity at a point horizontally distant from the gun 2200 yards, and what is the inclination of the curve at that point? Answers. 1683 f. s.; 0° 07' 12". 8. In the 6000-yard trajectory of a 6" gun (7 = 2300 f. s., w = 100 pounds), ^ = 5° 53'. What is the pseudo velocity after 1000 and after 2000 yards horizontal travel, and what are the ordinates of the trajectory at those distances ? Ansivers. 2010 and 1747 f. s.; 279 and 485 feet. 9. In the 3000-yard trajectory of a 3" gun (7 = 2800 f. s., w = 13 pounds), <^ = 1° 53'. What are the pseudo velocities and what the horizontal distances traveled after 1, 2 and 3 seconds flight? Answers. 2276, 1898 and 1619 f. s.; 840, 1529 and 2117 yards. 10. Determine the reduced ballistic coefficient for a 10", 500-pound projectile of standard form, if the temperature and barometric height at the gun be 84° F. and 29.12", and the time of flight be 16 seconds. If the initial velocity be 2000 f . s., given the above time of flight, find the range and the pseudo velocity at the point of fall. A7iswers. C = 5.432 ; i2 = 7927 yards; Wc.= 1151 f. s. 11. Determine the reduced ballistic coefficient for an 8", 250-pound projectile of standard form, the temperature being 46° F., and the barometer 30.11", the time of flight being 22 seconds. If the initial velocity be 2300 f. s. and the angle of departure be 11° 00', find Uc from the given value of the time of flight, and then find the range. Ansivers. C = 3.85S1; i/w = 974.3 f. s,; /? = 9947 yards. CHAPTEE 8. THE DERIVATION AND USE OF SPECIAL FORMULA FOR FINDING THE ANGLE OF DEPARTURE, ANGLE OF FALL, TIME OF FLIGHT AND STRIK- ING VELOCITY FOR A GIVEN HORIZONTAL RANGE AND INITIAL VELOCITY; THAT IS, THE DATA CONTAINED IN COLUMNS 2, 3, 4 AND 5 OF THE RANGE TABLES. INGALLS' METHODS. New Symbols Introduced. Uu. . . ' Pseudo velocity at the point of fall, t'o,. . . .Eemaining velocity at point of fall, or striking velocity. Su^ .... Value of the space function for pseudo velocity Uu. Tu^ .... Value of the time function for pseudo velocity Uu- Aj,^. . . .Value of the altitude function for pseudo velocity v^a. It,^.. . . .Value of the inclination function for pseudo velocity w^. Su^. . . .Value of the space function for pseudo velocity u^. Tu^. . . .Value of the time function for pseudo velocity u^. Au^. . . .Value of the altitude function for pseudo velocity Uq. /„„. . . .Value of the inclination function for pseudo velocity Uq. . . Difference between two values of the space function. . . Difference between two values of the time function. . . Difference between two values of the altitude function. . . Difference between two values of the inclination function. . . General expression for value of argument for Column 1 of Tabic TI. . . Special expression for value of argument for Column 1 of Table II. z = Z = B' = C, C^, C3, AT.. AA.. A/.. X x_ C " a. . h.. a' . . t' . . A.. B. . A'.. rpt A " A".. etc. 'General values of Ingalls' secondary functions. 'Special values of Ingalls' secondary functions for whole trajectory. , Successive values of C. The same system of notation by subscripts also applies for successive approximate values of other quantities where such use of them is necessary. 92 EXTERIOE BALLISTICS 136. As already deduced, the six fundamental ballistic formulte are : x = C{S,-Sr) (81) tan^zz:tan«/>- ^ {lu-Iv) (83) 3 COS" t = C seecl>{Tu-Tv) (84) V = u cos (f) see 6 (85) 137. It Avill be seen from the above that special formulas may be derived to fit all particular cases, which special formulas will contain only the quantities contained in the above fundamental equations ; that is, quantities that are either known or con- tained in the Ballistic Tables, or the values of which are to be found. Transforma- 138. Let US apply these formula to the special case under consideration, that is, tic formula' to the derivation of special formulas for computing the values shown in Columns 2, 3, 4 and 5 of the range tables. For the complete horizontal trajectory we may substitute in the fundamental equations as given above, as follows: x=Xj y = ^, t = T, v = V(o and 6= —0). If we make these substitutions we get: From (80) C=fi^^ (86) From (81) S,^ = Sy+ ^ (87) From (82) tan ^= ,^-^ (4^'^-^-- -^v) 2cos^cl>\ S,,^-Sy I 2sin<^cos-c^ ^^.^^ ^^Al»,^_^ \ ^gg) COS^ \bu^ — i^V I From (83) tan(-a>) ^tan - ^ (7^ -7^.) 2 COS" ^ ^ and substituting in this the value of tan given above we get tan a>= ;^-^ (7„^- i^-~iA (89) 2 cos- ci V ^ O,, , — '^F / From (84) T = C sec cl>{T„^-Tv) (90) From (85) Via = Uu cos 4> secot (91) 139. Considering the aboye expressions, we may note that, with the exception of the quantities that we desire to find, all the quantities contained in them are either known or else may be found in the Ballistic Tables (exclusive of Table II). Pro- fessor Alger uses them in this form, in his text book, for computing the values of the unknown quantities in those expressions. As an example of his methods we will noAV solve a problem by the use of the above formulae as they stand, and without using Taljle II of the Ballistic Tables. 140. For the 12" gun, 7 = 2900 f. s., w = 870 pounds, c = 0.61, atmosphere at method, standard density, to compute the values of the angle of departure, angle of fall, time of flight, and striking velocity, for a horizontal range of 10,000 yards (without using Table II) by Alger's method. Alger's PRACTICAL METHODS 93 A„.-A, 2 cos- (f>\ "'^ ^u^ — ^v I PGCw We cannot get a correct result without determining the value of /; but let us disre- gard that for the moment nevertheless, and proceed for the present as though /=1. The value of C for standard conditions could be taken from Table \1, and this will usually be done to save labor, but for this first problem we will compute it. w = 870 log 2.93953 c = 0.61 log 9.78533-10. .colog 0.31467 (^2 = 144 log 2.15836 colog 7.84164-10 C= log 0.99583 colog 9.00417-10 2: = 30000 log 4.47713 ^ =A»S =3039.0 log 3.48139 4„^ = 253.05 T„^ = 1.880 /S„^ =5048.4 j^l- 75,09 Tf = 0.625 7^ = 0.4388 Wa, = 2014.8 _ Al = 177.96 Ar = 1.255 ^A = 177.96 log 2.25033 A;S = 3029 log 3.48129 M :^.05875 log 8.76903-10 A*b /r = . 04388 ^A AS -/^ = .01487 log 8.17231-10 Ar= 1.255 log 0.09864 C= log 0.99583 log 0.99583 20 = 8° 38' 09" sin 9.16814-10 (^ = 4° 14' 05" sec 0.00119 T= 12.464 log 1.09566 Let us now determine the approximate maximum ordinate for the above trajectory. To do this we have 8 r=12.464 log 1.09566 2 log 2.19133 ^ = 32.2 log 1.50786 8 log 0.90309 colog 9.09691-10 F= 625.3 log 2.79541 |F = 416.87 feet, whence, from Table V, /= 1.0105. We will now repeat the preceding process, using the found value of / to correct the original value of C, in which / was considered as unity, and introducing con- secutive subscripts to the several symbols to represent successive approximate found values. 94 EXTERIOR BALLISTICS Ci= log 0.99583 /i = 1.0105 log 0.0045-i 02= log 1.00037 colog 8.99963-10 Z = 30000 log 4.47712 A^ = 2997.4 log 3.47675 - -^^ =tan 6+ -^ (99) X 2 cos- (f> 2 COS" - ^"''^^ ^ (100) 2 cos- (f> From (84) t = Cf sec cl> (101) From (85) t; = w cos sec ^ (102) 144. Xow Iniralls' Ballistic Tables, Table IT, give values for a, h, a' and t' for different values of V and of 2= -^ , so that, in any given problem, knowing V and x, we can compute the value of z from the proper formula, and then take the correspond- ing values of the secondary functions from Table II. 145. For a complete horizontal trajectory, however, certain other simplifications secondary become possible, if we have Table II available for use. The relations between <^, w, fo^^'ent^re T and fw, and the other elements of the trajectory involve the complete curve from the ^^^^^ °^^' gun to the point of fall in the same horizontal plane with the gun. Under these con- ditions we have that y = 0; ^=0; and^=-a, When this is the case our equations become : X = CiSu,-Sv) (104) tan ,^=^-^(4^"— ^-7,) 2 COS^ 0\ bu^^ — Oy / or, as 2 tan ^ cos- = 2 sin cos <^ = sin 2 sin2 = cf 'l"^"f^ -ly] (105) V bu^ — by I tanco=-tan<^+^^(/„„-M 96 EXTERIOR BALLISTICS Substituting in this the value of tan <^ from the first expression used in deducing equation (105) we get C fr * A,,-Ay T = C sec (T,,^-Ty) (107) t;^ = ?/„ cos <^ sec w (108) 146. Now let us take another set of secondary functions, or rather a set of special values of the regular secondary functions, as follows : A_^u.-Av_j^ (109) Su . — 5' B = In^-i'^^-^ (110) A' = A+B = I„^~Iv (111) T' = T,^-Tv (112) ingaUs' 147. ISTow by combining these special values of the secondary functions given in formula, equations (109) to (113) inclusive with the formula? in (103) to (108) inclusive, we get X = C{Su,-Sv) (114) sin2 (117) ■y^ = 7/^ cos ^ sec w (118) 148. It will readily be seen, as already stated, that, as A, B, A' and T' are only special values of a, h, a' and t', the values of both sets of secondary functions, the general and the special, may be taken from Table II, provided we enter the table with the proper argument in each case, that is, with the corresponding values of Y and of X X z= ytOt Z= -^ , z being of course for any point in the trajectory whose abscissa is x, and Z being for the entire horizontal trajectory, where the abscissa is the range X. And, as stated above, values of log B may also be taken from Table II as required, with the same arguments. 149. We therefore see that, by certain somewhat involved mathematical processes. Colonel Ingalls put tlie ballistic formulae into such form that their use involves the least possible amount of logarithmic work. By the use of equations (113) to (118) inclusive, we can find the value of ^, w, T and i'^, by Ingalls' methods, provided we have Table II from which to take the values of the secondary functions, and provided PRACTICAL METHODS 97 we make no correction for altitude. Before we proceed to computations, however, we must find out how to determine the value of /, and, as a preliminary to this, we must consider the values of the elements at another important point of the trajectory, that is, at its highest point, summit, or vertex. This question is more fully discussed in a later chapter, but a certain preliminary consideration of it is necessary here in order to enable us to correct for / in using the formulse just derived. The coordinates, etc., at the vertex are denoted by the symbols already used, with the subscript zero. 150. At the vertex (Xq, y^), we have that 6 = 0; hence at this point the funda- mental equations become : In^alls' formulae for vertex. C: (119) By putting ^ = in the expression for tan 9 we get tan <^ = C 2 cos- cj) {Iu,-Iv), or sin 2cj> = C{Iu,-Iv) or Iu=Iv + sin 2cf> C cc^ = C(Su-Sv) ^=tan-- ^ Xq ' 2 COS- \ Su^ — Sy t, = Cseecf>(Tu,-Tv) Vq = Ufy COS ^ -/, (120) (121) (122) (123) (124) 151. Alger solves for the elements at the vertex by the use of the above expres- sions and of Table I. Ingalls, however, simplifies the work of the computer by some further preliminary reductions, as he did in the preceding case. The important point at issue in the present problem is to find a simple expression for finding the value of the ordinate of the vertex, in order to find the value of /. Ingalls, by a somewhat involved mathematical process which we will not follow through in this chapter, finally reduced the formula to give the following expression for the value of the maximum ordinate: r = A"C tan (^ (125) 152. The values of A" for difi'erent conditions he computed and tabulated in Finding- value of Table II, with values of Y and of ^^^ ^ as arguments; the latter, for reasons that '°™ will be explained later, being taken as the value of a\ at the vertex. That is, com- pute the value of ^^^ ^ - from the given data ; and then use it, under the proper page in the table for the known value of V, as an argument in the A' column, to find the value of A" from its own column. Then solve (125) to determine the maximum ordinate. A full explanation of the reasons for using this value of — j~ as an argu- ment in the A' column of the table will be given in the next chapter.* * In further explanation of this point let us recall the fact that we have derived certain formulae relating to the trajectory, and that each of these formulae has in it one or more somewhat involved integral expressions. In reducing these formulae to serviceable shape we replaced these integral expressions by certain symbols, each such symbol representing one particular one of these integral expressions. Now by the labor of previous investi- gators, notably Ingalls, the values of each one of these integral expressions has been computed for all possible useful conditions and the results tabulated in the several columns of the Ballistic Tables, under the several symbols which we used to represent these integral expressions. (Note continued at foot of next page.) 98 EXTERIOR BALLISTICS "Use of tables. 153. Before proceeding further, let us now investigate the manner of using Table II. Suppose we have Z = 2760 and 7 = 1150, and we wish to find the value of the secondary function A. Looking in the table for y=1150, and working from ^ = 2700, we have, by the ordinary methods of interpolation, A -.07643+ -00322x60 _^07643 + .00193 = .07836 Now sujDpose we had had Z = 2700 and y = 1175, we would have had ^ = .07643+ ^~'^^t^^^^ ^^^ =:.07643-.00248 = .07395 50 Now suppose, to combine the two, we had had Z = 2760 and y=1175, then we would have had Interpolation formula. A n-r>iQ , .00322x60 , ( -.00496) X 25 ^-.0.643+ —^^ + ^ ^ : .07643 + .00193 - .00248 = .07588 Expressing this algebraically, we would have had Vt A: At+^^^ZA+'^^ Af.4 100 ~"" ' \V where At is the next lower tabular value of A below the desired value, Zt and Vt Let us consider the trajectory OlsiPQ in Figure 13. The ballistic formulae repre- sent certain relations existing between the elements of the trajectory at any point. For the point P, for instance, the equations would contain certain integral expressions which we have called the " secondary functions," and have represented by a, a', b, b' (of which the logarithm is always used), and t', and also the pseudo velocity u. Now for this par- ticular point, P, the integral expressions in each of the formulse must be integrated between the proper limits for that particular point and their numerical values for this special case determined; a space integral being given the limits x and for instance. Instead of hav- ing to actually perform this integration in each case, however, Colonel Ingalls has already done the work for us; and we simply compute the value of the argument z = '-^ , and knowing V, we can take from Table II each of the numerical results of integration between the proper limits that we desire. So for each point of the curve there is a special numerical value of each of the integrals represented by the symbols a, a', etc. Figure 13. In a subsequent chapter we will use values so found for different points of the curve, but for the present we are dealing with the entire trajectory, and want to find the values of the functions for the point of fall, Q. Therefore, x becomes X, the range, and our integral expressions are represented by A, A', B, B' and T', and u becomes «jj. Knowing V, we therefore compute the value of Z = -^ , and thence from Table II we may find the correct values of each of the integral expressions represented by the symbols A, A', etc., and also of the pseudo velocity at the point of fall, xi^. Keep it firmly in mind that these particular symbols and values. A, A', etc., are only special symbols and values of the gen- eral symbols a, a', etc., at one particular point of the trajectory, that is, at the point of fall, or, expressed in another way, that they pertain to the entire trajectory and not to any part thereof or to any other point of the curve than the point of fall. (Note continued at foot of next page.) PEACTICAL METHODS 99 are the next lower tabular values of Z and Y below the given values ; AT is the differ- ence between successive tabular values of F (that is, 50 f. s. for the table 7=1150 to 1200 f, s., and 100 f. s. for all other tables given in the tables to be used with this book). \Z is always 100 for all tables, and it is therefore allowed to remain in numerical form in the above algebraic expression, ^.za and Atm are the differences given in the proper line of the A columns pertaining to A. Care must be taken to use the proper signs for all the quantities given in the above expression. It will also be seen at once that any other one of the secondary functions may be substituted for A in the above expression provided we exercise due care in regard to the signs. It must also be noted that the formula applies for the next lower tabular values only. If we work from the nearest tabular values, there would be a change of signs in the expres- sion if the nearest tabular value happens to be the next higher tabular value in any case. Similarly, for the vertex, where x = Xo, we have another set of special values, but here another symbol (which also represents an integral expression) A" has been introduced. So we would have as our special symbols for the vertex Co, a./, A", 6o, W and U, and the pseudo velocity i which is a process employed in a later chapter. From this found value of Zo we may also take from the tables the values of the other secondary functions for the vertex, a^, &o. log be,' and t,/, and of the pseudo velocity at the vertex, u,,. But to get the correct value of A" it must be taken out by cross interpolation as already described, and not by finding first 0,, and then working with a found value of Zn as an argument. There are points in the explanation contained in this foot-note which are perhaps not mathematically perfect, but it is hoped that these explanations will nevertheless lead the student to a better practical understanding of the reason why, when we have found a numerical value for A for the whole trajectory, we go over with it to the A' column when we wish to interpolate to get the value of the integral expression A" and that of So = Y^ , both of which values pertain solely to the vertex. 100 EXTERIOE BALLISTICS Interpolation Eepeating the expression just derived, and also solving it for Z and Y , we get formulae. AF 100 Z-Zt , 1 U--4.)-^^*Ar.4] (128)* Az From the first of these we find the value of A, given Z and Y (or of any other of the secondary functions in place of A). From (127) we can find Y , and from (128) we can find Z, given Z or Y respectively, and A. or any other secondary func- tion in its place. The expressions of course simplify greatly when working with a tabular value of either Z or Y , in which case Z — Zt or Y —Yt becomes zero. Cross inter- 154. In practice it is often necessary to find the value of A" corresponding to a formula found value of A', knowing Y , in finding the value of the maximum ordinate, as will ^^A" be shown later, under circumstances when we do not care to know the value of Z or of any other of the secondary functions. To do this, that is, to find the double inter- polation formula for crossing direct from A' to A" without finding Z, knowing Y , substitute in (126) expressed for A", the value of Z found from (128) expressed for A' . The resultant expression is A" = A/'-i^X^^^^-^-t-^^f^Ar^<,+ (l'-A/)^' (129)* AK Az^' Ak i^zA' If we are working with a tabular value of Y , which is fortunately generally the case, then Y —Yt — ^, and the above formula simplifies greatly, becoming A" = A/'+(A'-A/)4^^ (130)* As examples, suppose we have 7 = 1175 and Z = 2760, and desire to find the corresponding values of the secondary functions. Then 7— 7^ = 25 and Z — Zi = 60, and we have A= 07G43-h •QQ^^^X'^Q + (-•0Q^Q'3)x25 _,07643-f-. 00193 -.00248 = .07588 .0072X60 ( -.0096) X 25 ^.i63l + .00432-.0048 = .16262 A" = 1435-F^^^ 4- ^^ = 1435 + 33 + 4 = 1472 100 oO B- 0866+ -Q^"^^^^ + (--00^6) X 25 ^,0866 + . 0024 -.0023 = .0867 locr5'= 05465+ •^^^^^^'^^ + -00522x25 ^,05465 + .00089 + .00261 = .05815 » ■ 100 oO ^ = 948.5+ i^;?)-^^^ _^ 22J3X25 ^948.5-3.18 + 11.3 = 956.62 100 0^ y,_ ^3 .106x60 (-.08)x25 ^2.613 + .0636-.04 = 2.6366 7)'-117+ 122160 1^6) X 25 ^117 + 6-3 = 120 ^ ^ 100 50 * It must be noted that the interpolation formulae here derived for use with Table II of the Ballistic Tables, neglect second and higher differences. They therefore give results that are accurate only within certain limits, which limits are sufficiently narrow to permit the formulae to be used for the purposes for which employed in this text book. A caution must be given, however, against using them for other purposes without ascertain- ing whether or not they will give sufficiently accurate results for the purpose in view. A case where they cannot be successfully used is given in Chapter 14. (See foot-note to paragraph 239, Chapter 14.) PRACTICAL :\[ETHODS 101 155. Suppose we have .1" = 14T2, as given above, and know T = 11T5, and wish to find the value of Z. From (1'38) Z = 2:00+ ^ f37- ^-15) =2:00+ 34^ =2760 55 \ oO / 00 Suppose we had been given w = 956.G2 and Z = 2760, and wanted to find the value of Y , knowing it to be between 1150 and 1200 f. s. From (127) And we could proceed similarly with the other secondary functions. N"ow suppose that we know that 7 = 2775 and that A' = .20235, and desire to find the corresponding value of A" . Then from (129) we have ,„_,oo- 75^ ^ (-.0143) X 74 , 75x11 , .00225x74 A -4Joo- ^^^ x -p^„^ + ^^^ + ^.0^56~ ^" = 4935 + 141.7 + 8.3 + 29.7 = 5114.7 Suppose we find the value of A" from the above data by first finding the value of Z by (128) and then of A" from that by (126) expressed for A" . Z = 7G00+-^?;; f. 00225- ^~ '^^tfl ^'^'^ l =7600 + 231.7 = 7831.7 .OO06 \ 100 / ,„ .^Q., , 31.7x73 , 13x75 ,.^. ^ -^ =^^^^''^+-^00- + -100-=^^^^-^ which shows that the results obtained by the two methods difl^er slightly. To get results to coincide with those obtained in this text book (129) must be used.* 156. Eeturning now to our formulfB, and having found an expression for the value of the maximum ordinate, on the basis that the mean density of the air through which the projectile travels is the same as that at a point two-thirds the maximum ordinate above the gun (which is absolutely true for a uniformly varying atmosphere and for a trajectory that is a true parabola) ; and starting Avith Chauvenet's dis- cussion of atmospheric refraction, Colonel Ingalls, by another rather involved mathe- * It must be noticed here, as well as elsewhere throughout this text book, that interpo- lations are carried out to more decimal places than is strictly justifiable for the limits of accuracy obtainable with the ballistic and logarithmic tables used. The rule adopted for instruction of. midshipmen has been to have them carry out all interpolations to five working decimal places, as five place logarithmic tables are standard at the Naval Academy; thus 2534.7, 2.5347, .0025347, etc. This has been found advisable, not because it is expected to attain more accurate results thereby, but in order that a comparison of the relative accuracy of the tabular and logarithmic work of all midshipmen may be secured. Due consideration was given to all possible methods of attaining this object, and the only available rule that could be given them that would not depend upon individual judgment in any case and yet that would fit all cases was the one that is set forth above. Under it, with a given set of data, all midshipmen should secure the same results (within decimal differences in the last place), and therefore their results can be compared and relative marks fairly awarded. In service, however, this practice would simply be an apparent effort to attain an impossible degree of accuracy, which might in some cases, indeed, introduce small errors into the work that would otherwise be avoided by a less rigorous rule for interpolations. Only an experienced mathematical judgment can tell to what degree each process of interpolation should be carried in each separate case, and this is not possessed by midshipmen. Some simple, absolute rule was therefore necessary for them, and this being a simple, invariable rule, it is believed that the end justifies the means in this particular. This note should be a caution against attempts in service to attain impossible accuracy of results by excessive carrying out of interpolations. 102 EXTERIOR BALLISTICS matical process which it is unnecessary to follow through, derives an equation for determining the value of / as follows : loglog/ = logF + 5.01765 -10 (131) ingaUs' soiu- 157. With the aid of the formulae of Colonel Ingalls already given, we may now cessive^ap'- proceed to the solution of the original problem already solved by Alger's method, proxima ions. ^^ g^yg^ jj^ paragraph 140, by the methods used in preparing the range tables by the expert computers of the Bureau of Ordnance. The data is the same as before. — y- ; argument for Table II, Z — -^ cd- C^= -I'L . argument for Table U, Z = ^; sin 2ct> = A0 wz=870 log 2.93952 f = 0.61 log 9.78533-10. .colog 0.21467 d- = lU log 2.15836 ..colog 7.84164-10 C,= log 0.99583 colog 9.00417-10 Z = 30000 loo- 4.47712 ^, = 3029 log 3.48129 For 7 = 2900 (table for 7 = 2900 to 3000), for Z = 3029, we have Ai = .014882 /I = .014882 log 8.17266-10 Ci= log 0.99583 2<^i = 8° 28' 34" sin 9.16849-10 ^1 = 4° 14' 17". .. .First approximation, disregarding /. Having obtained the preceding approximation to the value of 6, we may now proceed to determine a second ap]oroximation to the value of the same quantity, and this time we can correct for a value of /, using equations (125) and (131). Y = A"C tan 4> loglog/ = log F+ 5.01765 -10 For finding A" from Table II we use a^'^ sm^o = 4° 13' 13". .. .Second approximation. Having obtained the above second approximation to the value of ^, we see that it differs from the first approximation by over one minute of arc, and we therefore cannot assume that the second value is sufficiently accurate. We therefore repeat the above process to get a third approximation. From above work ^2 = 2983. 8 and Ao = ao/ = . 014559, which gives, from Table II, A;' = 793+ :000899x56 ^g3g_^ ^/' = 838.7 log 2.92360 C„= log 1.00235 0^ = 4° 13' 13" tan 8.86800-10 J\_= log 2.79395 Constant log 5.01765-10 /.,= log 0.00648 loglog 7.81160-10 C\= log 0.99583 C,= log 1.00231 colog 8.99769-10 X=z 30000 log 4.47712 ^3 = 2984.1 log 3.47481 Hence from Table II, as before, A3 = .014601 A3 = .014601 log 8.16438-10 C,= log 1.00231 2 Vu = Uu cos sec w From Table II log5' = .10371 r = 1.2333 w^ = 2026.1 B'= log 0.10371 C= log 1.00231 <^ = 4° 13' 14" tan 8.86803-10. .sec 0.00118 cos 9.99882-10 r' = 1.2332 log 0.09103 w^ = 2026.1 log 3.30666 , = 5° 21' 11" tan 8.97174-10 sec 0.00190 r = 12.43 log 1.09452 ra, = 2029 log 3.30738 Hence we have as the solutions to this problem «/) = 4° 13' 14". a> = 5° 21' 11". T= 12.43 seconds. i;a; = 2029 f. s. These are the correct and final results, and it will be observed that they are the values which appear in Columns 2, 3, 4 and 5 of the range table for this gun, for a range of 10,000 yards. We have therefore learned how to compute the values for these columns in the range tables. In a later chapter will be given the forms used by the computers in actually computing the data for the range tables. Note. — The mathematical processes carried througli in the preceding cliapters may be briefly and generally described as follows: 1. Considering the forces acting on the projectile in flight, that is, the force of gravity and the atmospheric resistance, and dealing with differential increments at any point of the trajectory, certain equations are derived (from the laws of physics governing motion) which show the relations existing between these differential increments in the different elements of the trajectory at the given point. These are not equations to the curve itself as a whole, but simply express the relations between the differential increments referred to above. Could they be integrated in general form, they could be generally used for solu- tions, but such integration is impossible owing to fractional exponents, and some other method must be adopted. The accepted method is known as Siacci's method, from its deviser, its essential point being the introduction into the computations of a new quantity known as the " pseudo velocity," which is defined by saying that " the pseudo velocity at any point of the trajectory is the component of the remaining velocity at that point in a direction parallel to the original line of projection." By the introduction of this quantity, it becomes possible to reduce the differential equations to certain others that are known as the " ballistic formulfe." which are used in the practical solutions of ballistic problems. Each of these formulae contains certain integral expressions, which are represented in the formulae by the symbols A, I, S and T (the altitude, inclination, space and time func- tions), and the values of these functions for any given velocity, whether real or pseudo, may be found in Table I of the Ballistic Tables. That is, the tabulated values of these PRACTICAL METHODS 105 functions are simply the values of the given integral expressions when integrated between the proper limits for the given velocity. 2. As stated above, these several functions are merely values of certain rather involved integral expressions, the values of which for any given velocity may be found in Table I. Different subscripts to the symbols are used to represent the values at different points of the trajectory; thus -S'^^is the value of the space function for the point of fall, So for the vertex, etc. 3. Professor Alger took the ballistic formulae as they stood after the reductions described above, and put them in the form desired for any particular problem. Any neces- sary changes for this purpose were simply algebraic and trigonometrical transformations in order to get the value of the desired unknown quantity expressed in terms of the ones that are known. He then solved in each case, taking the necessary values of the integral expressions I, A, S and T from Table I. 4. The successive approximation feature (and this description fits every such case, whether Alger or Ingalls) becomes necessary because, when we start a ballistic problem we usually do not know the maximum height of flight of the projectile. We therefore work the problem by first disregarding this height, that is, by considering that the density of the air throughout the flight is constant and equal to that at the gun. We know this to be wrong, however, and that our first result can therefore be only approximate. Therefore by using our first result we determine the maximum ordinate (Alger by one formula and the use of a table; and Ingalls by the use of different formulae but by computation without the use of any table) and from that the altitude factor of the ballistic coefficient. This approximate value of / we apply to our first value of the ballistic coefficient, and then repeat our computations, thereby getting a second result, which is still approximate, but more nearly correct than the first one. By continually repeating this process we will finally get to a point where repeated computations make no change in the value of the bal- listic coefficient, and at that point the limit of accuracy of our methods, whatever they may be, has been reached, and our result is as nearly correct as it is possible to get by the adopted methods. Taking the final value of the ballistic coefficient thus obtained as cor- rect, we can then proceed to the final solution of the problem. 5. Ingalls further simplified the ballistic formulae so that their use would be less difficult. In these formulae there are certain integral expressions involving the values of 7, A, S and T for the point of fall and for the vertex, that is, I^^, !„, etc., and certain con- stantly repeating combinations of these integrals. Ingalls substituted for these con- stantly repeating combinations of integrals certain other quantities which he called " secondary functions," and represented by the symbols A, A', A", B, B', T', etc., and thereby derived new and simpler formulae involving those secondary functions and the pseudo velocity, u. He also computed the values of the secondary functions with the expressions integrated between all useful limits (that is, of the integral forms which they represent) and tabulated them in Table II of the Ballistic Tables. To solve by his methods, we therefore take his forms of the ballistic formulae, properly transposed to put all known quantities in the right-hand member and the desired unknown quantity as the left-hand member of each expression, and then solve; taking the values of the secondary functions that we need from Table II. To use this table we must know for use as argu- ments the value of the initial velocity and also the quotient of the horizontal distance traveled divided by the ballistic coefficient, that is, ot z = -^- or Z =-^ The above paragraphs of this note indicate the manner in which a student shouhl try to retain in his mind the general features of the mathematical processes described in this text book. Each student, in addition to learning the text of each chapter in detail should endeavor to formulate in his own mind a general under- standing of the processes described in the chapter in accordance with the general method illustrated above for the fundamental processes of exterior ballistics. 106 EXTERIOE BALLISTICS EXAMPLES. 1. Given the values of V and Z contained in the first two columns of the follow- ing table, take from Table II of the Ballistic Tables the corresponding values of A, A', A", B, log B', u, r and D' . DATA. ANSWERS. Problem. ^=4- V. A. A'. A". B. log 5'. u. 7". D'. 1 3370 1150 .09855 .21279 1809.9 .11431 .06440 914.07 3.3323 192.10 2 1763 1150 .04755 .09963 922.02 .0.5214 .03935 1002.5 1.6510 46.780 3 6982 11-50 .23947 .54002 .30961 .11151 763.85 7.6714 1055.3 4 1326 2000 .01200 .02550 700.56 .01,354 .04989 1683.3 0.72.534 9.2600 5 4173 2000 .04992 .12123 2454.9 .07132 .15484 1175.2 2.7711 136.57 6 7652 2000 .12736 .33625 4753.3 .20885 .21477 919.40 0.1727 786.12 7 1943 2000 .01087 .02365 1047.9 .01274 .06869 20.50.8 0.84207 11.860 8 3756 2600 .02489 .05904 2173.0 .0.3414 .13693 1619.7 1.8427 59.240 9 9743 2600 .12187 .35927 6438.1 .23739 .28950 921.85 7.0604 1139.1 10 10742 2600 .14759 .43913 7131.0 .29149 .29561 873.90 8.1739 1599.4 11 1818 2700 .00931 .02005 974.44 .01074 .06345 2169.1 0.7.5228 9.1800 12 4747 2700 .03210 .07981 2840.9 .04774 .17323 1484.6 2.3925 101.82 13 5561 2700 .04095 .10630 3421.5 .06536 .20376 13.32.6 2.9718 162.49 14 7937 2700 .07613 .21928 5183.0 . 14322 .274.50 1050.0 .5.0092 514.14 15 9541 2700 . 10868 .32305 6336.1 .21501 .29617 948.95 6.6211 988.58 16 1856 2900 .00823 .01777 994.92 .009.59 .06326 2331.8 0.71408 8.5600 17 2942 2900 .01434 .03253 1643.0 .01818 .10215 2037.1 1.2126 24.840 18 3839 2900 .02034 .04814 2216.7 .02787 .13558 1813.6 1.6795 .50.170 19 4815 2900 .02809 .06988 2879.5 .04176 .17277 1595.8 2.2.535 90.750 20 8634 2900 .07677 .22861 5735.5 .15189 .29651 1044.3 5.. 301 6 592.50 21 3231 3100 .01404 .03213 1819.5 .01808 .10982 2119.6 1.2626 27.620 22 5742 3100 .03182 .08283 3534.7 .05106 .20430 1.5.30.2 2.6633 129.36 23 8841 3100 .06943 .21036 .5923.6 .14094 .30760 1076.3 5.1291 553.84 24 10305 3100 .09566 .30074 7027.7 .20504 .33126 974.70 6.5632 998.90 Note for Instructor. — In exercising class in these interpolations, give to each mid- shipman one problem from each of the tables given in tliis and in the following five examples. 2. Given the values of V and Z contained in the two first columns of the follow- ing table, take from Table II of the Ballistic Tables the corresponding values of A, A', A", B, log B', u, r and D' . DATA. ANSWERS. Problem. -I- V. A. A'. A". B. log B'. u. T'. D'. 1 2200 1162 .05974 .12640 1160.4 .00666 .04798 981.92 2.0769 73.800 2 5500 1173 . 172.57 ..38758 3051.4 .21499 .09544 826.98 5.7299 .581.96 3 8100 1187 .28019 .65643 .37621 .12808 735.41 9.0164 1492.5 4.. 1800 2030 .016.54 .03592 970.00 .01936 .06810 1606.6 0.99900 17.400 5. . 4200 7700 2057 2082 .04764 . 12004 .11600 ..32104 2475.3 4821.9 .06828 .20103 .15644 . 22389 1200.5 932.58 2.7176 6.0.389 131.02 6 755 . 90 7 3100 2618 .01902 .04359 1749.0 .02456 .11167 1779.7 1.4399 35.460 8.. 7.300 2643 .068.32 .19222 4702.7 . 12.389 .25853 1087.5 4.5167 403.81 9 9400 2663 .108.54 .32104 6222 . 3 .21254 .29194 950.71 6.5537 961.42 10 5100 2730 .03494 .08853 .3090.0 .05364 .18625 14.33.0 2.6048 122.00 11. 6100 2750 .04.581 .12240 3818.0 .07660 .22305 1271.0 3.3235 205.50 12 8700 2779 .08.543 . 25263 57.59.1 .16723 .29158 1012.8 5.5986 668.13 13 5500 2913 .03410 .08787 3368.0 .05378 .19836 1465.8 2 . 6889 131.. 57 14 8200 2954 .06667 .19629 5415.4 .12959 .28890 1093.6 4.7975 471.02 15 9700 2982 .09156 .28154 6545.6 .19002 .31698 989.70 6.1909 860.34 16 4600 3140 .02210 .0,5408 2722.0 .03198 .16032 1804.6 1.9410 66.000 17 7000 3150 .04291 .119.30 4486.0 .07645 .25060 1322.5 3.4925 2.32.00 18 9900 3160 .08423 .26398 6741.8 .17978 .32934 1009.4 6.0338 819.60 PEACTICAL METHODS 107 3. Given the values of V and Z contained in the two first columns of the follow- ing table, take from Table II of the Ballistic Tables the corresponding values of A°A', A", B, log B', u, r and B' . DATA. ANSWERS. Problem. .=4. F. A. A'. A". B. logB'. u. r. D'. 1 2730 1157 .07670 .16392 1452.6 .08716 .05583 9.50.07 2.0336 119.16 o 5980 1169 . 19258 .43558 3336.81.24299 .10109 806.72 6.3303 713.94 3 8730 1182 .31173 .73709 .42530 .13503 712.68 9.9097 1814.7 4 1936 2028 .01807 .03947 id49.2 .02138 .07.330 1576.1 1.0858 20.100 5 4757 2048 .05773 .14368 2846.4 .08599 .17296 1131.5 3.2098 187.20 6 7915 2063 .12780 .34164 4955.5 .21386 .22372 918.22 6.3151 833.56 7 3342 2628 .02082 .04836 1903.9 .02753 .12076 1731.1 1.5724 42.840 8 7539 2644 .07243 .20534 4879.8 .13292 .26385 1067.8 4.7373 450.35 9 9526 2684 .10970 .32588 6319.0 .21624 .29471 946.98 6.6413 994.18 10 5433 2733 .03848 .09921 3328.8 .06078 .19856 1373.2 2.8391 147.01 11 6214 2748 .04734 .12721 3902 . 4 .07993 .22720 1253.1 3.4172 218.04 12 8848 2763 .08944 .26494 5861.8 . 175.52 29272 1000.7 5.7785 719.75 13 5584 2925 .03464 .08970 3429.3 .05506|. 20142 1457.0 2.7347 136.13 14 8282 2944 .068.53 .20229 .5475. 9;. 13369 .29045 1084.3 4.8913 492.76 15 9748 2962 .09391 .28840 6574. 2 '.194.52 .31627 983.28 6.2813 890.22 16 4632 3148 .02221 .05445 ,2744.1 .03224 .16136 1802.8 1.9536 67 . 200 17 7148 3155 .04430 .12447 '4601.4 .08010 . 25606 1300.2 3.5998 248.31 18 9923 3163 .08449 .26497 ,6759.5 .18052 .32979 1008.6 6.0511 825.65 4. Given the values of V contained in the first column and of the secondary func- tions contained in the second column of the following table, take from Table II of the Ballistic Tables the corresponding values of Z. 108 EXTERIOR BALLISTICS 5. Given the value of Z contained in the first column, the value of the secondary function contained in the second column, and the limits near which the vahie of Y lies contained in the third column of the following table, take from Table II of the Ballistic Tables the corresponding value of Y . Problem. DATA. 1 1732 2 3 4140 5615 4 1232 5 43S1 6 8175 7 2222 8 4444 9 8888 10 2551 11 5743 12 9107 13 3232 14 6474 15 9876 16 13.34 17 4321 18 8448 Secondary function. A = 04632 A' = 27837 A" = 3127 5 B = 01273 u = 1154 7 T' = 6 7324 A = 01278 A' = 07735 A" = 5796 7 B = 01693 u = 1298 4 T' = 6 2333 A = 01607 A' = 12087 A" = 6593 3 B = 00555 u = 1835 2 r = 4 7867 Limits of Y.* 1150- 1150- 1150- 2000- 2000- 2000- 2600- 2600- 2600- 2700- 2700- 2700- 2900- 2900- 2900- 3100- 3100- 3100- -1200 -1200 -1200 •2100 -2100 ■2100 ■2700 ■2700 ■2700 •2800 ■2800 ■2800 -3000 -3000 -3000 -3200 -3200 -3200 ANSWEKS. 1154.9 1137.0 1195.6 1962.4 2007.4 2007.5 2595 . 9 2602.0 2469.1 2715.0 2693.0 2668.9 2909.6 2016.8 2727 . 6 3080.0 3089.0 3090.1 * These limits determine the table to be used; In some cases it will be found that tlie interpolation gives a value of Y lying outside of the limits indicated. 6. Given the values of Y and of A' contained in the two first columns of the following table, take from Table II of the Ballistic Tables the corresponding values of A", without determining the corresponding value of Z. Problem. DATA. AXSWERS. Y. A'. A". 1 1150 1179 1192 2000 2053 2086 2600 2677 2689 2700 2750 2772 2900 2932 2988 3118 3150 3173 0.19787 0.32995 0.40843 0.04932 0.15563 0.37.543 0.05837 0.12647 0.27563 0.02543 0.10023 0.00995 0.03613 0.13333 0.30057 0.20475 0.27777 0.02975 1699.6 9 2692.0 3 3234.5 4 1236.5 5 3009.1 6 5293.2 7 21.54.4 s 37.52 . 9 5805 . 7 10 1194.3 11 3376.6 1^ 547.19 13 1786.5 14 4357.5 15 6760.3 16 5880.7 17 6877.5 18 1774.1 PEACTICAL METHODS 109 7. Compute by Ingalls' method for standard atmospheric conditions, using successive approximations, the values of the angle of departure, angle of fall, time of flight and striking velocity in the following cases. DATA. ANSWERS. Problem. Projectile. Velocity. Range. 0. T. V(j3. d w . f. s. Yds. w Sees. f.s. In. Lbs. c. A 3 13 1.00 1150 2130 5° 39.1' 6° 46' 6.56 867 B 3 13 1.00 2700 3720 2 59 . 3 5 33 6.94 1074 C 4 33 0.67 2900 3825 1 4.S.4 2 20 4.98 1843 D 50 1.00 3150 4370 2 11.4 3 44 6.33 1408 E 5 50 0.61 3150 4465 1 44.8 2 25 5.44 1941 F 6 105 0.61 2600 12690 11 03.4 19 38 24.70 1104 G 6 105 1.00 2800 3875 1 00. / 2 41 5.34 1712 H 6 105 0.61 2800 3622 1 32.3 1 51 4.46 2134 I 7 165 1.00 2700 7230 5 00.7 8 31 12.32 1221 J 7 165 0.61 2700 7357 3 57.4 5 30 10.54 16.50 K 8 260 0.61 2750 8390 4 15.3 5 49 11.62 1735 L 10 510 1.00 2700 10310 6 49.8 11 09 17.05 1293 M 10 510 0.61 2700 11333 6 07.6 8 30 16.30 1653 N 12 870 0.61 2900 21650 12 30.9 19 55 33.59 1441 13 1130 1.00 2000 10370 11 15.2 16 09 21.45 1168 P 13 1130 0.74 2000 11111 10 58.0 14 44 21.57 1281 Q 14 1400 0.70 2000 14220 14 48.8 20 15 28.68 1251 R 14 1400 0.70 2600 14370 8 32.4 11 55 21.76 1577 8. Given the data contained in the first eight columns of the following table, compute in each case the values of <^, w, T and v^^, by Ingalls' method, using Table II, and using in each case the value of / from Table V corresponding to the maximum ordinate given in the table below. DATA. ANSWERS. Problem. Projectile. Atmosphere. f.s. R. Yds. Max. T. Sees. d. 10. Bar. Ther. ord. leet. 0. . w. f.s. In. Lbs. c. In. °F. A 3 13 1.00 Standard 11.50 2550 265 7° 03.4' 8° 40' 8.07 832 B 3 13 1.00 Standard 2700 3450 158 2 36.4 4 42 6.20 1122 C 4 33 0.67 Standard 2900 4000 112 1 49.8 2 31 5.27 1802 D 5 50 1.00 29.00 20 3150 3870 108 1 52.1 3 06 5.46 1475 E 5 '50 0.61 29.50 22 31.50 3850 80 1 28.0 1 58 4.61 2011 F 6 105 0.61 30.00 25 2600 14530 3798 15 42 .,4 28 31 32.04 1036 G 6 105 1.00 30.1^ - 27 2800 4570 169 2 33.0 3 55 6.82 1476 H 6 105 0.01 30.25 30 2800 4030 101 1 47.2 2 14 5.11 2010 1 7 165 1.00 .30.. 33 33 2700 6030 363 3 ,55.2 6 22 9.85 1301 J 165 0.61 30.50 35 2700 6540 328 3 28.3 4 46 9.29 1676 K 8 260 0.61 30.67 40 2750 8080 485 4 10.0 5 45 11.32 1698 L 10 510 1.00 31.00 45 2700 9090 807 5 .52.9 9 28 14.80 1.320 LI 10 510 0.61 30.75 50 2700 10070 784 5 20.3 7 17 14.28 1691 N 12 13 870 0.61 30.33 30.25 60 70 2900 2000 22030 10560 4801 1937 13 11 23.2 39.1 21 16 56 51 35.37 22.08 1377 1130 1.00 1154 P 13 1130 0.74 29.50 80 2000 11050 1830 10 45.6 14 19 21.24 1299 Q 14 1400 0.70 29.00 90 200C 14020 3204 14 09.7 19 00 27.67 1284 R 14 1400 0.70 28.75 100 2600 14590 1960 8 23.2 11 23 21.63 1645 110 EXTERIOE BALLISTICS 9. Compute by Alger's method, without using Table II, the values of 4>, w, T and Vu, from the data contained in the following table, correcting for altitude in each case by successive approximations. DATA. ANSWERS. Prob- Projectile Atmosphere. Wind* lem. V. f.s. R'nge. Yds. T. Sees. f.s. d. w. Bar. Ther. Value pon't. f.s. 0. w. In. Lbs c. In. °F. of 8. 1 3 13 Standard 2800 2000 None 1° 00.9' 1° 25' 2.79 1671 2 5 50 Standard 2550 3000 None 1 54.4 2 49 4.76 1439 3 6 100 Standard 2300 4000 None 3 07.5 4 31 6.97 1321 4 8 250 Standard 2300 4000 None i 2 45.1 3 36 6.38 1552 5 12 850 Standard 2250 5000 None 3 26.0 4 15 7.88 1626 6 11.024 760.4 1.0306 1733 2260 + 19| 2 21.5 2 36 4.27 1480 7 11.024 760.4 1.0058 1733 6788 — 14 8 25.7 11 01 14.64 1173 8 6 100 30.05 70 2900 9700 None 8 53.3 17 58 19.80 987 9 12 850 30.19 59 2S27 11566 None i 6 55.0 11 14 18.16 1364 10 11.024 760.4 1.0174 1733 11207 — 12; 17 32.7 24 33 28.06 1046 11 15.75 2028 Standard 1805 1094 None 57.8 1 00 1.86 1712 12 15.75 2028 Standard 1805 3281 None 3 06.2 3 27 5.91 1542 13 15.75 2028 Standard 1805 5468 None 36.4 6 42 10.43 1391 14 3 13 Standard 2800 2000 None 1 00.7 1 26 2.79 1672 15 3 15 Standard 2628 1883 None 1 00.0 1 21 2.66 1720 16 5 60 Standard 2900 3000 None 1 21.2 1 50 3.91 1837 17 5 55 Standard ■■ 2997 3095 None 1 21.7 1 00 4.02 1796 * The sign + means a wind with the flight of the projectile, and a — sign a wind against it. Therefore, in problem 6, say, in order to get the desired range we would have to proceed as though the initial velocity were really 1733 — 19 = 1714 f. s. and there were no wind, and compute results accordingly. Note. — The above problems in Example 9 are taken from Alger's text book, and cover guns of older date, both U. S. Navy and foreign. Note the difference between this data and modern weights and velocities; and observe care to use correct data as given in the table. X = tan -Iv) and by el iminatiiig Iv from the above we get a; 2 cos- \ - — -Ay -Sv. 159. Equation (76) is CHAPTEE 9. THE DERIVATION AND USE OF SPECIAL FORMTJL^ FOR FINDING THE COORDINATES OF THE VERTEX AND THE TIME OF FLIGHT TO AND THE REMAINING VELOCITY AT THE VERTEX, FOR A GIVEN ANGLE OF DEPARTURE AND INITIAL VELOCITY, WHICH INCLUDES THE DATA GIVEN IN COLUMN 8 OF THE RANGE TABLES. 158. Equations (74) and (75) are BaUistie ^ ^ ' ^ ' formula. (132) (133) (134) t = C sec tan^ = tan(^-— ^^ (137) 2 cos- cf> t = Ct' sec (f> (138) by the introduction of the general forms, a, a', h and /' of Ingalls' secondary func- tions, as explained in the last chapter. 161. Equations (136) and (137) may be written Transforma- tion of equa- l=ta„^(l--^) (139) """■ X ^ \ sm 2(f)/ tan^ = tan<^fl--A?-') (140) \ sm 26J Substituting in these the value of sin 2^ = .4C' from (115), we get l=^(.4-a) = — ^(.4-a) (141) X A ^ ^2 cos' (^ ^ ^ ^ ^ tane=^-^^ (A-a') = —^ (A -a') (142) A 2 COS" <^ 162. Xow by taking the iirst two members of each of the above equations, that is, Equations for ordinate and i.„„ I inclination, y=^^^iA-a)x (143) tan^= ^^ (A-a') (144) we can readily find the values of y and 9 corresponding to any given value of x for any given trajectory; that is, by computing the ordinates and angles of inclination corresponding to any necessary number of abscissse, we are in a position to actually plot the trajectory to scale, provided we have determined or know the values of <;!>, 113 EXTERIOR BALLISTICS V, X and C for that trajectory; for a knowledge of the values of V and oi Z= -^ is necessary to enable us to use Table II. 163. The quantity a varies with x, and must be taken from the " A " column of Table II with V and 2= yp as arguments. Similarly, a' must be taken from the " A' " column with the same arguments. 164. Eor the vertex, we know that 6 = 0, and (14^) therefore becomes, for that particular point, i^(l-ao')=0 A. and, as — ~ cannot be equal to zero, then we must have A-ffo' = or A=ao' (145) 165. Also, if we suppose 6= —(j> at some point in the descending branch of the trajectory, which point manifestly exists, as in that branch the value of 6 varies from zero at the vertex to — w at the point of fall, and we have seen that w is always numerically greater than , equation (142) will become for that point tan(-<^) = ^(4_^'_^) or A—a'_^=—A or a'_^ = 2A (146) whence, from (145) and (146) ao=A=U'_^ 166. Substituting a^' for A in (141) and designating symbols relating to the vertex by the subscript zero, we get -% = ^^^^0 tan =^tan Xq ttQ CIq whence 2/o = -^ tan cf> = C -^ tan The secondary 167. Now if we let A" = ^ = ^% in which A" may be taken from Table II, function A" tt^L tt^ using V and ao' = A= - — j~ , we will have the expression for the summit ordinate, or ordinate of the vertex i/, = r = A"C'tan<^ (147) It will be observed that a^' is a special value of A', this latter symbol referring to the entire trajectory. This value of the ordinate at the vertex, y^, is ordinarily denoted by Y in work. 168. We have already shown that, for the whole trajectory, A = - — ^y > ^^, V and C, we can compute this particular value of A', namely, ag, from the ex- pression a(,'= - — -^ ; and then, as this is a special value of A', we may look for it in the A' column of Table II, and by interpolation in the usual manner we may take PRACTICAL METHODS 113 from that table the corresponding value of A" ; and, in fact, the corresponding values of any other of the secondary functions for the vertex. This explains the reasons for the method of determining the value of A" described and used in the last chapter.* 169. We also find, in the Z column corresponding to the above interpolation. the value of Zq = -yf , and we therefore have , = Cz„ (148) 170. Assembling the equations already derived, we see that our formulas for Equations >^ i -^ ' for vertex. finding the elements at the vertex are y, = Y = A"C tan cl> Xo = Czo t^^CtQ sec Vq = Wq cos 4> (149) (150) (151) (153) 171. Let us now proceed with our standard problem, the 12" gun, for which "F = 2900 f. s., ?t' = S70 pounds and c = O.Gl, for which, at 10,000 yards range, we have already determined in the last chapter that log (7 = 1,00331, Z=:2984.1 and <^ = 4° 13' 14". tQ=:Ct' sec (f> Vo=:U(yCOS Y = A"Ct&nc(> From Table II Xn — yyZn 4 = .01408+»^ =.014601 4" = 849- 1.99x56 13 = 838.87 'for the entire trajectory, which equals a^' for the vertex. This could also be determined by solving A = — j~ for the above values instead of taking it from the table. by using the value of A' given above as an argument in the A' column, and working with the nearest tabular value. (Ordi- narily work from the next lower tabular value, however.) 500 _ ,000199X100^ J. gj^g io = .565 + Wn = 2435- .0011 .041x81.9 100 30x81.9 = .59858 100 = 3410.4 C= log 1.00231 log 1.00231. .log 1.00231 .1" = 838.87 log 2.92370 <^ = 4°13'14" ...tan 8.86803-10 sec 0.00116 cos 9.99882-10 Zo = 1581.9 log 3.19918 V = .59858 log 9.77712-10 Mn = 2410.4 loff 3.38209 r= 622.36 log 2.79404 a:o = 15903.3 log 4.20149 ^0 = 6.034 log 0.78061 " fo = 2403.9 log 3.38091 a:o = 5301.1 yards. ^o = 6.034 seconds. F= 622.36 feet. fn = 2403.9 foot-seconds. * See foot-note to paragraph 152. 8 114 EXTEEIOE BALLISTICS 172. Had we not known the correct value of C\ as corrected for altitude, but had only known loglog / = log F + 5.01765 -10 x^ = Cz^ tg = Ctf^' sec (f> Vq = UqC0S(}) w = S70 log 2.93953 c=0.61 log 9.78533-10. .colog 0.21467 d- = U4: log 2.15836 colog 7.84164-10 C,= log 0.99583 colog 9.00417-10 2 = S° 26' 28" sin 9.16670-10 Go/ = .01482 log 8.17087 - 10 From Table II, with .01482 in the A' column as an argument, A," = 8id+ ^^^ X ^ =849 + 1 = 850 A," = SoO log 2.92942 C\= log 0.99503 = 8° 26' 28" sin 9.16670-10 flo/ = .0146 log 8.16439 - 10 As these last two successive values of a^ are equal, we have evidently reached the limit of accuracy in our approximations, and we have for the remainder of the problem a ' = .0146 and log C = 1.00231 PEACTICAL METHODS 115 Also, from log Yn as found above, we have that F = 622.43 feet, and from Table II 2o = 1581.8 V = -59854 Mo = 2410.5 C= log 1.00231 log 1.00231 (fy = 4° 13' 14" sec 0.00118 cos 9.99882-10 2o = 15S1.8 log 3.19915 V = . 59854 log 9.77709-10 w„ = 2410.5 log 3.38093 a-, = 15902.3 log 4.20146 L = 6.0336 lose 0.78058 t'o = 2404.0 log 3.38093 a;o = 5300.7 yards. r = 622.43 feet. ^0 = 6.0366 seconds, i;^,^ 2404.0 foot-seconds. 173. If we desire to plot any particular trajectory to scale, we can determine the ordinate corresponding to any given abscissa, and also the angle of inclination of the curve at the given point as follows : We have from (143) and (144) y= :t^ (A-a)x (153) and tan e= '^-^ (A-a') ' (154) A Suppose we wish to plot the 10,000-yard trajectory for our standard problem, for which we now know that c/> = 4° 13' 14" and log (7 = 1.00231. 2cj> = 8° 26' 28" sin 9.16670 - 10 C= colog 8.99769-10 1 = .014601 log 8.16439-10 a^' = A = .01i601 ^.^,,3^.009^01^X56 ^g33_g, ,, = 1500+ ^^^ X^-^^^^^"^ =1581.9 ^" = 838.87 log 2.92370 C= log 1.00231 log 1.00231 <^ = 4° 13' 14" tan 8.86803-10 2„= 1581.9 log 3.19918 y/„ = F = 622.35 log 2.79404 a:o = 15903 log 4.20149 The coordinates of the vertex are therefore 5301 yards in range and 622.35 feet in altitude. 174. In the equations given we now find the value of /^ for the given trajectory, having the value of A as above. <^ = 4° 13' 14" log 8.86803-10 A = .014601 log 8.16439-10 colog 1.83561 t_anj. ^5 Q5^ j^^ 0.70364 and our equations become ^ = 5.054(.014601-a)x tan ^ = 5.054(.014601-a') 116 EXTERIOE BALLISTICS 175. The following table gives the results of work with these equations for abscissa varying by 1000 yards for this trajectory, some of the cases being worked out below : Abscissae. Ordinates. Remarks. Yards. Feet. ti. 4° 13' 14" Origin. 1000 203.58 3 31 57 2000 370.06 2 48 55 3000 497.25 2 02 20 4000 ,581.62 1 10 37 5000 620.81 17 11 5301 622.35 00 00 Vertex. 6000 610.34 (— )0 41 35 7000 547.66 (-)l 43 08 8000 427.21 (— )2 50 43 9000 246.44 (— )4 04 16 10000 (-)5 21 34 Point of fall; 6 = - — w. Work for 3000 yards : C= colog 3.99769-10 .^ = 9000 los: 3.95^:34 2 = 895.21 log 2.95193 a = .00326+ -00043x95.21 ^Qogggg ^' = .0067+ -0009x95.21 ^00^55^ A = .014601 a = .003669 100 yl = .014601 a' = .007557 100 4-a = .010932 log 8.03870-10 A -a' = . 007044 ." lo^ 7.84782-10 tan (^ A = 5.054 log 0.70364. : = 9000 losr 3.95424 .los: 0.70364 2/ = 497.25 log 2.69658 6 = 2° 02' 20" tan 8.55146-10 Work for 8000 yards : C= colog 8.99769-10 a; = 24000 log 4.38021 2 = 2387.3 log 3.37790 a=.01059+ •00056X87^_3 ^^^11079 a' = .0233+:^013x87J ^^^^^^^g^ 4 = .014601 o = .011079 100 ^ = a' = 100 .014601 .024435 4-a=.003522 log 7.54679-10 A -a'=(-). 009834 (-)log 7.99273-10 tan <^ :5.054 log 0.70364, los: 0.70364 a; = 24000 log 4.38021 i/ = 427.21 log 2.63064 6={-)2° 50' 43" (-)tan 8.69637-10 PRACTICAL METHODS 117 The work for the point of fall, that is, for an abscissa of a:= 10,000 yards, becomes : C= colog 8.99769-10 a: = 30000 log 4.47712 2 = 2984.1 log 3.47481 ,^.01408+ m^l^ ..014601 a' = .0319+ ^^^^^^ =-033162 A = .014601 A= .014601 a=. 014601 a'= .033163 4-a = A_a'=(-).018561 (-)log 8.26860-10 ifl^^=5 054 log 0.70364 A ^=(-)5° 21' 34" (-)tan 8.97224-10 y = 176. A reversal of the original formulge ■would enable us to find the abscissa Reverse corresponding to any given ordinate; but there are some practical difficulties in the way of a simple use of the formulae for this purpose, and as it is not a usual case it is not considered necessary to go into the matter here. EXAMPLES. 1. Given the data contained in the following taljle, compute the values of To, y^iY), ^0 and t^o by Ingalls' method, using Table II, and correcting for altitude in each case by computing successive approximations to the value of C. DATA. ANSWERS. Problem. Projectile. Atmos phere. (159) t?tj = Ujj cos ^ sec (0 (160) 178. As no account is taken of the altitude factor in the above expression, we cannot use our standard problem at present, so we will take a different case for our first solution, and one at such a short range that the altitude factor may be neglected without material error. Let us therefore compute the values of X, w, T and v^ for an angle of departure of 1° 02' 34", for the 5" gun for which y = 3150 f. s., w = 50 pounds and c = 0.61, for a barometer reading of 30.00" and a thermometer reading of 50° F. From Table VI. E= log 0.51570 8= 1.035 loff 0.01494 C= log 0.50076 colog 9.49924-10 2(^ = 2° 04' 48" sin 8.55984-10 4 = .01146 log 8.05908-10 From which, from Table II, we get Z = 2700+ nnnL f '^'^^n^n^^"^ + .01146 - .01119') = 2700 + 121.1 = 2821.1 log B' = .0945 + :00||8M - .521|p = .09449 «„ = 3235- ^^-j^ + 52^ =3270.8 r = 1.064+ ^^ - :M5X50 ^j^„.^Q J. V \J J.UU PEACTICAL METHODS. 121 C= log 0.50076 log 0.50076 Z = 2821.1 log. 3.45042 <^ = 1°02'24" tan 8.25894-10. .sec 0.00007. .cos 9.99993-10 i?'= log 0.09449 ^^. = 2270.8 log 3.35618 r = 1.056 loff 0.02366 X = 8936.7 los 3.95118 . = 1° 17' 34" tan 8.35343-10 sec 0.00011 r = 3.3457 loff 0.52449 ra, = 2271 log 3.35622 Z = 2978.9 yards. T = 3.3457 seconds. (0=1° 17' 34". t'<^ = 2271 f. s. 179. Or, with perhaps no more labor, we may avoid the double interpolation necessary in the above solution by working the problem for y = 3100 f. s. and then again for 7=: 3200 f. s., and then get our final results by interpolation between those obtained for the two velocities. In this case, as 7 = 3150, our final results should be half way between the results obtained for the two values of V with which we work. The value of C and that of A are of course the same as in the preceding problem, so starting from that point, with J. = .01146 and log C = 0.50076, we have For F = 3100 f. s. Z = 2751.9 logS' = .09282 ^^. = 2248.5 r = 1.0428 C= log 0.50076 log 0.50076 Z = 2751.9 log 3.43963 (> = 1°02'24" tan 8.25894-10. .sec 0.00007. .cos 9.99993-10 B'= log 0.09282 «^ = 2248.5 log 3.35190 T' = 1.0428 log 0.01820 X = 8717.4 log 3.94039 . = 1° 17' 16" tan 8.35176-10 sec 0.00011 r = 3.3039 log 0.51903 r^ = 2248.75 log 3.35194 For 7 = 3200 f. s. Z = 2904 log5' = .09674 m„ = 2288.9 T'=1.0738 C= log 0.50076 log 0.5007G Z = 2904 log 3.46300 c> = l° 02' 24" tan 8.25894-10. .sec 0.00007. .cos 9.99993-10 B'= log 0.09674 w<, = 2288.9 log 3.35963 r = 1.0738 log 0.03092 X = 9199.4 lo? 3.96376 w = l°17'5S" tan 8.35568-10 sec 0.00011 r = 3.4021 log 0.53175 r« = 2289.1 log 3.35967 122 EXTEEIOE BALLISTICS Our results then are For 7 = 3100 f . s. For F = 3200 f . s. For 7 = 3150 f . s. (By interpolation be- tween the results obtained for 3100 and 3200 f. s.) Z 2905.8 yards. 3066.5 yards. 2986.1 yards. ^ 1° 17' 16". 1° 17' 57". 1° 17' 37". T 3.3039 seconds. 3.4022 seconds. 3.3531 seconds. v^ 2248.75 f . s. 2289.1 f . s. 2268.9 f . s. 180. We will now take our standard problem, introduce the altitude factor, and solve. This for the 12" gun, for which 7 = 2900 f. s., w = 870 pounds and c = 0.61. For this problem we will take the angle of departure as 4° 13' 14", which we already know corresponds to a range of 10,000 yards, and will consider the atmospheric con- ditions as standard. Proceeding in a manner similar to that employed in originally computing the angle of departure, that is, by performing the work first without con- sidering / until we have gone far enough to enable us to determine the value of / by a series of approximations, and then introducing it, we have, by the use of the formulae employed in the preceding problem, and in addition of F = 1"C tan = 4:° 13' 14" tan 8.86803-10 1^3= log 2.79404 Constant loir 5.01765-10 /3= log 0.00648 logiog 7.81169-10 C,= W 0.99583 C,= log 1.00231 and as 0^ = 0^, we see that we have reached the limit of accuracy in determining the value of C, and we therefore proceed with the work with log (7=1.00231, 4=ao' = .014601, F = 2900, and from Table II Z = 2984, log i3' = . 10371, r = 1.2332 and t/^^ = 2026.2. The further work then becomes C= log 1.00231 log 1.00231 Z = 2984 log 3.47480 <;!> = 4° 13' 14" tan 8.86803-10. .sec 0.00118. .cos 9.99882-10 B'= log 0.10371 r = 1.2332 log 0.09103 Wa, = 2026.2 W 3.30668 2: = 29999 loir 4.47711 > = 5° 21' 11" tan 8.97174-10 sec 0.00190 T = 12.431 locr 1.09452 t'^ = 2029.55 log 3.30740 A comparison between these results and those obtained in Chapter 8, where we computed the values of the same elements with V and X as the data, gives an inter- esting measure of the accuracy of the methods employed. Tabulating these results for comparison, we have: Value by work under Element. Chapter 8. This Chapter. R 10,000 yards. 10,000 yards. CO 5° 21' 11". 5° 21' 11". T 12.43 seconds. 12.431 seconds. Vu 2029.0 foot-seconds. 2029.6 foot-seconds. 124 EXTERIOR BALLISTICS EXAMPLES. 1. Given the data contained in the following table, compute the values of R, w, T and v^, by Ingalls' methods, using Table II, and determining the value of / by- successive approximations, and applying it to get the correct value of the ballistic coefficient. DATA. ANSWERS. Prob lem. Projectile. Atmosphere. Ve- locity. f. s: 0. Range. Yds. u. T. Sees. Vo3. f. 8. a. If. Bar. Ther. In. Lbs. In. °F. A.... 3 13 1.00 28.00 1150 7° 13' 36" 2564 8° 59' 8.22 816 B.... . 3 13 1.00 28.10 ^ 2700 3 45 36 4072 7 18 8 22 988 C... 4 33 0.67 28.50 10 2900 1 35 36 3547 2 09 4.61 1850 D.... 5 50 1.00 28.67 15 3150 2 02 42 4088 3 29 5.90 1412 E.... .5 50 0.61 29.00 20 3150 1 30 12 3934 2 02 4.71 2004 F.... 6 105 0.61 29.83 25 2600 12 30 06 13201 22 36 26.98 1067 G.... 6 105 1.00 29.75 35 2800 1 48 36 3646 2 30 5.01 1716 H... . 6 105 0.61 30.00 43 2800 1 00 00 24H6 1 08 2.94 2306 I.... . 7 105 1.00 30.20 47 2700 3 59 54 6189 6 28 10.06 1311 J.... . 7 165 0.61 30.50 51 2700 3 00 24 5929 3 58 8.15 1781 K... 8 260 0.61 .30.75 58 2750 3 31 54 7214 4 39 9.73 1773 L.... . 10 510 1.00 31.00 65 2700 6 17 48 9665 10 10 15.81 1313 M... . 10 510 0.61 30.45 75 2700 5 43 54 10806 7 50 15.34 1692 N ... . 12 870 0.61 30.20 80 2900 10 45 00 19751 16 32 29.40 1516 0.... . 13 1130 1.00 30.00 85 2000 10 12 24 9810 14 20 19.73 1205 P.... . 13 1130 0.74 29.50 90 2000 10 40 48 11086 14 06 21.16 1315 Q.... . 14 1400 0.70 29.00 95 2000 13 45 18 1.3838 18 17 27.01 1302 E.... . 14 1400 0.70 28.67 100 2600 8 19 00 14553 11 14 21.44 1657 2. Given the data contained in the following table, compute the values of R, w, T and f ^ by Ingalls' methods, using Table II, and correcting for / by the use of the maximum ordinate given in the table. A.. B.. C. D.. E.. F.. G.. H. I.. J. . K. L.. M. N . O . P.. Q.. R.. DATA. Projectile. Atmosphere. Ve- locity. d. 10. c. Bar. Ther. f.s. In. Lbs. In. °F. 3 13 i.do 28.10 100 1150 3 13 1.00 28.40 95 2700 4 33 0.67 28.25 90 2900 5 50 1.00 29.00 87 3150 5 50 0.61 29.20 85 3150 6 105 0.61 29.50 80 2600 6 105 1.00 29.75 77 2800 6 105 0.61 30.00 73 2800 < 165 1.00 30.10 70 2700 7 165 0.61 30.20 60 2700 8 260 0.61 31.00 50 2750 10 510 1.00 30.50 40 2700 10 510 0.61 30.30 35 2700 12 870 0.61 30.00 31 2900 13 1130 1.00 29.75 20 2000 13 1130 0.74 29., 50 22 2000 14 1400 0.70 29.00 20 2000 14 1400 0.70 28.50 10 2600 14' 09 02 55 41 05 03 11 55 18 44 54 45 05 40 18" 24 06 06 27 42 24 18 18 48 00 48 48 18 36 24 18 12 12 00 18 Maxi- mum ordi- nate. Feet. 150 115 43 79 45 2481 L59 95 638 471 532 1214 824 5363 1488 1511 3450 1569 ANSWERS. Ranee Yds. 2042 3245 2757 3479 3035 12755 4447 3912 7298 7449 81.32 10073 9991 21989 9220 9978 14043 13017 T. Sees. 6° 08' 6.14 3 35 5.38 1 20 3.29 2 09 4.46 1 16 3.36 19 16 24.. 58 3 20 6.32 2 03 4.86 8 42 12.50 5 43 10.79 5 46 11.38 11 37 17.08 7 16 14.19 22 52 35.96 14 13 18.94 13 05 19.24 21 01 28.92 10 40 19.60 Vol, f. s. 900 1263 2179 1759 2333 1115 1616 2094 1213 1619 1702 1245 1685 1345 1167 1284 1219 1585 CHAPTEE 11. THE DERIVATION AND USE OF SPECIAL FORMULiE FOR FINDING THE ANGLE OF ELEVATION NECESSARY TO HIT A POINT ABOVE OR BELOW THE LEVEL OF THE GUN AND AT A GIVEN HORIZONTAL DISTANCE FROM THE GUN, AND THE TIME OF FLIGHT TO AND REMAINING VELOCITY AND STRIKING ANGLE AT THE TARGET; GIVEN THE INITIAL VELOCITY. New Symbols Introduced. (f>x. . ' .Angle of departure for a horizontal distance x. 181. As we know, the secondary function a refers to the point (x, y) in the Transfor- ' "^ \ • T mation of trajectory (A being its special value when x = X and y = 0) ; but since the pseudo formulae. velocity is independent of the height of this point, and dependent only on x, or the horizontal distance from the muzzle of the gun, we may consider x as the horizontal range of another trajectory having the same initial velocity, whose angle of departure may be designated by cjix. For this case then, we have, from (115) _ sin 2(^j a- ^~ (161) as well as A=^^^ (163) 182. We have also derived, in (141), the expression y-= --^ (A-a) (163) X 2 cos^ <^ ^ ^ ^ ^ 183. Now substituting in (163) from (161) and (162), we get y _ C /sin 2(f> _ sin 2 cf).A ~x~ 2 cos"" [~~C~~ C~l y _ sin 2<^ — s in 24>x X 2 cos^ 184. iSTow if p be the angle of position, we know that tan p= ^-- (IGo) (164) X and combining (164) and (165) we get sin2^-sin^. ' 2 cos^ — cos 2 — /;) — sin » • ^^ or - — >^— c — i^ ^ =sm2 — p. Elevated target. 126 EXTERIOR BALLISTICS 185. Now for any given point (x, y) we may compute the value of z from z= ^ , and then, with z and 7. as arguments, we may take the value of a from the A column of Table II. 186. We may then compute the value of sin 2x from sin 2(f>x= —aC and thence of ^ from (167). Also, with the same arguments, we may take from Table II the values of a', u and t' for the point (x,y). 187. Assembling the formulae deduced in this chapter, and also the other necessary formulae previously deduced as given in (85), (113), (138) and (1-14), we have, for the solution of this problem, tanp=y- (169) (170) sm2cf>a; = aC (171) sin(3<^-;5) =sin p(l + cot p sin 2cf>a;) (172) ^ = ct>-p (173) A='-^ (174) tau^=;^^ (.1-a') ■ (175) ^ = Cfsec (176) t; z= M cos ^ sec ^ (!''"<') . 188. Let us now compute these several elements for our standard problem 12" gun, y = 2900 f. s., w=:870 pounds, c = 0.61, for a target at a horizontal distance of 10,000 yards from the gun, and 1500 feet above the level of the gun, the barometer being at 29.00" and the thermometer at 90° F. Also, instead of computing the alti- tude factor, we will take it from Table V as being sufficiently accurate for this purpose. Taking the mean altitude as two-thirds of 1500 feet, that is, 1000 feet. Table V gives us that /= 1.026. Taking K from Table VI. K= log 0.99583 /=1.026 log 0.01115 S = .921 log 9.96426-10 colog 0.03574 C= log 1.04272 colog 8.95726-10 ?/ = 1500 log 3.17609 a- = 30000 log 4.47712 log 4.47712 p = 2° 51' 45" log 8.69897-10 ! = 2719.0 log 3.43440 PRACTICAL METHODS 127 From Table II a=.01299 a' = Md2 w = 2095.1 f = 1.104 C= log 1.04272 a = .01299 log 8.11361-10 2<^,,= sin 9.15633-10 p = 2° 51' 45" cot 1.30-103 cot p sin 2(^^ = 2.8665 log 0.45736 1 + cot ;? sin 2cf>^ = 3.8665 log 0.58732 j) = 2° 51' 45" sin 8.69844-10 2(p-p = ir 07' 59" sin 9.28576-10 p= 2° 51' 45" cf> = 6° 59' 52" 20=13° 59' 44" p = 2° 51' 45" cf>= 6° 59' 52" ip = 4:° 08' 07" 2(^ = 13° 59' 44" sin 9.38356-10 C= colog 8.95728-10 A= .02192 log 8.34084-10 a'= .02920 A-a'=(-).0072S (-)log 7.86213-10 (f, = 6° 59' 52" tan 9.08908-10. .sec 0.00324. . . cos 9.99676-10 A = .02192 eolog 1.65916 r=1.104 log 0.04297 u = 2095A log 3.32120 C= log 1.04272 ^=(-)2° 20'05" ..(-)tan 8.61037-10 sec 0.00036 ^ = 12.273 log 1.08893 v = 2081.2 log 3.31832 189. Xow suppose that, instead of the conditions worked out above, the gun had Depressei been in a battery on the hill and the ship had been the target, all other conditions "^® ' being the same. The work would have been the same down to and including the determination of the values of a, a', u and f, except that y is negative, and therefore p={ — )2° 51' 45". We then proceed as before, but with this negative value of p instead of the positive one employed before, and the subsequent work becomes : cot p sin 2x = ( — ) 2.8665 l + cot/Jsin2^=(-)1.8665 (-)log 0.27103 p={-)2° 51' 45" (-)sin 8.69844-10 2cl>-p = o° 19' 32" ( + )sin 8.96947-10 p=(-)2° 51' 45" = 1° 13' 54" 2= 2° 27' 47" p=(-)2° 51' 45" «;!,= 1° 13' 54" ij;= 4° 05' 39" 128 EXTERIOR BALLISTICS 2<^ = 2° 27' 47" sin 8.63321-10 C= coloff 8.95728-10 A = a! — .00390 log 7.59049-10 .02920 A-a'=(-).02530 (-)log 8.40312-10 <^ = 1°13'54" tan 8.33243-10. .sec 0.00010. . . cos 9.99990-10 A = .0039 colog 2.40951 f = 1.104 log 0.04297 w^2095.1 log 3.32120 C= loff 1.04272 '=(-)7° 57'01" ..(-)tan 9.14506-10, sec 0.00419 ^ = 12.184 W 1.08579 i' = 2114.9 log 3.32529 190. Assembling the results of these last two problems for comparison, we have : Value for Ship attacking battery. Battery attacking ship. t^' 4° 08' 07". 4° 05' 39". 6 i-)2° 20' 05". (-)7° 57' 01". t 12.273 seconds. V 2081.2 f. s. 12.184 seconds. 2114.9 f. s. In working problems similar to the above, great care must be taken to carry through consistently the signs of the several quantities and logarithms. Figure 14. 191. From Figure 14, in which (a) represents the first case and (&) the second, we plainly see that in (a) the force of gravity acts to reduce the velocity of the pro- jectile, and in (h) to increase it from what it would be in the horizontal trajectory. Therefore we would expect to find it necessary to give the gun a greater elevation relative to the line of sight in order to hit in (a) than in (b), and the results of the work show that such is the case. 192. Also, from the figures we can see that the angle of inclination of the curve to the horizontal at the point of impact would be greater in (&) than in (a), which is again shown by the work. PEACTICAL METHODS 129 193. Also, as gravity in (o) reduces and in (&) increases the velocity, we would expect to have the remaining velocity less and the time of flight greater in (a) than in (b), and again the work shows this to be the case. 194. The angle of elevation resulting from the work is of course the angle at which the gun must be pointed above the target in either case, that is, above the line of sight AB. The sight drums are marked in yards, however, and not in degrees of elevation ; so to practically set the sights we look in the range table of the gun and find in Column 2 an angle of departure equal to our found angle of elevation, and find in Column 1 the range in yards corresponding to that angle of departure. We then set our sights in range to that number of yards, and point the gun at the target, that is, bring the line of sight to coincide with the line AB of the figure. The gun is then elevated at the proper angle above the line AB, \p from the work, and at an angle of departure above the horizontal of = xp-\-p. 195. In the problem shown in Figure 14(a), we have by a simple interpolation between Columns 1 and 3, that the range corresponding to an angle of departure of 4° 08' 07" is 9841 yards, which is the range at which the sight should be set. 196. Similarly, in Figure 14(6), the sight should be set for an angle of departure of 4° 05' 39", that is, at 9764 yards. 197. In Chapters 8, 9 and 10, and in this chapter, we have shown the methods and formula to be employed in solving certain of the more common and more important ballistic problems. Those selected for the purposes of this book are the ones most likely to be encountered in naval practice, but there are a large number of others that may arise under special circumstances, which may be solved by similar methods. Some of the more important of these are enumerated below, to show the scope of the methods that have been taught, for they are all solved in similar ways. In each case the solution consists of a preliminary transformation of the fundamental ballistic formula, in a manner similar to those shown in the preceding pages of this book, in order to fit them for use in the particular problem under consideration ; and then the necessary computations may be made from the resultant equations. It should also be borne in mind that all our work has so far applied only to direct fire, as do also the problems enumerated below, and that when problems incident to mortar fire and other special classes of work are added, the number of problems that may present themselves becomes very large. Beside the problems already explained in these pages, some of the simpler direct fire problems that may be readily solved by similar methods are: (a) Knowing X, C and Va,; to compute V. (b) Knowing V, C and Voj', to compute X. (c) Knowing V, X and C; to compute T. (d) Knowing V, T and C; to compute Vu. (e) Knowing V, X and 0; to compute C. (f) Knowing V, C and v^; to compute X, , w and T, (g) Knowing V, C and w ; to compute X, , T and Vo). (h) Knowing X, and C; to compute V. (i) Knowing T, and C; to compute V. 130 EXTERIOE BALLISTICS EXAMPLE. 1. Given the data contained in the following table, compute the values of ip, t, V and 6 for the given values of x and ij, both when ij is positive and when it is negative ; and, whenever the range tables available permit, tell how to set the sight in elevation in each case in order to hit. Correct for / from the data given. Problem. DATA. Projectile. d. In. A B C D E F G H I. J K L M N P Q R 3 3 4 5 5 6 6 6 7 7 8 10 10 12 13 13 14 14 w. Lbs. 13 13 33 50 50 105 105 105 165 165 260 510 510 870 1130 1130 1400 1400 1.00 1.00 0.67 1.00 0.61 0.61 1.00 0.61 1.00 0.61 0.61 1.00 0.61 0.,61 1.00 0.74 0.70 0.70 Atmosphere. Bar. In. 31.00 30.90 30.75 30.33 30.00 29.80 29.50 29.25 29.00 28.50 28.25 28.00 28.10 28.07 29.00 29.15 29.75 30.00 Ther. °F. 5 10 20 27 33 37 32 30 40 50 55 60 75 80 83 87 93 97 Ve- locity f. s. 1150 2700 2900 3150 31.50 2600 2800 2800 2700 2700 2750 2700 2700 2900 2000 2000 2000 2600 Chart dis- tance from gun to target. Yds. 2500 4000 3300 4200 3700 7300 3800 3100 6800 7200 7800 9700 10800 20500 10500 11000 13000 13500 Height of target above or below gun. Feet. - 200 : 350 : 400 : 450 - 475 - 600 : 500 - 460 - 7.50 : 800 - 825 - 850 : 900 -1500 -1100 ■1000 - 950 -1200 Maximum ordinate for trajectory of range x. Feet. 252 251 70 144 75 522 109 57 518 424 456 995 952 3994 1937 1830 2637 1630 ANSWERS. Prob- lem. When y is positive. When y is negative. ^P. Set sight at: Yds. e. t. Sees. V. f. s. i. Set sight at: Yds. e. t. Sees. V. f. s. A B C D E F G H I J K L M N P Q R 7° 11.4' 3 56.9 1 28.3 2 11.0 1 23.3 4 38.8 1 54.9 1 17.5 4 36.0 3 49.1 3 49.0 6 02.2 5 35.5 11 14.3 11 15.6 10 39.1 12 54.0 7 42.8 4308 3388 4364 3764 7452 3854 3117 6869 7170 7744 9543 10G04 20180 10373 10879 13353 {—) 7° 34.4' (_) 6 09.4 20.3 (— ) 1 48.3 36.4 (— ) 5 25.1 (— ) 11.6 1 18.5 (— ) 5 37.5 (— ) 3 07.8 (— ) 3 01.5 (— ) 7 46.7 ( — ) 5 55.3 (— )16 02.3 (— )14 13.5 (— )12 30.4 (— )15.57.0 (— ) 8 48.8 8.13 8.38 4.27 6.24 4.38 11.60 5.29 3.77 11.42 10.23 10.54 15.40 15.09 30.72 21 . 62 21.12 25.39 19.91 795 948 1866 1345 2059 1412 1679 2192 1244 1675 1815 1377 1741 1502 1169 1299 1287 1645 7° 08.8' 3 56.0 1 28.1 2 10.6 1 22.9 4 37.5 1 54.6 1 17.3 4 34.4 3 48.0 3 47.9 5 59.9 5 33.6 11 07.5 11 05.8 10 31.5 12* 44.9 7 39.0 4299 3382 4358 3753 7429 3846 3114 6845 7145 7716 9505 10560 20046 10271 10786 i.S272 (— )10°26.1' (— ) 9 22.5 (— ) 4 16.4 (— ) 5 51.3 (_) 4 17.0 (— ) 8 27.0 (— ) 5 11.2 (— ) 4 20.6 (— ) 9 40.2 (— ) 7 17.2 (— ) 6 59.1 (— )10 55.3 (_) 8 58.2 (— )18 16.0 (— )17 29.5 (— )15 27.7 (— )18 07.7 (— )11 .55.8 8.07 8.. 35 4.26 6.22 4.37 11.55 5.27 3.76 11.35 10.18 10.49 15.31 15.00 30.41 21.31 20.87 25.09 19.75 807 960 1875 1355 2069 1428 1691 2203 1263 1695 1835 1398 1763 1535 1206 1331 1318 1675 CHAPTEE 12. THE EFFECT UPON THE RANGE OF VARIATIONS IN THE OTHER BALLISTIC ELEMENTS, WHICH INCLUDES THE DATA GIVEN IN COLUMNS 10, 11, 12 AND 19 OF THE RANGE TABLES. New Symbols Introduced. AX .... Variation in the range in feet. AR .... Variation in the range in yards. A(sin 2 = c(\ ^Su — Sv J These involve the range, X; the angle of departure, ^; the ballistic coefficient, C; and the initial velocity, V; either directly or through their functions as given in Table I. These four are the elements in which variations from standard may be expected, as indicated in the preceding paragraph ; as a change in the density of the atmosphere involves a corresponding change in the value of C, and as a change in the weight of the projectile involves a corresponding change in both the initial velocity and ballistic coefficient, as will be explained later. 200. For our present purpose, therefore, we wish, if practicable, to derive from (178) and (179) a single differential equation in which all four of the quantities enumerated shall appear as variables. By a noteworthy series of mathematical com- binations and differentiations, which need not be followed here. Colonel Ingalls has derived such an equation, his result being A{sm 2) = -CAvA-{B- A) AC + BC ^ (180) In this equation the symbol A indicates a comparatively small difference in value or differential increment (either positive or negative) in the value of the quantity to which prefixed. C is the ballistic coefficient; <^ is the angle of departure; A and B are Ingalls' secondary functions as they appear in Table II ; and A^a is the quantity contained in Table II in the A^ column pertaining to A, where Afa is for ± 50 or It 100 f . s. according to the table used. For 100 f . s. difference in velocity between successive tables, the solution of the above equation would give the proper result as it stands ; but for 50 f . s. difference in velocity, fifty one-hundredths, or one-half, of the variation should be taken, as shown later. Great care must be exercised not to confuse the three quantities represented by the symbols Afa as given above, SV, which represents a differential increment of the initial velocity and A7, which represents the difference in velocity between two successive tables in Table II. (Ingalls' tables PEACTICAL METHODS 133 as originally computed ; not the abridged tables reprinted for use with this text book. The values of V given at the top of each table show the value of AF for that table.)* 201. Having the above general differential equation (180) involving all four fn'inltfii'"*"* variables with which we wish to deal, we now wish to apply it to the several cases at velocity, issue; and we will first consider the variation in range resulting from a small varia- tion in the initial velocity ; in other words, we will find out how to compute the data contained in Column 10 of the range table. For this case the initial velocity and the range are the only variables, and AC and A (sin 2^) therefore become zero. Equation (180) then becomes AXv=^X (181) X) in which AAV represents the change in range in feet resulting from a change of SV in initial velocity. To use this formula we must first compute the value of Z = -^f , and, then, from Table II, with V and Z as arguments, we may take the value of B.f 202. Suppose that we desire to compute the change in range at 10,000 yards resulting from a variation in the initial velocity of ±50 foot-seconds from the standard, for our standard problem 13" gun (7 = 2900 f. s., w = 870 pounds, c=0.61). In paragraph 157 we found that for this problem Z = 2984.1 and log (7 = 1.00231. We desire our result in yards, and as there is no quantity appearing in the equation in * In determining the value of A y^ from the table for use in the formula, the tabular value must be corrected by interpolation for the exact values of Z and V, as follows: (a) Suppose we had V = 2800 f. s. and Z = 3773.2. The next lower tabular value of AvA is .00149, and the difference between this and the next tabular value above it in value is .00155 — .00149 = .00006. Therefore our value for use in the formula would be ^,, = . 00149 + :^M6^^^ ^.0015339 which is carried out for the full limit of use with our log tables. (b) Suppose we had V = 2750 f. s. and Z = 3770.5. The next lower tabular value of 70 5 Afa is .00167, and as in (a) the correction for Z would be (.00173 — .00167) -~ . The next lower tabular value of A^^, as given above, is .00167, which is for Z = 3700; and turning to the table for V = 2800 f. s.. we see that the value of A^ for Z — 3700 f. s. is .00149. Therefore the variation in A,-^ for 100 f. s. increase in V would be .00149 — .00167 = — .00018, and for 50 f. s. it would be half that. Our complete interpolation for this case would therefore be A ,. = .00167 -f ^^-XIM _ ,000^X10 ^ _^,^,223 (c) In the case of the 5" gun for which V = 3150 f. s., this interpolation is further complicated by the fact that we have no table from which to determine the value of A y^ for y = 3200 f. s. The rate of change of A^^ at this point for an increase of 100 f. s. in initial velocity may be obtained with sufficient accuracy for every ordinary purpose as follows: Suppose y = 3150 f. s. and Z — 3770.5 For Y = 2900 f. s. and Z = 3700 we have Ar^ = .00137 For Y = 3100 f. s. and Z = 3700 we have Ar^ = .00111 Therefore, for a change of Y of 200 f. s. we here have a change in Arx of .00111 — .00137 = — .00026. Assuming that the same rate of change continues for the next 100 f. s. increase in V, which assumption is not greatly in error, we would have that the change in the value of Ar4 between Y = 3100 f. s. and y = 3200 f. s. would be H— -00026) = — .00013. Our interpolation would therefore become A,., = .00111 + :0000^X70,5 _ -00013^X50 ^ ^^^^^^3 f The convention employed in this chapter relative to the double sign (±) is that a positive sign in a result means an increase in range and a negative sign a decrease. 134 EXTEEIOE BALLISTICS feet, we may use the range in yards, which will give a result also in yards. Also the difference Ay a at this point in the table is for a difference AT'' in velocity between two successive tables (A7=100 f. s., between tables 2900 to 3000 f. s., and 3000 to 3100 f. s.), therefore we apply a factor ot —y l^r^ ^'^ this casej. Therefore if we let ARv represent the change in range in yards for a variation of 8T^ = 50 f. s. in the initial velocity, the expression becomes ARy=^X^XB The work then becomes, from Table II : ArA = . 00099+ '^^^^/qq ^^'"^ =.001024 5=.0178+ -^^^^^^^-^ =.01856 ArA = .001024 log 7.01030-10 87= ±50 ±log 1.69897 i?= 10000 log 4.00000 £ = .01856 log 8.26857-10 colog 1.73142 A7 = 100 log 2.00000 colog 8.00000-10 ARv= ±276 yards ±iog 2.44069 and the signs show that an increase in initial velocity will give an increase in range, and the reverse, which was of course to be expected. For variation 203. Again, suppose that the density of the air varies from standard, as it ^"atmosphere, generally does, and we wish to determine the resultant effect upon the range. We know that C = J-^n: , and in this case the only variables are X and 8, as = the angle of departure. Z^' = Ingalls' secondary function for drift, to be found in Table II, with Y and Z as arguments. Z) = drift in yards for the given range and angle of departure. VARIATION OF THE TRAJECTORY FROM A PLANE CURVE 147 217. It has also been found necessary in some cases to multiply the results obtained by the use of the above formula by a certain empirical multiplier in order to get correct results. The data required for the drift computation is therefore : DATA. Problem. Gui 1 and projectile. Velocity. f. s. fi. X h ' n. Multi- d. In. w. Lbs. c. plier. A 3 3 4 5 5 6 6 6 7 7 8 10 10 12 13 13 14 14 13 13 33 50 50 105 105 105 165 165 260 510 510 870 11.30 1130 1400 1400 1.00 1.00 0.67 1.00 0.61 0.61 1.00 0.61 1.00 0.61 0.61 1.00 0.61 0.61 1.00 0.74 0.70 0.70 11.50 2700 2000 3150 3150 2600 2800 2800 2700 2700 2750 2700 2700 2900 2000 2000 2000 2600 0.53 0.53 0..53 0.53 0.53 0..53 0.53 0.53 0..53 0.53 0.53 0..53 0.53 0.53 0.53 0.53 0.53 0.53 0..32 0..32 0.32 0.32 0.32 0.32 0..32 0.32 0.32 0.32 0..32 0.32 0.32 0.32 0.32 0.32 0.32 0.32 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 1.0 B 1.0 c 1.0 D 1.5 E 1.5 F 1.5 G 1.0 H 1.5 I 1.0 J 1.5 K 1.5 L 1.0 M 1.5 N 1.5 1.0 p 1.5 1.5 E 1.5 218. Returning to our standard problem, we will compute the drift for the 12" gun (7 = 2900 f. s., w = 870 pounds, c = 0.61) for a range of 10,000 yards, for which conditions we found in Chapter 8 that the angle of departure was 4° 13' 14", Z = 2984.1 and log C = 1.00231. The computation then becomes 1..5 log 0.17609 f^ = 0.-53 log 9.72428-10 = 0.32 log 9.50515-10 /i = 25 log 1.39794 colog 8.60206-10 C= log 1.00231 2 log 2.00462 Z;' = 25.6 log 1.40824 (/) = 4° 13' 14" sec 0.00118 3 sec 0.00354 Z) = 26.5 yards log 1.42398 219. Guns are usually, and naval guns always, pointed by directing what is sights. called the " line of sight " at the target. Originally the upper surface of the gun itself was used as the line of sight; this was called " sighting by the line of metal," and resulted in giving to the gun an angular elevation corresponding to the differ- ence in thickness of metal at the breech and at the muzzle. Later on, a piece of wood, called a " dispart," was secured to the muzzle, so as to give a line from breech to muzzle parallel to the axis of the gun. Such a line of sight had to be directed more or less above the target according to the range. Early in the last century came into use the method of having one fixed and one movable sight, so that the line between them, which is the line of sight, could be adjusted at any desired angle with the axis of the gun. The rear sight was usually the movable one. At the present time the most approved form of sight, and practically the only one in use in the navy, is that 148 EXTERIOR BALLISTICS Theory of bar sights. in which a telescope has been substituted for the old pair of sights, front and rear. This telescope is so mounted that it is capable of being set at any desired angle with the axis of the gun, within necessary limits. The principles involved in the tele- scopic sight are the same as in the old bar sights, but in the former they are not so clearly apparent or so easily studied as in the latter. For this reason we will take up the theory of sights from the point of view of the old system of bar sights, rear sight adjustable, and the application of these theories to the telescopic sight will be plainly apparent. 220. The rear sight being movable, it is customary to graduate its bar in yards of range (and sometimes with the elevation in degrees corresponding to the range in yards), and sometimes there is added the time of flight in seconds corresponding to each range, this last information being for use in setting time fuses when using shrapnel, etc. This information is ordinarily not placed on the range scale of a telescopic sight, which shows only the range in yards; and if such information be wanted it must be taken from the range table for the gun wJiich is now furnished to ships. AT' -.--H^' Figure 16. Sight bar height. -J -yy„ 221. Figure 16 represents the usual arrangement of bar sights, AC being the movable graduated rear sight bar, at right angles to the axis of the gun, and B the fixed front sight. C'B is the line of sight, being a line from the notch in the rear sight C" to the top of the front sight B, and CB is the position of the line of sight when it is parallel to the axis of the gun, the rear sight notch being then lowered to C, usually its lowest position. The distance GB^l is called the " sight radius " of the gun, and the line GB is sometimes called the " natural line of sight." It will be seen that when the rear sight notch is raised to C", and the line of sight C'B is directed at the target P, the axis of the gun, which is parallel to CB, is elevated at the angle CBC = \\i, or the angle of elevation above the target. As we will deal only with hori- zontal trajectories, and disregard jump, the angle of departure will be equal to the angle of elevation, so CBC' = ip = (j>. The distance CC' = h is the " sight bar height " for the angle of departure (f>, and it is evidently given by the equation h = l tan (194) VARIATION OF THE TRAJECTORY FROM A PLANE CURVE 149 When the heights for the range graduations of the sight bar are to be computed the above formula is used. 222. In Figure 16 the trajectory is represented as though the axis of the gun coincided with the line of sight, instead of being, as it really is, offset from it by at least the radius of the breech of the gun. No appreciable error, however, results from making this assumption in the ordinary use of guns, except in the case of turret guns, where the pointers' sights may be located at a considerable distance from the axis of the gun, in the turret or in the pointers' hoods. In this case this distance must be allowed for in figuring on the fall of shot, and it is customary in bore sight- ing these guns to so adjust the sights that the line of sight of each of them will intersect the axis of the bore prolonged at the most probable fighting range. At the proving ground, where extreme accuracy is necessary, as in attacking armor plates, etc., it is customary to use bore sights in sighting the gun, thus eliminating the error due to the offset of the line of sight of the regular sights from the axis of the gun. 223. Besides the movement of the rear sight up and down to enable the gun to sliding leaf, be pointed with the proper elevation, it is desirable to have some means of moving it sideways, so that the line of sight may be adjusted at any desired angle with the axis of the gun, within reasonable limits, in the horizontal plane as well as in the vertical. This is for the purpose of allowing for drift or other lateral deviations of the projectile, by causing the gun to point the proper amount to one side of the target at which the line of sight is directed. In the bar sight this is usually done by forming the rear sight notch in a " sliding leaf," a piece mounted on the head of the sight bar and movable by suitable mechanism at right angles to the sight bar and to the axis of the gun. 224. In Figure 16, D and D' represent two positions of the sliding leaf on opposite sides of the central position C". Evidently, if the line of sight B'B be directed at the target P", and there be no deviation of the projectile in flight, the latter will fall at P; and so a movement C'V = d^ of the sliding leaf to the left will cause the projectile to fall P"P = D^ to the left of the point aimed at P". And, similarly, the moving of the sliding leaf C'D = d^ to the right will cause the projectile to fall P'P = D2 to the right of the point aimed at, P'. Therefore to correct for a deviation of the projectile due to drift, wind, motion of the gun or target, or any other cause, the sliding leaf is moved in the opposite direction to the deviation, and we have the general rule : Move the sliding leaf (or rear, or eye end of the telescope) to the side toivards which you ivish the projectile to go. Also, for the relation between the motion, d, of the sliding leaf and the resulting deviation, D, of the projectile, we have from similar triangles ^^_ FP d D C'B PB ^^ lseccl> ~ X whence d= —^"^ V (195) The error which results from putting see4> = l in (195) is inappreciable for the small angles of departure required in the ordinary use of modern guns, being only one-half of 1 per cent for (f> = G° ; and, as it is only a little over 3 per cent for (f> = lo°, the limit of elevation possible with our usual naval gun mounts ; and, as the value of D to be allowed for is seldom as closely known as that, it is evident that in direct fire, under all ordinary circumstances, we may use d=~D (196) * D and X must be in the same units, either both feet or both yards, d will then come in the same units as I, usuallj^ inches. 150 EXTEEIOE BALLISTICS Permanent angle. 225. It was common practice with naval guns and bar sights to correct for the greater part of the drift automatically by inclining the bar sight in a plane per- pendicular to the axis of the bore of the gun, so as to make with the vertical an angle i called the " permanent angle." Eef erring to Figure 17, we see that the three points D, C and C are in the same plane, which is perpendicular to the line CB, the angles CC'D and BCD being right angles ; the points B, C and C are all three in the same plane, which is at right angles to DCC; B, C and D are in the same plane, which is at right angles to BCC; and, similarly, for P, P' and Q. Then we have that CB = l, being the natural line of sight, and CD = h is the sight bar height for the angle of departure ^, and is now given by h = l tan sect (197) Then if PF be the drift in yards, at the range R, from the similar triangles we HP' T) have :^ = ^ ; but DC = h smi = I tan <^ tan i, and CB = I sec <^. Therefore C B R ^^^ "^ ^^" ^ =sin , which it is not far from being, setting the sight bar at the permanent angle i given by (198) would exactly compensate for drift at all ranges. Actually, however, D increases a little more rapidly in propor- tion than X sin ^, and so the sight should be more inclined for long than for sliort ranges. In practice, when bar sights are used, it is customary to compute the value * D and R must be in the same units. In the above equation they are both expressed in yards. VAEIATION OF THE TRAJECTORY FROM A PLANE CURVE 151 of i for an assumed average fighting range, and to set the bar at that permanent angle, leaving any uncompensated drift at other ranges to be corrected by setting the sliding leaf while using it to correct for deviations due to wind and speed. 227. The telescopic sight is now almost universally adopted, its great advantage Telescopic being that with it the line of sight, which is the optical axis of the telescope, is much more clearly defined and can be directed with much greater accuracy than is the case with the old bar sight. Also, as the eye is held close to the telescope in pointing, which coaild not be done with the old sights, the parallactic errors, which it was formerly almost impossible to avoid, are now practically eliminated. The theory of the telescopic sight in no way differs from that of the bar sight as explained in the preceding paragraphs, however, but the mechanical features of the bar sight are such that the bar itself actually establishes the system of triangles with which we have been dealing and from which the mathematical relations are plainly apparent. With the telescopic sight, however, with both ranges and deflections marked on rotary drums, circular discs, etc., the motion of which is transferred to the telescope itself by gearing, or any similar devices ordinarily in use, the triangles are not readily appar- ent, although the relations arising from them of course still exist; the sight radius in the bar sight being replaced in the telescopic mounting by the distance from the pivot of the sight yoke to the circle of graduations on the sword arm, that is, by the radius of curvature of the scale on the sword arm. 228. In the telescopic sight the telescope is so mounted that it can be set at any desired vertical angle with the axis of the gun, within practical limits, thus enabling the gun to be pointed at the proper elevation, the elevating scale being marked in yards in range computed for the corresponding angle between the line of sight and the axis of the bore. The telescope can also be rotated, within reasonable limits, about its vertical axis, which corrects for deviation in the horizontal plane exactly as did setting over the sliding leaf of the bar sight ; the rotation of the telescope about its vertical axis being recorded on the deflection scale, which is marked in " knots," the knots thus indicated corresponding to speed of target. The reasons for this graduation and the method of using it will be explained later, in the chapter describ- ing the use of the range tables. Drift is not compensated for in the mounting of this sight, but as the gun is elevated the pointer on the deflection scale moves up or down over the scale, and the line on the scale for each knot setting of the sight in deflec- tion, instead of being a straight line, is a curve so computed and laid on the scale that when the deflection pointer is on it the drift is compensated, no matter what the range may be. This system, while not automatic, gives perfect compensation for drift at all ranges, which the old permanent angle system did not, as we have already seen, and introduces no troubles or errors into the actual process of setting the sight which would not exist without it. (See Appendix C for a description of the system of arbitrary deflection scales now in use for turret guns.) 153 EXTEEIOR BALLISTICS EXAMPLES. 1. Compute the drift in yards for the following conditions, taking the angle of departure and the maximum ordinate from the range tables. Conditions standard. DATA. ANSWERS. Problem. Projectile. Velocity. f. s. Range. Yds. Drift. d. In. to. Lbs. c. Yds. A 3 3 4 5 5 6 6 6 7 7 8 10 10 12 13 13 14 14 13 13 33 50 50 105 105 105 165 165 260 510 510 870 1130 1130 1400 1400 1.00 1.00 0.67 1.00 0.61 0.61 1.00 0.61 1.00 0.61 0.61 1.00 0.61 0.61 1.00 0.74 0.70 0.70 1150 2700 2900 3150 3150 2600 2800 2800 2700 2700 2750 2700 2700 2900 2000 2000 2000 2600 2500 4000 3500 4000 3700 10500 3800 3200 7000 6700 8000 10000 11000 22000 10000 11000 14000 14500 7.5 B 8.5 c 2.4 D 5.8 E 3.3 F 69.4 G 3.2 H 2.6 I 17.6 J .... K 15.7 21.5 L 33.9 M 44.3 N 233.8 52.1 P 82.9 148.1 R 89.3 2. Compute the sight bar heights in inches, and the distance in inches that the sliding leaf must be set over for the data given in the following table, conditions being standard. DATA. ANSWERS. Problem. Gun. Range. Yds. <(>■ Sight ra- dius or radius of curva- ture of sword arm. In. Deflection to be com- pensated. Yds. Sight bar height. In. Set of sliding leaf. Cal. In. I.T. f. s. o Right or left. A B C D E F G H I J K L M N ...... . P Q R 3 3 4 5 5 6 6 6 7 7 8 10 10 12 13 13 14 14 1150 2700 2900 3150 3150 2600 2800 2800 2700 2700 2750 2700 2700 2900 2000 2000 2000 2600 1700 3600 3100 4000 4500 13500 4100 3400 7000 6700 7800 9700 10900 23500 9600 10800 13600 14000 4° 19' 06" 2 48 54 1 18 42 1 53 12 1 45 48 12 20 18 2 05 18 1 25 42 4 44 48 3 28 48 3 51 12 6 11 42 5 48 18 14 12 12 10 03 18 10 32 42 13 54 00 8 14 06 24.750 28.625 45.900 58.500 42.625 42.625 42 . 625 42.625 55.850 62.500 41 . 125 44.675 44.675 47.469 61.094 61.094 36.219 36.219 25 R. 20 L. 40 L. 50 R. 30 R. 75 L. 50 R. 30 L. 100 R. 75 L. 50 R. 60 L. 50 R. 150 L. 70 R. 60 R. 50 L. 75 L. 1.869 1.408 1.051 1.927 1.312 9.324 1.5.54 1.063 4.637 3.801 2.770 4.849 4.542 12.014 10.833 11.373 8.963 5.242 0.365 0.159 0.592 0.732 0.284 0.242 0.520 0.376 0.801 0.701 0.264 0.278 0.206 0.313 0.452 0.345 0.137 0.196 Left Right Right Left Left Right Left Right Left Right Left Right Left Right Left Left Right Right Note. — The data given in the sight radius column is approximate only, and must not be accepted as reliable. The computed sight bar heights and the set of the sliding leaf are for the old bar sight. To use them for the telescopic sight they must be transformed as necessary into the proper distances for marking on sword arm scales or range or deflec- tion scales, as the case may be. These supplementary computations have to do with the mechanical features of the sight only, and not with the principles of exterior ballistics, and are therefore not considered here. CHAPTER 14. THE EFFECT OF WIND UPON THE MOTION OF THE PROJECTILE. THE EFFECT OF MOTION OF THE GUN UPON THE MOTION OF THE PRO- JECTILE. THE EFFECT OF MOTION OF THE TARGET UPON THE MOTION OF THE PROJECTILE RELATIVE TO THE TARGET. THE EFFECT UPON THE MOTION OF THE PROJECTILE RELATIVE TO THE TARGET OF ALL THREE MOTIONS COMBINED. THE COMPUTATION OF THE DATA CON- TAINED IN COLUMNS 13, 14, 15, 16, 17 AND 18 OF THE RANGE TABLES. New Symbols Introduced. W . . . . Eeal wind, force in feet per second, /?.... Angle between wind and line of fire. Wx . . . • Component of W in line of fire in feet per second. Wi2a!- • • • Wind component of 12 knots in line of fire in feet per second. Wz. . . . Component of W perpendicular to line of fire in feet per second. TFi2«- • • • Wind component of 12 knots perpendicular to line of fire in feet per second. X . . . . Range in feet without considering wind. X' . . . . Range in feet considering wind. V . . . . Initial velocity in foot-seconds without considering wind. y . . . . Initial velocity in foot-seconds considering wind. <^ . . . . Angle of departure without considering wind. ' . . . . Angle of departure considering wind. T . . . . Time of flight in seconds without considering wind. T" . . . . Time of flight in seconds considering wind. AAV- • • • Variation in range in feet due to Wi. AA'^ott- • • • Variation in range in feet due to a wind component of 12 knots in line of fire. ABw • • • Variation in range in yards due to Wx- LR^ow- • • • Variation in range in yards due to a wind component of 12 knots in line of fire. y . . . . Angle between trajectories relative to air and relative to ground. Dw- • • • Deflection in yards due to wind component Wg perpendicular to line of fire. D^^w- • • • Deflection in yards due to wind component of 12 knots perpendicular to line of fire. G. . . . Motion of gun in feet per second. Gx- • • ■ Component of G in line of fire in feet per second. G'lox- . • • Motion of gun of 12 knots in line of fire in feet per second. Gz. . . . Component of G perpendicular to line of fire in feet per second. Gjos. . . . Motion of gun of 12 knots perpendicular to line of fire in feet per second. AXg .... Variation in range in feet due to Gx. AX120. • • • A'ariation in range in feet due to a motion of gun in line of fire of 12 knots. Ai^G. . . . Variation in range in yards due to Gx- AR^2G- - • . Variation in range in yards due to a motion of gun in line of fire of 12 knots. 154 EXTEEIOE BALLISTICS Dg • • • . Deflection in yards due to a motion of gun of Gz perpendicular to line of fire. Di20- • • • Deflection in yards due to a motion of gun of 12 knots perpendicular to line of fire. T . . . . Motion of target in feet jser second. Tx- . ■ ' Motion of target in line of fire in feet per second. Tiaa- • • • • Motion of target of 12 knots in line of fire in feet per second. Tg. . . . Motion of target perpendicular to line of fire in feet per second. Ti2« .... Motion of target of 12 knots perpendicular to line of fire in feet per second. AXt .... Variation in range in feet due to T. AX^2T' • . • Variation in range in feet due to a motion of target of 12 knots in line of fire. ARt- . . . Variation in range in yards due to T. AR^2t- ' • • Variation in range in yards due to a motion of target of 12 knots in line of fire. Dt .... Deflection in yards due to a motion of target Ts perpendicular to line of fire. DisT. • . . Deflection in yards due to a motion of target of 12 knots perpen- dicular to line of fire. a. . . . Angle of real wind with course of ship. a' . . . . Angle of apparent wind with course of ship. TFj .... Velocity of real wind in knots per hour. TFo • • • • Velocity of apparent wind in knots per hour. Section 1. — The Effect of ^Yind Upon the Motion of the Projectile. 229. In considering the effect of wind upon the flight of the projectile, we are obliged, for want of a better knowledge, to assume that the air moves horizontally only, and that its direction and velocity are the same throughout the trajectory as we observe them to be at the gun. Actually the wind is never steady, either in force or in direction ; its velocity usually increases with the height above the gun, and its motion is not always confined to the horizontal plane. Moreover, lateral wind pressure alters the drift due to rotation. 230. It is for these reasons that the deviations caused by the wind can only be roughly approximated; and, consequently, that experiments for determining any of the ballistic constants, to be of value, must be made when it is calm or very nearly so. Primary 231, Let US denote by W the velocity of the wind in feet per second, and by YVx and YV~, respectively, the components of that velocity in and at right angles to the plane of fire. Also let us call Wx positive when it is with the fiight of the projectile, and negative when it is against it. Let us also call Wz positive when it tends to carry the projectile from right to left of an observer looking from gun to target, and negative in the opposite case. In Figure 18 let us denote by ji the angle between the direction from gun to target and the direction towards which the wind is blowing, measuring the angle to the left from the first direction around to the second. Then in Figure 18(a), fi is in the first quadrant, and \Yx is blowing with the projectile and is positive, and Wz causes lateral motion to the left and is also positive. In Figure 18(&), ^ is in the second quadrant, and Wx is negative and Wz is positive. In Figuro 18(c), )8 is in the third quadrant, and both Wx and Wz are negative. In Figure 18 ((i), p is in the fourth quadrant, and Wx is positive and Wz is negative. Note especially that, in the system of notation adopted, fi is the angle between the plane of fire and the direction towards and not that from which the wind is blowing. VARIATION OF THE TRAJECTORY FROM A PLANE CURVE 155 (In a later chapter dealing with practical service problems concerning the wind, it will be found that the wind is generally stated as coviing from a given direction, but the reverse convention is used in this chapter. Bear the difference constantly in mind and do not be confused by it.) J^^ (a) Wx is + Wz is + (6) Wx is — Wz is + .'7 ^. ,--/S^ (c) id) Wx is — Wx is # Wz is — We is — Figure 18. >~^ 232. By an examination of the figure we see that, no matter what the direction of the wind, we shall always have its components in the two primary reference planes given with their proper signs by the two expressions: F^ = Fcos^ (199) W„, = Fsin^ (200) 233. The respective values of the wind components being thus found, they are hereafter treated as though independent of each other, each producing its own effect in its own primary plane only. 234. To find the effect of the wind upon the range, let ^Vx be the wind com- ponent in the plane of fire in feet per second, positive when with the flight of the pro- jectile and negative when against it ; let V and <^ be the initial velocity and angle of departure, respectively, relative to still air or to the ground, which is of course station- ary; and let X be the range in feet for this initial velocity and angle of departure, that is, relative to the ground or to still air. 235. Let us now designate by X', V and ^' the range, initial velocity and angle of departure, respectively, relative to the moving air, of a projectile fired with a range of X, initial velocity of V, and angle of departure of 4> relative to still air or to the ground, and by dV and dct> the differences between V and V and <^ and <}>', respect- ively. Let us designate by T" and T the corresponding times of flight. Now let us suppose that the wind component along the line of flight, Wx, is blowing with the flight of the projectile; in which case, while the projectile is in flight, the air through which it is traveling moves in the same direction, carrying the projectile with it a distance WxT'. Then the total horizontal distance traveled by the projectile relative to the ground or to still air will be X' + WxT'. As the normal range relative to the ground corresponding to V and ^ is X, then the difference between the two ranges, or the change in range caused by the wind, would be ^X^X' + WxT'-X (201) Effect in range. 156 EXTEEIOE BALLISTICS Force diagrams. 236. In order to use the above equation it is necessary to determine the values of X' and T', and we can do this by methods previously explained if we can determine the corresponding values of V and (f)'. This we can do if we can find the values of dV and dcf). To do this let us draw the triangle of forces acting in this case (and also for a negative wind) . We would have the results as shown in Figure 19. Figure 19 (a) is for a positive wind, for which Wx is positive (being drawn in the proper direction for constructing a triangle of forces with all parts of proper relation to one another), and from the diagram it will be seen that OA = V combined with H^^c/ AB— TF^ gives 05= F, which is less than OA = "T by the amount dY = AC—\\'x cos <^. Also the angle, BOH = (^' is greater than the angle AOH = cf> by the angle d^ = AOB=^ = ^^^^ (assuming that for this small angle the sine and the circular measure of the angle are equal). In other words, the forces acting would produce a trajectory relative to the moving air for which the initial velocity is V' = V — dV and the angle of departure is ' = (f> + d(l>. Similarly, from Figure 19(&), where Wx is negative or against the flight of the projectile, we would have V greater than V by the amount dV=Wx cos (j), and cf>' less than 4> by the amount f/0 = -^ = ^''^^ ^ . Thus, in both cases we can obtain the values of the changes in V and with their proper signs from the expressions dV=-Wx cos cf> (202) d=^^^^ (203) The negative sign is arbitrarily introduced into the second term of (202) to ensure that a positive value of Wx shall always produce a negative value of dV, and that a negative value of Wx shall always produce a positive value of dV, as is seen from the triangles of forces must always be the case. 237. To determine the effect of a wind Wx, therefore, we compute dV by (202) and d(f> by (203) ; compute the range Z given by V and ^ by methods heretofore explained; compute the range X' and the time of flight T' given by V'=V + dV and <^' = (ji + d(i> by methods heretofore explained, and then by (201) we can compute the change in range due to the wind. VARIATION OF THE TRAJECTORY FROM A PLANE CURVE 157 238. An examination of Figure 20 will help to reach a clear understanding of the foregoing method of determining the effect of wind upon the range. Let represent the stationary gun and M the stationary target which would be hit if there were no wind at the range OM = X, by a projectile fired from with an initial velocity V and an angle of departure 4>, the flight being through still air or considered relative to the ground with no wind Mowing. Then OSH represents the trajectory relative to still air or to the ground; that is, the trajectory that would be given in still air by the initial velocity V and the angle of departure <^', in which the projectile would fall short of M by the distance HM. But since the air is moving with the projectile with a velocity Wx, and is carrying the projectile with it with the same velocity during the time of flight T, when the projectile reaches the ground the trajectory OSH will have moved to the position O'S'W, the actual point of fall will be at //', and the >^A^ >: Figure 20. actual range over the ground will be OH' = X' -\-^VxT'. Therefore, instead of falling at the target M, the projectile will really strike at the point H', a distance beyond M of ^X = X'+^Ya:T'-X 239. The process just explained is not only a somewhat lengthy and inconvenient one, but the methods of interpolation used with the ballistic tables were not devised with this particular process in view, and do not produce results sufficiently accurate to determine the small differences in range with the precision necessary in this class of problem.* A carrying out of the process just described may therefore not bring correct results, and it is desirable to reduce the formulae, if possible, to some form more convenient for practical use and that will not involve the use of the ballistic tables. Although, as already stated, the above formulge are not useful for obtaining practical results in this case, they are nevertheless theoretically correct, and from them will now be derived the formula that is actually used in practice in computing the data contained in Column 13 of the range tables. From Chapter 4 we see that (when a = 2, which is sufficiently accurate for present purposes) the relation between the range in air and the angle of departure is - 9J^ - 9^ y2 V' sin2: (I + PZ) (204) Taking logarithmic differentials of this expression, that is, differentiating and then dividing by the original equation, and considering and X as the only variables (the small angle 2(j> being considered as having its natural sine equal to its circular measure) we get OJj. 1 I 41. V /7V (205) 2dcf> _l+^kX ^ dX tan2 l+^JcX X * See foot-note to paragraph 153, Chapter 8. An effort to use the formulae just derived for the value of the effects of wind upon the flight of the projectile, by the use of Ingalls' method, with the interpolation formulae given in Chapter 8, will not be successful, because those interpolation formulae neglect second and higher differences; and the limits of accuracy within which these results would have to be obtained in this case are too narrow to permit such higher differences than the first to be neglected in using Table II. 158 EXTERIOR BALLISTICS Now l + ^kX = n; whence l + |-^•.Y = 2?^— 1, and so (205) becomes 2dci > _ 2n-l ^ dX tan 24> n X whence — = ^^ X -^^ (206) ^ ^® Z 2n-l tan2(;S ^ ^ 240. In this and in similar expressions the value of d(j) must of course be ex- pressed in circular measure (l' = . 0002909), but when the form is the ratio -^ , the value is the same whether (/<^ and 4> are expressed in minutes of arc or in circular measure. Now returning to (204) and writing it in the form V^sin2cb = gX{l + ikX) and taking logarithmic differentials with regard to V and X as variables, we get 2dV _l + ^kX ^dX .on7\ ~T^-T+pZ^ X ^^"^^^ which, substituting n for l + p-.Y and 2n — l for 1 + ^hX, becomes 24L We have found in (208) an expression for the variation in range due to a variation in initial velocity, and in (206) an expression for the variation in range due to a variation in the angle of departure ; and we have already seen that the effect upon the range of a wind component in the plane of fire is to give, so far as results are concerned, an apparent variation in both V and 4>- Therefore Z' — Z is nothing but the change in range which would result from increasing V hj dV and <^ by c?(^, dV being determined by (202) and d by (203). Then by employing (208) and (206) we see that the change in Z due to simultaneous changes dV= —Wx cos ^ and Wx sin ■ ■ 1 ±1 dcj) = — ^, — — IS given by the expression Wx sin cf) _ Wx cos <^ \ ('909') 2n-l\V'taji2^ V ) ^'' ' Now we may put V for V in the preceding expression without material error, because jy is always very small compared with V^ and the expression then reduces to X'-X 2n ^WJsm _ . AT sin (b , , ( tan d> -, Now T ^ - cos = ^ = — , . • ., cos 2<^ cos-<^ — sm-^ 1 , / tan d) -,\ , /sin d> ^ cos- <^ — sin- <^ ., whence cos (/> ( -^ ~j~ — 1 I = cos <^ ( — ^ X ^ . . — r — 1 tan 2<^ / \cos<^ 2 sin <^ cos ^ , /cos- , , , ^s whence, neglecting tan- c^ in comparison with unity, '2n — l V and substituting this in (201) we finally get for the change in range due to the wind component Wx AX=W, (t--^^ X ^^^) * (311) In the above equation, T, although actually the time of flight for V and 4>', may be taken as the time of flight for the actual firing data, V and , without introducing any material errors. This formula is the one employed in computing the data in Column 13 of the range tables, giving the change in range resulting from a wind com- ponent of 12 knots in the plane of fire. 242. ^STow let us compute the data for that column for our standard problem, the 12" gun, 7 = 2900 f. s., w = 870 pounds, c=0.61, i2 = 10,000, r = 12.43, and = 4° 13' 14". We have the formula given in the preceding paragraph and r-sin2, and if we substitute this value, instead of dropping the tan' (p, we would get as a final result ^ = ^^^(^-27^x^^r^) which is equally easy for work, and is more in keeping with the form of the expression for determining the deflection due to wind given in equation (212). The difference in results is not material however. IGO EXTERIOR BALLISTICS Lateral 243. To determine the lateral deviation due to wind, let Tf ^ be the lateral wind aeviation due to wind, component in foot-seconds, positive when it blows from right to left across the line of fire, and negative when it blows in the reverse direction ; let V and <^ be the initial velocity and angle of departure relative to still air or to the ground, and Z be the corresponding range, that is, the range wheri there is no wind. Then if we draw the triangle of forces for this case, we may obtain the initial velocity and direction of flight relative to the moving air. Thus referring to Figure 21(a), which represents the case of a negative wind, the resultant of OA = V with OC= — Ws is OB=V', which is very slightly greater than V; the angle BOD = (j>', which V makes with the horizontal, is very slightly less than AOE = (f>; and V is inclined to the left so as to make with V the small angle BOA, z ^ yi-tan y (b) Figure 31. 244. Now since Y' and ' differ so little from Y and <^, and since the effect of the increase in Y is offset by that of the decrease in ^, we may take the range X' corresponding to Y' , . Therefore the only essential difference between the trajectory relative to the moving air and that relative to the ground or still air is that the plane of the former makes the angle DOE = y with the plane of the latter. Referring now to Figure 21(h), in which represents the gun and M^ the target at the range OMq = X, we see that tan y: ; and OM' is the horizontal trace of the trajectory relative Y cos to the moving air. But while the projectile moves through the air from to M', the air itself has moved WzT to the right, and so the projectile really strikes to the right of the target by the distance Mjr =W,T-X tan y= W, It- ^ — \ \ Y cos <^/ Thus the lateral deviation caused by the wind component VF« normal to the line of fire is given by the expression X Dw=W. T Y cos <^ (212) VARIATION OF THE TRAJECTORY FRO^^I A I'LANE CURVE 161 in which T, though really the time of flight corresponding to V, ', mav without appreciable error be taken as the time of flight for the actual firing data. 245. Now let us return to our standard problem and find the data for Column 14 in the range table; deviation for lateral wind component of 13 knots; for our 12" gun at 10,000 yards. We have the above formula, and also W^.^= ^'^^^^ "^l vards per second. GO X 60x3 X = 30000 log 4.47712 7 = 2900 log 3.4G240 colog 6.53760-10 <^ = 4° 13' 14" sec 0.00118 10.37 log 1.01590 T= 12.43 2.06 log 0.31387 12 log 1.07918 6080 log 3.78390 60x60x3 = 10800 log 4.03342 coW 5.96658-10 Djow = 13.9 yards log 1.14353 Section 2. — The Effect of Motion of tlie Gun Upon the Motion of the Projectile. 246. As in the case of the wind, we resolve the horizontal velocity of the gun due to the ship's motion into two components, Gx in the plane of fire, and Gg at right angles to that plane; and determine their effects separately, the first affecting the range only and the second the lateral deflection only. P---- Figure 22. 247. Let Gx be the resolved part of the speed in the line of fire in foot-seconds, positive when with and negative when contrary to the flight of the projectile. Then evidently the true initial velocity of the projectile is the resultant of Gx and V, and, as shown in Figure 22, 7 receives the increment i^V=Gx cos , while 4> is decreased by A<^ = ^^^ — ^. But by equations (208) and (206), these two changes in 7 and <^, respectively, will cause a change in range given by Change in range due to motion of gun, AXg_ 2n X 2n-l Gx cos 4> _ Crx sin <^ \ 7 7 tan 2cf>l AXo X 2n Gx cos ^ /g _ 2 tan <^ \ 2/i-l 7 \ tan 2c}> ) 11 162 EXTEEIOE BALLISTICS -KT J. o , 2 tan (b JNowas tan 26:= — ■ — ■ — x_ ^ l-tan-(^ the above expression becomes ^;« = -^ X ^-^^ (l + tan= 0) A 2 ?l — 1 V which, when (f> is small enough to make tan- <^ negligible in comparison to unity, reduces to AX a — n Gv cos <^ or X 3/1-1 T X cos 4> rs * AXo=^^X^^-^G.* (213) 248. As an illustration, let us return to our standard problem 12" gun, and compute for a range of 10,000 yards the data contained in Column 14 of the range table; change of range for motion of gun in plane of fire of 12 knots. We have the above formula and F-sin2 = 4:° 13' 14" cos 9.99882-10 F = 2900 log 3.46239 colog 6.53761-10 12 log 1.07918 6080 log 3.78390 60x60x3 = 10800 loi? 4.03342 colog 5.96658-10 A7?,,rv = 57 yards log 1.75770 Lateral 249. Now let Gz be the resolved part of the motion of the gun at right anodes to deviation . ^ '" . '" . due to mo- the line of fire in foot-seconds. Then, in addition to the initial velocity V in the line tion of gun. of the axis of the gun, the projectile on leaving the gun has a lateral velocity Gz, and so, as may be seen from Figure 21(&), the real plane of departure makes an angle with the vertical plane of the gun's axis given by tan y= . ^^ — , and the resultant deviation at range X is given by Z)G = Xtany or Dc= ^, ^ , G, (214) K cos * The above is the formula actually employed. There seems to be no good reason, however, for neglecting the tan^ tp, for tan- 4, + 1 = sec^ (p, and if we substitute this value, instead of dropping the tan^ 0, wa would get as the final result ■^o = 2n — 1 X Vcos^ ^ ^-^ which is equally easy for work, and more in keeping with the form of the expression for determining the deflection due to motion of the gun given in equation (214). The differ- ence is not material however. VARIATION OF THE TRAJECTORY FROM A PLANE CURVE 163 250. Taking our standard problem again, we have the above formula and ^ 13x6080 1 1 G~= „^ \^ — 7i yards per secoiul. ~ 60x60x3-^ ^ Z = 30000 log 4.47712 7 = 3900 log 3.46240 eolog 6.53760-10 (/> = 4° 13' 14" sec 0.00118 12 log 1.07918 6080 log 3.78390 60x60x3 = 10800 log 4.03342 colog 5.9665S-10 D,.o=^0.1 yards log 1.84556 Section 3. — The Effect of the Motion of the Target Upon the Motion of the Projectile Relative to the Target. 251. Motion of the target evidently has no effect upon the actual flight of the projectile, but it is equally clear that it will affect the relative positions of the target and of the point of fall of the projectile, as the target has been in motion during the time of flight of the projectile. 252. Evidently, if the target be moving in the line of fire with the velocity Tx, Effect of in order to hit it the sight must be set for a range greater or less than the true range target, at the instant of firing by the distance which the target will traverse in the time of flight, or T^T. So, also, if the speed of the target at right angles to the plane of fire be Tz, the shot will fall TzT to one side of the target unless that much deviation is allowed for in pointing. Once more we consider the motion as resolved into two components, one in and the other normal to the plane of fire, and consider the two as producing results entirely independent of each other. And it is readily seen that, for the effect of the motion of the target we must correct the range and deviation by the quantities given by the expressions AXt = TxT (215) Dt = T,T (216) 253. For our standard 12" gun, again, for 10,000 yards, to compute the data in Columns 15 and 18 of the range tables, for 12 knots speed of target, the work would be r= 12.43 log 1.09452 12 log 1.07918 6080 log 3.78390 60 x 60 X 3 = 10800 log 4.03342 colog 5.96658 - 10 R,._T = D^.T = 84: yards log 1.92418 Section 4- — The Effect Upon the Motion of the Projectile of All TJiree Motions Combined. 254. In the preceding sections we nave discussed the effects upon the motion of the projectile of the wind and of the motions of the firing and target ships. The resultant combined effect of all three of these causes of error would of course be obtained by computing them separately and then performing the necessary algebraic 164 EXTEEIOR BALLISTICS additions, first for all range effects to get the total effect upon the range, and then of all deflection effects to get the total effect in deflection. Note. — Professor Alger appends to this chapter the following foot-note: The method herein adopted for the treatment of the problem of wind effect was first set forth, so far as I am aware, in General Didion's Traite de Ballistique though it has been generally accepted since. It is mathematically correct only for spherical projectiles, to the motion of which the air offers a resistance which is independent of the direction of motion. With elongated projectiles it will be seen that the initial motion relative to the air is not exactly in the line of the projectile's axis, so that we have no right to assume, as we do, that the flight relative to the air is the same when the air is moving as when it is still. It has been supposed by some writers that the lateral wind component produces the same pressure on the side of the moving projectile as it would if the projectile were sta- tionary, and that the deviation can be computed upon that basis. If this were true, the deviation would be proportional to the square of the lateral wind component, whereas it is really much more nearly proportional to its first power. Actually the pressure is much greater when the projectile is moving at right angles to the wind current than when it is stationary, on account of the increased number of air particles which strike it. EXAMPLES. 1. Compute the errors in range and in deflection caused by the wind components as "iven below. DATA. ANSWERS. Projectile. Wind component in k per hour. nets V. f.s. Range. Yds. In line of fire. Perpen to line dicular In line of fire. Perpen to line iicular of fire. of fire. H d. In. 10. Lbs. c. o Knots. With or against. Knots. To the right or left Yards. Short or over. Yards. To the right or left. A.. 3 13 1.00 1150 3000 8 With 6 Right 16.5 Over 0.3 Kight B.. 3 13 1.00 2700 4500 10 Against 8 Left 34.5 Short 19.1 Left C. 4 33 0.67 2900 4000 11 With 9 Right 12.7 Over 5.7 Right D. 5 50 1. 00 3150 4500 13 Against 11 Left 26 . 8 Short 14.4 Left E.. 5 50 0.61 3150 4500 14 With 13 Right 17.1 Over S.S Right F.. 6 105 0.61 2600 13600 15 Against 14 Left 146.9 Short 88.8 Left G.. 6 105 1. 00 2800 4500 16 \\itli 15 Right 25 . 9 Over 14.0 Right H. 6 105 0.01 2800 4000 17 Against 16 Left 13.3 Short 6.4 Left I.. 7 165 1.00 2700 7500 18 With 17 Right 74.2 Over 44.3 Right J.. 7 165 O.OI 2700 7500 19 Against 18 Left 47.0 Short 24.8 Left K. 8 260!0.6I 2750 8500 20 With 10 Right 51.5 Over 26.9 Right L.. 10 510 1. 00^2700 10500 10 Against 20 Left 100.4 Short 64.7 Left M. 10 510 0.61 2700 11500 18 Witli 19 Right 69.0 Over 40.2 Right N. 12 870 0.61 2000 10500 17 With 18 Lelt 141.5 Over 87.0 Left 0.. 13 iisoli.oo 2000 10500 16 Against 17 Right 93.3 Short 54.8 Right P.. 13 ll;W,0.74 2000 11500 15 With 16 Left 79.6 Over 44.4 Left Q.. 14 1400 0.70 2000 14500 14 Against 15 Pvight 105.6 Short 58 . 1 Right R.. 14 14000.70 2600 14000 13 With 14 Left 03.7 Over 37.3 Left VAEIATION OF THE TEAJECTORY FEOM A PLANE CUEVE 161 2. Compute the errors in range and in deflection caused by the motion of the gun as given below. Conditions standard. DATA. ANSWERS. Pr ojeetile. Speed component in knots per hour. r. f.s. Range. Yds. In line of fire. Perpen to line dicular In line of fire. Perpen to line dicular of fire. of fire. s d. In. Lbs. c. 1 Knots. With or against. Knots. To the right or left. Yards. Short or over. Yards. To the right or left. A.. 3 13 1.00 1150 2000 6 Against 8 Left 14.7 Short 23.6 Left B.. 3 13 1.00,2700 3500 7 With 8 Right 10.2 Over 17.5 Right C. 4 33 0.67 21)00 3000 8 Against 9 Left 11.5 Short 15.7 Left D. 5 50 1. 0013150 3500 9 With 10 Right 12.4 Over 18.8 Right E.. 5 50 0.61j3150 3800 10 Against 11 Left 16.4 Short 22.4 Left F.. 6 105 0.6l'2600 12600 11 With 13 Left 58.3 Over 108.4 Left G.. 6 105 1 . 00 2800 4000 13 Against 14 Right 24.1 Short 33.8 Right H. G 105 O.Gl 2800 3000 14 With 15 Left 22.2 Over 27.1 Left I.. 7 165 1.00 2700 6500 15 Against 16 Right 43.1 Short 65.2 Right J.. 7 165 0.61 2700 6700 16 With 17 Left 52.8 Over 71.4 Left K. 8 260 0.61,2750 7500 17 With 18 Right 62.6 Over 83.1 Right L.. 10 510,1.00 2700 9500 18 Against 19 Left 76.3 Short 113.5 Left M. 10 510 0.612700 10500 19 With 20 Right 97.6 Over 132.0 Right N. 12 870 0.61 2000 23000 20 Against 19 Left 182.3 Short 262.0 Left 0.. 13 1130 1 . 00 2000 10000 19 With 18 Right 118.0 Over 154.7 Right P.. 13 1130 0. 7412000 11000 18 Against 17 Left 128.4 Short 160.8 Left Q.. 14 1400 0.7012000 14000 17 With 16 Right 149.5 Over 195.4 Right R.. 14 1400 0.70,2600 13700 16 Against 15 Left 109.4 Short 134.8 Left 3, Compute the errors in range and in deflection caused by the motion of the target as given below. Conditions standard. DATA. ANSWERS. Projectile. Speed component in k per hour. nots In line of fire. Perpendicular to line of fire. V. f.s. Range. Yds. In line of fire.- Perpen to line dicular of fire. B d. In. w. Lbs. c. 0) 3 o Knots. With or against. Knots. To the right or left. Yards. Short or over. Y'ards. To the right or left. A.. 3 13 1.00 1150 1800 7 With 8 Right 21.4 Short 24.4 Left B.. 3 13 1.00 2700 3300 8 Against 9 Left 26.2 Over 29.4 Right C. 4 33 0.67 2900 2800 9 With 10 Right 17.3 Short 19.2 Left D. 5 50 1.00 3150 3300 10 Against 11 Left 24.0 Over 26.4 Right E.. 5 50 0.613150 3400 11 With 13 Right 24.1 Short 28.5 Left b\. 6 105 0.612000 13800 13 With 14 Left 204.3 Short 220.1 Right a.. 6 105 1.00 2800 3800 14 Airainst 15 Right 41.1 Over 44.0 Left H. 6 105 0.612800 2800 15 With 16 Left 28.1 Short 30.0 Right 1.. 7 105 1.00 2700 6300 16 Against 17 Right 91.1 Over 96.9 Left J.. 7 165 0.612700 6700 17 Witli 18 Left 89.5 Short 94.7 Right K. 8 260 0.612750 7300 18 Against 19 Left 99.2 Over 104.6 Right L.. 10 510 1.00 2700 9300 19 With 20 Right 157.9 Short 166.2 Left M. 10 510 0.6l'2700 10300 20 Against 19 Left 162.7 Over 154.6 Right N . 12 870 0.61 12900 20200 19 With 20 Right 326.7 Short 343.9 Left 0.. 13 1130 1.00 2000 9300 18 Against 19 Left 188.6 Over 199.1 ! Right P.. 131130 0.74 2000 10300 17 With 18 Right 187.6 Short 198.6 1 Left Q.. 14 1400 0.70 2000 13300 16 Against 17 Left 236.8 Over 251.6 Right R.. 14 1400 0.70 2600 13700 15 With 16 Right 172.4 Short 184.5 Left CHAPTER 15. DETERMINATION OF JUMP. EXPERIMENTAL RANGING AND THE REDUCTION OF OBSERVED RANGES. Jump. 255. Primarily and in its narrowest sense, jump is the increase (algebraic, and generall}^ positive) in the angle of elevation resulting from the angular motion of the gun in the vertical plane caused by the shock of discharge, as a result of which the projectile strikes above (for positive jump; below for negative) the point at which it theoretically should for the given angle of elevation. A definition which thus con- fines jump to the result of such angular motion is a narrow and restricted one, how- ever, and other elements may enter to give similar results, all of which may be and are properly included in that resultant variation generally called jump. For instance, in the old gravity return mounts, the gun did not recoil directly in the line of its own axis, as it does in the most modern mounts, but rose up an inclined plane as it recoiled. As the projectile did not clear the inuzzle until the gun had recoiled an appreciable distance, this upward motion of the gun imparted a similar upward motion to the projectile, which resulted in making the projectile strike slightly higher than it would otherwise have done. This small discrepancy, unimportant at battle ranges, but necessarily considered in such work as firing test shots at armor plate at close range, was properly included in the jump. Also most modern guns of any considerable length have what is known as " droop," that is, the muzzle of the gun sags a little, due to the length and weight of the gun, the axis of the gun being no longer a theoretical straight line; and this causes the projectile to strike slightly lower than it otherwise would, and introduces another slight error which may properly be included in the jump. Also it is probable thai this droop causes the muzzle of the gun to move slightly in firing as the gun tends to straighten out under internal pressure, and perhaps this motion tends to produce another variation, " whip," in the motion of the projectile, which would modify the result of the droop. All these may therefore be properly included in the jump. 256. This matter has a direct bearing, under our present system of considering such matters, upon the factor of the ballistic coefficient which we have designated as the coefficient of form of the projectile, and which is supposed, under our previous definition, to be the ratio of the resistance the projectile meets in flight to the resist- ance that would be encountered in the same air, at the same velocity, by the standard projectile; that is, by a projectile about three calibers long and similar in all respects except in possessing a standard head, namely, one whose ogival has a two-caliber radius. Imagine that the gun jumps a little and increases the range in so doing. It gives the same range as a similar gun firing without jump a projectile exactly similar in all respects except in possessing a slightly lower coefficient of form. Sup- pose a gun droops and shoots lower. The coefficient of form calculated back from the range obtained by actual firing would work out a little large. And in practice we would proba])ly have both jump and droop afi'ecting the range, but by our method of determining the coefficient of form from actual firing, by comparing actual with computed ranges, all such effects are hidden in the found value of the coefficient of form. Broader defi- 257. As a matter of fact, as intimated above, the value of the coefficient of form coefficient is determined by firing ranging shots, and then computing its value from the results. of form. ^ VARIATION" OF THE TRAJECTORY FROM A PLANE CURVE 1G7 This coefficient of form therefore includes not only the results of variations in form of projectiles, etc., but also variations in range resulting from jump, droop, whip, etc., In fact, in the sense in which we now use the term, coefficient of form might better be defined as " that value which, if substituted for c in the usual formula?, will, for the given elevation, velocity, weight of projectile, etc., make the computed range come out in agreement with that actually attained in firing, after making all correc- tions to the firing results for atmospheric density, etc." 258. Thus a person looking over the range table computations for the first time would say off hand tliat jump, droop, etc., were neglected. Closer study, however, makes it evident that the adopted procedure amounts to taking jump, etc., into consideration as actually found to exist; not in assuming that it is zero, and, in fact, not greatly concerning ourselves as to just what its value really is (as we know it to be small), but still following a method that, for a given jump, etc., gives a correct com- puted result at a given range, and which checks well at all other ranges. If it be objected that different guns of the same type may jump, etc., differently, it may be answered that the coefficient of form used is an average of the values obtained in a great number of firings of different individual guns of the same type, and is really preferable to that obtaiiied by a complete ranging of a single gun. As a matter of fact, variations in the value of the coefficient of form obtained do not seem to go with certain individual guns more than with others, so the range tables are equally good for all guns of the type. In other words, this method of procedure produces results that are within the limits of accuracy obtainable, and any errors that follow its use must necessarily be included in those inherent errors of the gun which must always exist, and which, after all possible precautions have been tt:ken, will inevitably make it impossible to have all the shot from the same gun, when fired under exactly similar conditions, always strike in exactly the same spot. 259. Having explained how jump, etc., even if it exists, is looked out for by our methods, we may now go on and state that, as a matter of fact, it does not seem to exist to any extent appreciable in the service use of guns, and it may therefore be said that it is a matter which does not concern an officer afloat. He should, however, have a knowledge of the principles laid down in this chapter, in order that he may recog- nize unusual and abnormal conditions should they be found to exist under special conditions. 260. To determine the value of the coefficient of form for a projectile for our practical standard problem 12" gun, the gun was mounted at the Proving Ground, and laid at tion of value an angle of elevation of 8°, using a gunner's quadrant. Correction for height of of form. trunnions above the water level, for sphericity of the earth, etc., makes this angle the equivalent of an angle of elevation of 8° 04'. The gun, when fired at this elevation, at 2900 f. s. initial velocity, should range about 16,140 yards, from the range talkie, when the observed fall has been reduced to standard atmospheric conditions, but of course this is not perfectly obtained. Say the projectile falls 100 yards short of the computed range. It then remains to determine the value of the coefficient of form which produced this variation, and this may be done by calculating back by the methods that have already been explained in this book. In practice, at the Proving Ground, however, in order to avoid the constant repetition of tedious computations, the method actually employed is to work out a few ranges for different values of the coefficient of form, and to make a curve for the results. The curve is made for argu- ments " coefficient of form " and " error from corrected range table range," and one curve is needed for each caliber and service velocity. Having such curves and the results of ranging shots, it is a quick and easy matter to take from the proper curve the value of the coefficient of form for each projectile. These values are tabulated, 168 EXTEEIOR BALLISTICS Computation of range tables. Keduction of observed ranges. and a running record is kept, so that a great number of results will be available as a cumulative check on the range table. For a new caliber, a curve of "corrected range '' and " coefficient of form " is kept until enough data has been collected with which to start a range table. For rough work, the formula for change of range resulting from a variation in the value of the ballistic coefficient may be used in the absence of curves. 261. Prior to the appearance of the present range tables, guns were ranged by firing experimental shots at a number of different angles of elevation, and a curve of angles and ranges was plotted. From these faired curves the angles corresponding to all ranges were taken and a range table was made up from the results. As more confidence in the mathematical process was acquired, through the accumulation of considerable data, we began to get our range tables by computation, gradually abandoning the old system of ranging by experimental firing; and the use of the value of the coefficient of form as unity, with the projectiles then in use, was found to give range tables that agreed with the results of experimental ranging. When different lots of projectiles are presented for acceptance, a few have to be tested for flight from each lot; and these are ranged at the Proving Ground at 8° elevation in all cases, in order to make comparisons possible. At this angle there are no dangerous ricochets, and variations in the coefficient of form and differences between different projectiles will show up best at these long ranges. With a coefficient of form accurately de- termined by firing at the longest practicable ranges, we can compute an extremely accurate range table extending down through the medium and short ranges. The method of ranging only at a single elevation was therefore adopted, an occasional check by firing at short ranges being made. 2S2. The process of experimental ranging, as formerly carried out, was to fire a number of shots at different angles of elevation. The results for these shots were reduced to standard conditions, and the reduced observed ranges were plotted as abscissae, with the corresponding angles of elevation as ordinates. A fair curve was then drawn through these points, and from this curve the angle of elevation corre- sponding to any range could be obtained. 263. The process of reducing observed ranges to standard conditions was carried out in accordance with the principles and formulae already explained in this book, and this still has to be done for every ranging shot fired ; but as this process is one that pertains purely to Proving Ground work and has no bearing on the service use of the gun, it is not considered necessary to go into it at length here ; nor is it considered necessary to further discuss the details of the methods used for deter- mining the actual magnitude of the jump, etc. PAET IV. RAXGE TABLES; THEIR COMPUTATION AND USE. IXTRODUCTIOX TO PART IV. Having completed the study of all computations connected with the trajector}^ in air, both as a plane curve and allowing for existing variations from that plane, we are now in a position to make use of our knowledge in a practical way. The practical and useful expression of the knowledge thus acquired takes the form of: first, the preparation of the range tables; and after that, second, their use. Part IV will be devoted to a consideration of the range tables from these two points of view : first, as to their preparation; and, second, as to their actual practical use in service. Each column in the tables will be considered separately, the method and computations by which the data contained in it is obtained will be indicated, and then consideration will be given to the practical use of this data by officers aboard ship in service. CHAPTER 16. THE COMPUTATION OF THE DATA CONTAINED IN THE RANGE TABLES IN GENERAL; AND THE COMPUTATION OF THE DATA CONTAINED IN COLUMN 9 OF THE RANGE TABLES. New Symbols Introduced. E^. . . .Penetration, in inches, of Harvcyized armor by capped projectiles. E.,. . . . Penetration, in inches, of face-hardened armor by capped projectiles. K. . . .Constant factor for face-hardened armor. K' . . . .Constant factor for Harveyized armor. 264. With the single exception noted in the next paragraph, we have now con- sidered in detail the formulge and methods by which the data in each of the columns of the range table is computed. Summarized, this is as follows : ^ , Ti rr 1 1 Chapter in tliis ,■ Column m Range Table ^ , . ,• ■ XT T\ i. n t- ^ -i book in which rso. Data Contained , • , explained 1. . . .Range. This is the foundation column for which the data in the "I No explanation other columns is computed. There are therefore no j necessary. computations in regard to it. 2 . . . . Angle of departure 8 3 Angle of fall 8 4 Time of flight 8 5 . . . . Striking velocity ii 6 Drift 13 7 . . . . Danger space for a target 20 feet high 5 8 . . . . Maximum ordinate 8 and 9 9. . . .Penetration of armor This chapter 10. . . .Change of range for variation of ± 50 foot-seconds initial velocity 12 11. . . .Change of range for variation of ± dw pounds in weight of projectile 12 12. . . .Change in range for variation of ± 10 per cent in density of air 12 13. . . .Change in range for wind component in plane of fire of 12 knots 14 14. . . .Change in range for motion of gun in plane of fire of 12 knots 14 15. . . .Change in range for motion of target in plane of fire of 12 knots 14 16 ... . Deviation for lateral wind component of 12 knots 14 17. . . .Deviation for lateral motion of gun perpendicular to line of fire; speed 12 knots. 14 18. .. .Deviation for lateral motion of target perpendicular to line of fire; speed 12 knots 14 19. . . .Change of height of impact for variation of ± 100 yards in sight bar 12 265. The subject of penetration of armor is one which does not properly Ijelong to the subject of exterior ballistics, but this text book is compiled from the special point of view of the computation and use of the range tables, and as Column 9 of each of these tables gives the penetration, the subject is covered here in a brief way, in order that the full range table computations may be covered together. 266. In the earlier range tables, the penetration of armor was given for Harvey- penetration ized armor, and formulas devised by Commander Cleland Davis, U. S. jST., were *°*'™"^*' employed in the computation. For later armor, the range tables give the penetration of face-hardened armor by capped projectiles, this data being computed by the use of De Marre's formula. The heading of the column in each range table shows which type of armor is referred to in that particular table. Given the penetration in Harveyized armor, the penetration for face-hardened armor may be approximately obtained by multiplying the known figure for Harveyized armor by 0.8. Davis's formulre for Harveyized armor are : 17S EXTERIOR BALLISTICS in which For projectiles without caps. ^0.5^0.75 E = i]-\e penetration of Harvey ized armor in inches. or ^0.75 ■_ VW" (217) t' = the striking velocity in foot-seconds. U'=:the weight of the projectile in pounds. fZ=:the diameter of the projectile in inches. lo^ A' = 3.34512. For capped projectiles. ^= .,,0.5 -^^ VIV ,0.5 or ^o.s_jii^ (218) in which log Z'= 3.25312, and the other quantities are as before. De Marre's formula for face-hardened armor is ^0.75^0. K or E'>-' = VlV (219) in which log Z= 3.00945, £' = the penetration of oil tempered and annealed armor that has not been face hardened. For face-hardened armor (for the range tables accompanying this book and marked as C, F, H, K, M, N, P, Q and Pk,), the results obtained by the use of the above formula must be divided by a divisor known as De Marre's coefficient, which has been found to be 1.5 for such purpose. (For the other range tables accompanying this book, the value of this coefficient was taken as unity.) 267. As an example of the work under Davis's formula, let us compute the penetration by a capped projectile of Harveyized armor by the 5" gun; F = 3150 f. s., w = h() pounds; for a range of 4000 yards, first for a projectile for which c — l, and next for a projectile whose coefficient of form is 0.61. For these two projectiles, the range tables give the remaining velocities at the given range as v-l = 1510 f. s. for c=1.00 and 1^2 = 2098 f. s. for c = 0.61. w=50 log 1.69897 0.5 log 0.84948 K'= log 3.25312 colog 6.74688-10 d = 5 log 0.69897 0.5 log 0.34948 0.5 colog 9.65052-10 10 ,loff 7.246S8-10 log 7.24688- ^.^ = 1510 log 3.17898 t;. = 2098 log 3.32181 ,&o.8 0.8 log 0.42586 0.8 log 0.56869 J5^i = 3.4067" log 0.53233 ii;2 = 5.1380" log 0.71086 Comparison 268. As the Coefficient of form does not enter into the above equation, we see short pointed that the Only thing that gives a long pointed projectile a greater penetration than projec I es. ^ ]j|^jj-,^ pointed one at the same range is the fact that, at that range, the long pointed projectile will have a greater striking velocity than the blunt one. As a matter of fact, as far as their effect upon armor plate is concerned, the two projectiles are the same ; for the main body of the projectile is the same in each case, the only difference between them being in the shape of the wind shield. In other words, that part of the projectile which really acts to penetrate armor is the same for both the standard and tor the long pomted siiell, but one has no wind shield and the other a sharply pointed one, the actual points of the two shells being equally efficient in their effect upon the penetration. No difference in penetration could therefore be expected for equal striking velocities. EANGE TABLES; THEIR COMPUTATION AND USE 173 269. Now let us take our standard problem and determine, by the use of De IMarre's formula, the penetration of face-hardened armor at 10,000 yards, for the 12" gun of the problem. The range table gives t' = 2029 f. s. for that range. j^o--= 4Ci. ill which log /i =3.00945 r = 2029 log 3.3072S w = 8;0 log 2.93952 0.5 log 1.46976 K= log 3.00945 colog 6.99055-10 d = 12 loff 1.07918. . .0.75 W 0.80938 0.75 colog 9.19062-10 tables. E"-' 0.7 log 0.95821 ^2X1.5 log 1.36887 1.5 log 0.17609 colog 9.S2391-10 ^, = 15.59" log 1.19278 270. Having learned how to compute the penetration of armor by a given pro- practical jectile at a given range, we are now in a position to discuss the practical methods of range °° used in actually making the computations for a range table. The labor involved is of course very great, so much care has been taken to get up special forms, and these are printed and kept on hand for the work. These forms are given in the follow- ing pages, the figures given in them being for the problems that we have already worked out item by item, and shown here as they would appear in the work of pre- paring the 12" range table with the data for which, at a single range only, we have been working. These forms therefore show only the computations involved for 10,000 yards range. In computing the actual table, the work is done first for 1000 yards, then for 1500 yards, etc., for each 500-yard increment in range, the values between the computed values being obtained by interpolations, which interpolations are not difficult, as in most cases the second and third differences are negligible. As the range table of the particular gun in question runs up to 24,000 yards, and as computations must be made for every 500 yards, it will be seen that the work must be repeated for every 500 yards from 1000 yards up, which will involve 47 com- plete computations like the one shown in the following forms. This will give some idea of the magnitude of the work involved, and of the necessity for having special forms prepared, and of otherwise reducing the labor and increasing the accuracy as much as possible. Therefore, if much of this kind of work is to be done, forms similar to the ones shown should be prepared before commencing it (if a supply of the printed forms be not at hand) ; but if only a single problem is to be worked, as will generally be the case in the instruction of midshipmen, then the forms given in the previous chapters of this book should be used, as showing more clearly the nature of the problems involved and the methods of solving them. In solving problems under this chapter the forms given below must be used. 271. The form given for determining the angle of departure for a given range provides space for only three approximations; if more approximations are necessary to get the correct result, the form is simply extended. 272. It is to be noted that, in order to get smooth curves on the deflection drums of the sights, it is necessary in some cases to fair the computed drift curve, and this produces, in some places, a small difference between the computed drift and that shown in the range tables. Thus the computed drift for our standard problem was 26.6 yards, while in the range tables it is given as 26.8 yards. 273. The problem before us then is to compute, for a range of 10,000 yards, the data for the columns of the range table for the 12" gun for which 7 = 2900 foot- seconds, «' = 870 pounds, and c = 0.61. The forms and work follow. 174 - EXTERIOR BALLISTICS STANDARD FORMS FOR COMPUTATION OF RANGE TABLE DATA. Specific Problem. Compute the data for all columns of the range table for a range of 10,000 yards for a 12" gun for which V = 2900 f. s., w = 870 pounds, and c = 0.61. j Form No. 1. For Computation of Angle of Departure. Column 2, Uncorrected value of C Ci = ^„ = 9.9045 cd- log 10 . . colog c. colog d^ log C, 2 9 3 9 5 2 2 1 4 6 7 7 8 4 1 6 4 1 9 9 5 8 3 i Z, = -^= 3U29 log X. . . colog Ci log Z, . 4 4 7 7 1 2 9 4 1 7 3 4 8 1 2 9 — 10 Ai = .01470 + .00063 X 20 = .014S82 sin 2(pi = A^Ci log A, 8 1 7 2 6 — 10 log (7, 9 9 o 8 3 log sin 2(pi . . 9 1 6 8 4 9 — 10 = 4 8° 28' 34" 14' 17" first approximation 82 V 57 849+ -^ —8.52.9 A/' Zs =3 2983.8 A 2=r :i;>3c>.a „ = .01408 + .838 X .00002 = .014599 log Ai log C, log sin 202 8 1 6 4 3 2 1 2 3 5 9 1 6 6 6 7 — 10 — 10 8° 20' 20" 4° 13' 13" second approximation 8.99 X 56 11 '' 2984.1 .01408 + .00002 X .841 = .014001 ., 8 1 4 3 8 1 2 3 1 9 1 6 6 6 9 — 10 — 10 2(^3 = 8° 26' 28" (p3^4° 13' 14" third approximation R = 10,000 yards X — 30,000 feet, loglog / = log Y — 5.01765 Y = A"C tan<^ losr A" I loi o, log tan 4>i ■ • ■ log consta-nt. loglog fi log /i . log C, . log C,. log X .. log Z,, 5 6 I 9 1 7 j 6 9 I 5 2 3 7 I 7 I 1 7 ! 4 i 7 — 10 — 10 — 10 Z, = 2983.8 log A.," log Co log tan 0o . . . 2 1 8 9 8 2 6 3 2 8 6 3 1 5 — 10 log constant. 5 1 7 6 5 — 10 loglog /s 7 8 1 1 6 1 — 10 log/= log Ci 9 9 6 5 4 8 8 3 log C3 1 2 3 1 log A- 4 4 7 7 1 2 logZ3 3 4 7 4 8 1 Z3 2984.1 From tlie above work, we have for our final values:

IK =13.9 yards X Dw=W,,, "Fcos0 log X. . .. colog V . . log sec (p. log Y cos

r= 12.43 2.06 log IF,,,, log 2.0G . . log Dw — 10 8 2 9 6 1 7 3 1 3 8 7 1 1 4 3 5 4 17. Gun motion effect in deflection, Dg— 70.1 yards X Dg. Y cos (p f^iiz log G... log Y cos(f>" log Dg 1 8 2 9 6 7 1 1 5 9 1 8 4 5 5 7 19. Change in height of impact for varia- tion of ± 100 yards in sight bar, H = 28 feet H = AX tan w aX = 300 feet log AX log tan cj. . . Ion H, 2 4 7 1 2 8 9 7 1 7 4 1 4 4 8 8 6 — 10 12 178 EXTEEIOPt BALLISTICS EXAMPLES. 1. For examples in determining the angle of departure corresponding to any- given range, the data in the range tables may be used, computing for standard atmos- pheric condition, and proceeding to determine the true value of the ballistic coefficient by successive approximations. (See also Examples in Chapter 8.) 2. As the process of successive approximations is somewhat long for section room work, the following are given. Given the data contained in the following table, compute the corresponding values of cf>, w, T and fa, of the drift, of the maximum ordinate, and of the penetration of armor by capped projectiles (Harveyized armor, by Davis's formula for guns A, B, D, E, G, I, J, L and ; face-hardened armor, by De Marre's formula for guns C, F, H, K, M, N, P, Q and R. De Marre's coefficient = 1.5). Atmosphere standard. DATA. ANSWERS. Penetra- Projectile. Multi- Maxi- tion. s Value of/. Y. f.s. Range. Yds. plier for drift. 0. 03. T. Sees. f.s. Drift. Yds. mum ordi- nate. Ft. o u d. w. In. Lbs. 1 3' 13 c. Harv. In. F.H. In. A... 1.00 1.004411.50 2500 1.0 6° 53.1' 8° 26' 7.89 837 7.5 253 0.96 B... 3 13 1.00 1.0034 2700 3600 1.0 2 48.9 5 10 6.61 1094 5.9 181 1.30 C... 4 33 0.67 1. 0011:2900 3000 1.0 1 15.3 1 35 3.71 2043 1.6 55 "iig D... 5 oO 1.00 1.0025 3150 4000 1.5 1 53.4 3 05 5.57 1511 5.8 126 3.40 .... E... 5 50 0.61 1.0024 3150 4500 1.5 1 45.7 2 26 5.49 1932 5.4 122 4.6 F... 6 105 0.61 1.0605 2600 14,000 1.5 13 27.3 24 17 28.88 1070 183.0 3513 .... '2.'9 G... 6 105 1.00 1.0022 2800 3800 1.0 1 52.6 2 35 5.21 1729 3.2 109 5.7 .... H.. 6 105 0.61 1.0015 2800 3500 1.5 1 28.6 1 46 4.29 2153 3.0 74 .... 1 7.9 I... 71 165 1.00 1.0095;2700 7000 1.0 4 45.6 7 59 11.78 1243 17.6 566 4.6 .... J... 7' 165 0.61 1.0083 2700 7500 1.5 4 04.1 5 42 10.821631 20.9 473 6.4 K.. 8 260 0.61 1.00S5 2750 8000 1.5 3 59.8 5 21 10.961771 21.5 484 "s.'i L... 10 510 1.00 1.014112700 9000 1.0 5 31.6 8 33 14.14 14C1 24.6 811 8.6 .... M.. 10 510 0.61 1.0137j2700 10000 1.5 5 10.5 6 55 13.95[1744 34.5 785 ' .... 10.4 N.. 12 870 0.61 1.11302900 24000 1.5 15 07.7 25 01 39.511359 309.9 6358 .... 8.8 0... 131130 1.00 1.0337 2000 10500 1.0 11 32.4 16 40 21.901157 59.9 1955 9.4 P... 131130 0.74 1.0316'2000 11000 1.5 10 52.1 14 37 21. 3611279 82.9 1845 .... "s.'9 Q... 141400 0.70i 1.057 12000 14000 1.5 14 37.1 20 02 28.281246 148.4 3246 9.3 R... 14 1400 0.70 1. 0342^2600 14500 1.5 8 41.7 12 13 22.10J1560 89.3 1975 1 1 1 12.8 3. Given the data and results contained in Example 2, compute the correspond- ing values of : 1. Danger space for a target 20 feet high. 2. Change in range resulting from a variation from standard of ±50 f. s. in the initial velocity. 3. Change in range resulting from a variation from standard of ±10 per cent in the density of the atmosphere. 4. Change in range resulting from a variation from standard as given below in the weight of the projectile : Gun C ± 1 pound. Guns F and // ± 3 pounds. Gun J ± 4 pounds. Gun K ± 5 pounds. Guns M,N,P,Q i^ndiR ±10 pounds. EAXGE TABLES; THEIE COMPUTATION AND USE ANSWERS. 179 Change in range Change in range Change in range Problem. Danger space. Yds. for variation in V. for varia- tion in density. for variation in w. Yds. Yds. Yds. A 45.8 75.2 200.6 ± 114.5 ± 74.4 -+- 83 . 2 ^ 46.3 T 163.9 T 63.0 B C Hi 33.6 D 127.3 ± 82.3 T 154.7 E 162.4 ■+- 104.4 T 125.6 F 14.8 ±279.8 H= 656.4 -H37.9 O 153.5 -4- 99 . =i= 104.7 H 229.5 =h 103.2 T 56.8 q=43.2 I 47.9 ■+- 159.1 T 284.4 J 67.4 -+- 199.9 T 214.7 T42.2 K 71.8 ±211.3 T 202.1 ^41.6 L 44.6 -+- 220.4 IP 319.9 ^I 55.3 ±275.9 :p 253.9 ii=55.4 N 14.3 ± 522 . =F 1010.3 T 9.2 22.3 -+- 339 . 9 :;: 333.7 P 25.6 ±389.5 =? 290.0 T 24 . Q 18.3 -H 479.8 h: 399.0 h:20.9 R 30.9 -+- 395.4 qi 425.2 H=22.5 •i. Given the data and results contained in Example 2, compute the correspond- ing values of (atmospheric conditions being standard) : Effect in range and deflection of wind components of 12 knots. Effect in range and deflection of a speed of gun of 12 knots. Effect in range and deflection of a speed of target of 12 knots. Change in position of point of impact in the vertical plane through the target for a variation of ±100 yards in the setting of the sight in range. ANSWERS. Value of n. Wind. Speed of gun. Speed of target. Change of point of im- Problem. Range. Yds. Deflection. Yds. Range. Yds. Deflection. Yds. Range and deflection. Yds. pact in ver- tical plane. Ft. A 1 . 3032 2.0577 1.2709 1.G929 1 . 4028 2.2622 1..39S1 1.1947 1.7830 1.4241 1.3609 1.6075 1 . 3558 1.8278 1 . 5459 1.3941 1 . 4446 1.4424 17.8 26.8 7.8 19.4 14.6 126.9 13.8 7.2 43.2 29 . 8 27.6 46.7 35.0 155.4 70.9 .59.0 86.1 63.8 8.9 17.6 4.1 11.9 8.1 82.9 7.7 3.6 26.9 16.7 14.0 27.7 18.9 93.2 .3!) . 4 30 . 8 44.4 35.0 35.5 17.8 17.3 18.3 22.5 68.1 21.4 21.8 36.4 43.3 46 . 5 4S.8 59.2 111.5 77.0 85.3 105.0 85.5 44.4 27.1 21.0 25.7 29.0 112.1 27.5 25.3 52.7 56 . 4 59.1 67.9 7."5.4 173.7 10S.6 113.5 146 . 6 114.3 53.3 44.7 25 . 1 37.6 37.1 195.1 35.2 29.0 79.6 73.1 74.0 95 . 5 94.2 266.9 147.9 144.0 191.1 149.3 -1- 44 5 R -t- -'7 1 C D E -+- 8.3 ± 16.2 ■+■ !•> 7 F G ± 135.4 ■+■ 13 5 H ■+■ 9 3 I. ... -+- 4? 1 J -t- ''9 9 K L ± 28.1 ■+- 45 1 M.. -J- 3(] 4 N -+- 140 -H 8!) 8 P ■+- 78 2 Q -+- 109 4 R -^ (35 CHAPTER 17. THE PRACTICAL USE OF THE RANGE TABLES, Range tables. 274. A range table should be so constructed as to supply all the data necessary to enable the gun for which it is computed to be properly and promptly laid in such a manner that its projectiles may hit a target whose distance from the gun is known. This condition is fulfilled by the official range table computed and issued by the Bureau of Ordnance for each of our naval guns. In its simplest form, such a range table consists of a tabular statement of the values of the elements of a series of com- puted trajectories pertaining to successive horizontal ranges, within the possible limits of elevation of the gun as mounted, which is generally about 15° for naval guns, with such ranges taken as arguments and with the ranges and corresponding data disposed in regular order for ready reference, so that any desired range may be quickly found in the table, and from it all the corresponding elements of the required trajectory. In other words, complete and accurate knowledge of all the elements of the trajectory for each range is essential to the efficient use of the gun, and to this must be added complete data as to the effect upon the range of any reasonable varia- tions in such of the ballistic elements as are liable to differ in service from those standard values for which the table is computed. There must also be added the neces- sary data to show the variations in range and deflection resulting from the velocity of the wind and from motion of the gun and target. We have seen in the preceding chapters how to compute all this data. Constants 275. The constants upon which a range table is based we have seen to be the variations. Caliber, weight and coefficient of form of the projectile, that is, the factors from which the value of the ballistic coefficient is computed ; the initial velocity ; and the jump, this last being habitually considered as practically non-existent in service unless there is reason to believe to the contrary in special cases. The initial velocity, as well as the characteristics of the projectile, constitute features of the original design of each particular type of gun ; and, although the values of some of them may -be somewhat modified as the result of preliminary experimental firings, they are fixed quantities when the question of sight graduations and of range table data is under consideration. Of course the initial velocity and weight of projectile may vary some- what from their assigned standard values, the amount of variation from round to round depending upon the regularity of the powder, the care taken in the manu- facture and inspection of the ammunition and in putting up the charges, etc. Two very possible and important causes of variation in the initial velocity are variations from standard in the temperature of the powder, and drying out of the volatiles from the powder. Both of these causes have a very marked effect upon the initial velocity, and to overcome them efforts are made to keep the magazines at a steady temperature and all at the same temperature; while each charge is kept in an air-tight case in order to prevent evaporation of the solvent remaining in the powder when it is issued to service. It may be pointed out that it is more important that all magazine temperatures shall be kept the same throughout the ship than that they should be kept at the standard temperature. If the charges for all guns are at the same temperature, then, so far as that point is concerned, the guns will all shoot alike if the battery has been properly calibrated; and the spotter can readily allow for variations from standard ; but if one magazine is at a high temperature and another at a low, then EAXGE TABLES; THEIR COMPUTATION^ A>^D USE 181 standard projectiles. the guns involved will have different errors resulting from this cause, and the spotter cannot hope to get the salvos bunched when the sights of the several guns are set for tlie same range. 276. The sights are marked and the range table computed for the mean initial ^^s^t ° . markings. velocity and the mean weight of projectile, and these are made identical with the fixed standard values as given in the range table. In preparing projectiles for issue to the service great care is exercised to bring the weight of each one to standard so that this cause of variation in range may not appear. 277. Up to within the past few years (that is, up to the adoption of the long pointed projectile) the value of the coefficient of form was taken as unity. This was its value, for which the ballistic tables were computed, for the type of projectile then standard in service, as described in the preceding chapters of this book. With the adoption of the long pointed projectile, however, the value of the coefficient of form has dropped below unity, and for the several guns and projectiles covered by the Eange and Ballistic Tables published for use with this text book, its value ranges from 1.00 for blunt pointed projectiles (radius of ogival of 2.5 calibers) to from 0.74 to 0.61 for long pointed projectiles (radius of ogival of 7 calibers). Its value, what- ever it is, must be used in computing the value of the ballistic coefficient^, so long as the present ballistic tables are used. Perhaps it may be advisable some day to recom- pute the ballistic tables with the long pointed projectile as the standard projectile of the tables, in which case the coefficient of form of such a projectile would then become unity for computations with the new tables ; and a coefficient of form whose value is greater than unity would have to be used for computations involving the blunt-nosed projectiles. Such recomputation of the tables has not yet been made, however, and it is unlikely that it will be done unless progress in the development of ordnance makes recomputation necessary by raising service initial velocities above the present upper limit of the ballistic tables. 278. To show the results obtained by the adoption of the long pointed projectile, let us compare the range tables for the 7" gun of 2700 f . s. initial velocity, weight of projectile 165 pounds, for a range of 7000 yards, for each of the two values of the coefficient of form. The two range tables give : Value of c. 1.00 0.61 Rangp for an angle of ele- vation of about 15°. Yards. 13100 16900 Time of flight. Sees. 11.76 9.89 Striking velocity. f. s. ■ 1247 1690 Danger space for target 20' high. Yards. ' 48 76 Maximum ordinate. Feet. 563 39.5 Penetration in Harve\'ized armor. Incnes. 4.6 6.7 From what has been studied in the preceding chapters, a glance at the above figures will show at once how vastly improved the performance of the gun has been in every • particular by the introduction of the long pointed projectile. 279. After the preceding preliminary remarks it is possible to proceed to a care- ful consideration of the uses to which the range tables may be put in service, and this will now be done, column by column. 280. Explanatory Notes. — The explanatory notes at the beginning of each range Explanatory table are in general a statement of the standard conditions for which the data in the "angl tlbies. columns is computed, and of the methods by which it is computed. There is one item given therein which is used in practical computations aboard ship, however, and that is the information in regard to the effect upon the initial velocity of variations in the temperature of the powder. The note in every case gives the standard temperature of the powder, which is generally taken as 90° F.; and then states that a variation from this standard temperature of ±10° causes a variation in initial velocity of about ±35 foot-seconds in the initial velocity in most cases, although in some cases the variation in initial velocity for that amount of variation in temperature is ± 20 foot- 183 EXTEEIOR BALLISTICS Col. 1, range. Col. 2, *. To lay gun at given angle of elevation. seconds instead of ±35 foot-seconds. For instance, with our standard problem 12" gun, we see that the variation in initial velocity for a variation of ±10° from standard is ±35 foot-seconds. Therefore, if the temperature of the charge were 80° F., our initial velocity would be 2865 foot-seconds and not 2900 foot-seconds. If the tempera- ture of the charge were 100° F., the initial velocity would be 2935 foot-seconds. A variation of ±5° in the temperature of the charge gives a proportionate change in 35 the initial velocity, that is, ±-j-rX5=: ±17.5 foot-seconds and if the temperature of the charge were 77° F., we would have a resultant initial velocity of 35 X 13 2900- 10 = 2854.5 foot-seconds and similarly for other variations. 281. Column 1. Range. — As already explained, this is the argument column of the table. The data in the other columns is obtained by computation for ranges beginning at one thousand yards and increasing by five-hundred yard increments, and that for intermediate ranges by interpolation from the computed results (using second or higher differences where such use would affect the results). Therefore, to obtain the value of any element corresponding to a range lying between the tabulated ranges as given in Column 1, proceed by interpolation by the ordinary rules of proportion. 282. Column 2. Angle of Departure = Angle of Elevation + Jump. — As has been said, although jump must be watched for and considered in any special case where it may be suspected or found to exist, still it is normally practically nonexistent in service, and the angle of departure and angle of elevation coincide for horizontal trajectories. C 283. To lay the gun at any desired angle of elevation, the sights being marked in yards and not in degrees; find the angle of elevation in Column 2, and the corre- sponding range from Column 1. Set the sight at this range, point at the target, and the gun will then be elevated at the desired angle. An example of this kind is given in paragraphs 188, 189, 190 and 191 of Chapter 11. As there seen, this process is necessary when shooting at an object that is materially elevated or depressed relative to the horizontal plane through the gun. Let us now see how correctly the range tables may be used to determine the proper angle of elevation to be used to hit an elevated target; assuming the theory of the rigidity of the trajectory as true within the angular limits probable with naval gun mounts. For this purpose we will consider the problem solved in paragraph 188 of Chapter 11, Avhich was for the 12" gun; 7 = 2900 f. s.; w = 870 pounds; c = 0.61; horizontal distance = 10,000 yards; elevation of the target=1500 feet above the gun ; barometer = 29.00"; thermometer = 90° F. In paragraph 188 we computed that the correct angle of elevation for this case is 4° 08.1'. Now let us use Column 12 and correct for atmospheric conditions, for which, from Table IV, the multiplier is -1-0.79 ; from which we have that for a sight setting of 10,000 yards the shell would range 10000+ (215 x .79) =10169.85 yards. There- fore in order to make the shell travel 10,000 yards we must set the sight in range for 9830.15 yards. From the range table, by interpolation, the proper angle of departure for this range is 4° 07.9'. If we use this for the angle of elevation desired (instead of the computed value of 4° 08.1') we will have an error of 2', that is, of about 7 yards short. Xow if we solve the triangle to determine the actual distance from the gun to the target in a straight line, we will find it to be 9857.8 yards (by use of traverse tables), using the horizontal distance corrected for atmospheric conditions as the base. From EANGE TABLES; THEIE CGMPUTATIOX AND USE 183 the rano:e table, the angle of departure for this range is 4° 08.6'. If we use this as the desired angle of elevation we will have an error of 5' in elevation, or of about 17 yards over. The above processes show, for this individual case, the degree of inaccuracy that would enter from the use of the range table for this pur])ose : and the errors introduced above are not great enough to prevent the first shot from falling within reasonable spotting distance. That this will be the case under all circumstances cannot be pre- dicted, and each individual case must be considered on its own merits. For instance, it is evident that the horizontal distance could not be used in attacking an aeroplane at a high angle with a gun so mounted as to enable such angles to be used. It is to be noted that, when we use the range table as described above, assuming that the trajectory be rigid, we get the same results whether the target be elevated above or depressed below the horizontal plane of the gun. This shows at once that the method is not theoretically perfect. 284. For subcaliber practice a one-pounder gun is mounted in oi' on a turret gun, subcaiiber sis'litin&r* the axes of the two guns being parallel; and it is desired to point the pair, using the sights of the turret gun, so that the one-pounder projectile will hit a target at a known range. A combination range table for this purpose is given on page 34 of the Gunnery Instructions, 1913, but if this be not available we may proceed as follows (and this is the method by which the table referred to was prepared) for our standard problem 12" gun : Suppose that our subcaliber target is 1750 yards away. Look in the range table for the one-pounder gun that is to be used, and it will be found that the angle of departure for that gun for 1750 yards is 4° 13'. The problem then becomes similar to that stated in the preceding paragraph. Look in the 12" range tables and it will be seen that for that gun the angle of departure of 4° 13' corre- sponds to a range of 10,000 yards. Set the 12" sights at 10,000 yards, point at the target with those sights, and the guns will then be so elevated that the one-pounder shell should hit at 1750 yards. 285. A problem that has come up a number of times in our service has been to Use of same sislits for use the si2:hts marked for the initial velocitv due to a full charge when firing with varying ' . . . , r . initial reduced charges and correspondingly reduced initial velocities. Suppose, for in- velocities, stance, we have our standard problem 12" gun and are to fire it with the reduced charge which gives 2100 f. s. initial velocity instead of the regular 2900 f. s. given by the full charge. Proceeding by the methods already explained, we would compute the angles of departure necessary to give the desired ranges with the reduced velocity. Having tabulated the results, suppose we find that, for a given range A, the proper angle of departure is 6° 08' 42". Looking in the range tallies for 2900 f. s., we find that the proper range at which to set the sights in order to get this angle of departure is 13,300 yards, and if we set the full charge sights for that range we should hit at A yards with the reduced charge. The results should be tabulated for all probable ranges, and the resulting table used by the spotter ; or else paper sight scales may be prepared for the reduced velocity and pasted on the sights over the old scales. 286. Knowing the possible amAe of elevation resulting from the mechanical Roii as umit- •- >- '^ '^ ing iire. features of the mount, Column 2 will also show what degree of roll will necessarily throw the line of sight off the target at, say, the bottom of the roll. We have seen that, with our standard problem 13" gun at 10,000 yards, we must have an angle of departure of 4° 13'. Most turret guns cannot be elevated more than 15°, therefore when a ship has rolled so as to depress her guns 10° the total angle of elevation required of the mount a:t the bottom of the roll would be 10° -f 4° 13' = 14° 13', which is so near the limit of elevation of 15° as to show that it wouLd probably be impossible to fire on the bottom of the roll under these conditions. 184 EXTERIOR BALLISTICS Col. 3, « 287. Column 3. Angle of Fall. — This column will give information as to the angle of inclination of the axis of the projectile to the face of the target at any given range which is known as the " angle of impact " ; and, also, its inclination to the sur- face of the water at the point of fall, and hence of the probability of a ricochet. 288. It is also used for computing the value of the danger space by the use of the formulge given in Chapter 5. Col. 4,r 289. Column 4. Time of Flight. — This column gives one of the elements that enter into the time interval necessary between successive shots from the same gun, that is, between salvos. For instance, with our standard problem 12" gun at 10,000 't u ^ yards, we see that the time of flight is 12.43 seconds. The time that must elapse ( , between salvos is therefore 13 seconds plus the additional time it takes to spot the shot ' and get ready for the next one. It will be seen from this that the longer the range the longer must be the interval between salvos, provided they be properly spotted. At the longer ranges ordinarily used the time between two successive salvos as deter- mined above is ample for loading, so that the loading interval does not enter under those conditions in determining the time between salvos. At short ranges, however, the loading interval might be greater than the time as determined above, and in such cases the loading interval would necessarily determine the salvo interval. 290. The information contained in this column is also necessary to determine the setting for time fuses Avhen using shrapnel. In order to have this information readily available at the gun, the range scales of the sights generally bear correspond- ing time scales showing the time of flight in seconds corresponding to any given range. 291. When guns of different calibers are being fired together at the same target^ or when different ships are firing together at the same target, the information con- tained in this column aids the spotter to identify the splashes of the shot which he is spotting. An assistant should start a stop watch at the instant the guns in question are fired, and then the shot that splash at the end of the tabulated time of flight are in all probability the ones in which the spotter is interesed. 292. Also, suppose a ranging shot goes well over, and its time of flight is measured. Its excess over the time of flight for the given range is a check on the amount of spot necessary to bring the next shot down to the target, and a check like this does much to increase the confidence of the spotter in ordering a large change in sight setting under such conditions. Col. 5, vu ^^^- Column 5. Striking Velocity. — The only practical use of this column is to enable a judgment to be formed of the effect of the projectile at any given range, and hence as to the advisability of using a given gun for a given purpose of attack at a given range. Col. 6, drift. 294. Column 6. Drift. — The drift being accurately compensated in sight setting with all modern telescopic sights, at all ranges, the data in this column is of no special value with modern guns, but of general interest only. With the old bar sights, however, in which the drift compensation was made by inclining the sight bar at a permanent angle, the compensation for drift was correct at a single range only, and at all other ranges the uncompensated portion of the drift had to be allowed for by the use of the sliding leaf. In such a case, if the accurate compensation be at 7000 yards, we would have to know the amount of drift compensated by the perma- nent angle at any other range at which we wish to shoot, say 10,000 yards, and the difference between that and the tabulated value of the drift at 10,000 yards would have to be compensated by the use of the sliding leaf. Col. 7, danger 295. Column 7. Danger Space for a Target 20 Feet High. — This column is very frequently used. In the first place, it gives us a quick general idea of the probability of making a hit, for the greater the danger space the greater the chance of success. space. RANGE TABLES; THEIB COMPUTATION AND USE 185 It also shows us the " danger range " or " point blank range " ; that is, the maximum range for which the projectile never rises higher than the top of the target. (Com- pare with the same information as given in Column 8.) For our standard problem 12" gun we see that this is the case up to and including 2100 yards, at which range the maximum ordinate is 20 feet. For targets of other heights than 20 feet, the danger space may be determined within reasonable limits of height by a simple proportion from the data contained in this column. Thus, for our standard problem 12" gun, at 10,000 yards, the danger space for a target 30 feet high would be about 30 72 X -^77= 108 yards. If the height be so great that these results are not sufficiently accurate, then the new danger space must be computed by the formulfe given, the shorter one, S = h cot w, being ordinarily sufficiently 'accurate. A wrong conception of the danger space is often acquired, namely, that it may be defined as the distance from the target to the point of fall of a projectile that just touches the top. of the target. This conception may apply fairly well at long ranges, but a reference to Column 7 of the range table for short ranges will show that it does not fit the data given in that column. Considering Column 19 of the range table, which gives the change in the height of the point of impact in the vertical plane through the target resulting from a variation of 100 yards in the setting of the sight in range, for long ranges the height of the target divided by the figures given in Column 19 gives the same result as the danger space given in Column 7, but this is not the case for short ranges. Bearing in mind that the danger space is the distance that the target can be moved from the point of fall directly toward the gun at the given range, and still be hit, it will be seen that, as the range is reduced, as soon as the maximum ordinate becomes equal to the height of the target, then the target may be moved all the way to the gun and still have the trajectory pierce the screen, so that the danger space is then equal to the range, and at the point where this happens we have the danger range. As a matter of fact Column 19 should be used for all practical computations, but there should be no confusion of thought as to the relation existing between Columns 7 and 19. 296. At long ranges a knowledge of the danger space shows immediately how far beyond the target a shot will fall that just touches the top of the target screen. 297. Column 8. Maximum Ordinate. — This column is valuable for use in coi. 8, y determining the value of the altitude factor, /, in making ballistic computations to obtain approximate solutions. 298. Column 9. Penetration of Armor (Harveyized or Face Hardened) with coi. 9. Ei Capped Projectiles. — The data in this column is of value only in determining the probable efficiency of attack upon armor at different ranges. The figures given in the column are for normal impact, and the angle of fall (as given in Column 3) must be taken into account in considering this subject in order to determine the angle of im- pact, the " angle of impact " against any surface being that angle less than 90° between the axis of the projectile at the moment of impact and the surface in question. Thus, for a vertical armor plate at the same level as the gun, the angle of impact is the complement of the angle of fall. For elevated or depressed targets the angle of inclination at the striking point must be used instead of the angle of fall given in the table. Up to a certain critical angle we get penetration in the usual manner, although a pronounced increase in the angle of impact probably reduces the efficiency of pene- tration. The critical angle referred to is that at which the ix)int of the projectile ceases to bite, and we no longer have the penetrative effect due to the shape of the point but simply the smashing effect due to the momentum, which effect is of course 186 EXTEEIOR BALLISTICS small as compared to penetration proper. This subject is not particularly well under- stood up to the present time. 299. The tables which give the penetration in Harveyized armor were computed * before the present form of face-hardened armor came into general use. A rough approximation to the penetration of face-hardened armor may be obtained from those tables by multiplying the penetration in Harveyized armor by 0.8. 300. For our standard problem 12" gun, at 10,000 yards range, the angle of fall, as given by the range table, is 5° 21'. The angle of impact against a vertical side armor plate would therefore be 84° 39'. The angle of impact against a protective deck plate inclined to the horizontal at an angle of 15° would be 20° 21', which is probably very near or less than the biting angle, and little if any penetrative effect could be expected ; in other words, the protective deck would probably deflect the pro- jectile, thus fulfilling the purpose for which it was designed. If, however, the ships being on parallel courses, the target ship were rolled 10° towards the gun at the moment of impact, the angle of impact against the vertical side armor would l^e 74° 39'; and that against the protective deck plate would be 30° 21', 301. For an elevated target, a vertical armor plate, taking the problem given in paragraph 188 of Chapter 11, the angle of inclination to the horizontal at the target is ^=2° 20.1', and the angle of impact would therefore be 87° 39.9'. For the prob- lem given in paragraph 189 of Chapter 11, the angle of impact against the vertical plate would similarly be 82° 03', as we have ^= ( — ) 7° 57' in this case. Col. 10. 302. Colunm 10. Change of Range for a Variation of ± 50 Foot-Seconds '■^*" Initial Velocity. — A number of causes tend to produce variations in initial velocity, one being a variation in the temperature of the charge, which has already been dis- cussed. The volatiles in the powder may dry out, giving a resultant quicker burning powder, with an increase in both pressure and initial velocity. A damp powder will burn more slowly and give a reduced initial velocity. Slight deterioration in the powder not sufficient in amount or of a character to cause danger may reduce the initial velocity (any deterioration that causes an increase in the initial velocity will cause increased pressure, and should be looked upon as dangerous). There is no means of determining the amount of variation of initial velocity due to these last two causes except experimental firing. Firing at a given range under known conditions, with all known causes of variation eliminated, as will hereafter be explained in the discussion of calibration practice, and a comparison of the resulting actual range with that given in the range tables for the given angle of elevation, would give an approxi- mate idea of any such change in initial velocity, by working backwards in this column. 303. For our standard problem 12" gun, suppose we know that our charge was at a temperature of 100° F. ; that the solvent had dried out enough to cause an increase in initial velocity of 15 f. s., and that deterioration of the non-dangerous kind had reduced the initial velocity by 20 f . s. Our final initial velocity would then be : Standard initial velocity 2900 f. s. Variation due to temperature of charge. ... -f- 35 f . s. Variation due to drying out of volatiles. ... -|- 15 f . s. Variation due to deterioration of powder. . . — 20 f. s. Total variation 4-30 f. s.. . 30 f . s. Actual initial velocity 2930 f. s. And at 10,000 yards range, this variation in initial velocity would give us, from 277 Colunm 10, a resulting actual range of 10000 + —..-^ X 30 = 10166 yards, or the gun OO EANGE TABLES; THEIR COMPUTATION A:NrD USE 187 would shoot 166 yards over the target unless allowance was made for this variation by setting the sight at 9834 yards. 304. These figures show the importance of keeping all powder charges at a con- stant temperature, and all the charges in the ship, particularly for guns of the same caliber, at the same temperature ; and also for keeping the volatiles from drying out and for keeping the powder from becoming damp ; as well as for using care to prevent deterioration, even if not of the dangerous kind. They also show how seriously results may be affected by keeping a powder charge in a hot gun before being fired long enough to let the temperature of the gun materially raise that of the charge; it being particularly necessary to look out for this point when carrying on a calibra- tion practice. 305. Working back with the problem given above, suppose the gun had been fired at an angle of elevation of 4° 13' 14", that is, sighted for 10,000 yards, and that the point of fall had been accurately determined by triangulation ; and, that, after the observed results had been reduced to standard conditions, the data showed an actual range of 10,100 yards, or 100 yards in excess of normal. This would tend to show that a drying out of volatiles had taken place sufficient to give an increase in initial velocity of — ir;^^:— =18 f. s. If this work be deemed reliable, we could then figure 277 on an initial velocity of 2918 f. s. for further firing. . 306. Column 11. Change of Range for a Variation of ±Aw Pounds in Weight coi. ii, of Projectile. — All projectiles, before issue to service, should be brought to standard weight, and it will be found that this has usually been done, and that there is ordi- narily little use for the data contained in this column. However, if for calibration or for any other form of experimental firing, we find that the projectiles are not of standard weight, and that it is not practicable or convenient to make them so, we can reduce the results to standard by the use of this column. (The regular service pro- jectile for the 14" gun is subject to a tolerance of ±4 pounds in weight; that is, these projectiles may weigh anywhere from 1396 to 1404 pounds. For ordinary firing this small variation in weight is considered as immaterial.) 307. It is important to note that, where there is no sign prefixed to the entry in this column, an increase in the weight of the projectile causes a decrease in range and the reverse ; but if the entry in the column carries a negative sign (as is the case in some parts of the table for the 6" gun for which F = 2600, w = 105, and c = 0.61), then an increase in weight causes an increase in range at all ranges for which the negative sign appears in this column. 308. With our standard problem 12" gun at 10,000 yards, a projectile weighing 42 V 7 877 pounds would travel 10000- V: =9970 yards. 309. With the 6" gun, for which 7 = 2600 f. s., w = 105 jDounds, and c = 0.61, if the shell weighed 110 pounds, for a set range of 12,400 yards, the travel would actually be 12400 + ^4^^ =12430 yards. 310. The physical reason why, under some conditions, an increase in weight of projectile gives a decrease in range at one range and an increase at another may be readily understood if we remember that the effect upon the range of an increase in the weight of the projectile is the result of two entirely independent causes. The first acts entirely before the projectile leaves the gun, and an increase of weight thus acting always causes a decrease in the initial velocity, and hence, so far as this part of the effect alone is concerned, an increase in weight would always cause a decrease in range. The second part of the effect, however, which acts entirely outside the gun. 188 EXTEEIOE BALLISTICS is that due to the momentum stored in the moving projectile; that is, it depends upon the weight. As the weight is a factor in Mayevski's expression for retardation, we see that an increase in weight increases the power of the projectile to overcome the atmospheric resistance, and hence increases the range. Therefore of two similar pro- jectiles, differing somewhat in weight but leaving the gun with the same initial velocity, the heavier would travel the further; and, so far as this part of the effect alone is concerned, an increase in weight of the projectile would always give an increase in the range. We do not have equal initial velocities in the case under consideration, however, and the increase in the weight of the projectile acts first to decrease the initial velocity, and then, after leaving the gun, to make the projectile travel further than would one of standard weight if fired with the same reduced initial velocity. It can readily be conceived from this line of reasoning that, under some conditions, this second effect might more than balance that due to loss of initial velocity, and in all such cases an increase in the weight of the shell will increase the range. The proper sign for the data in Column 11 is determined by a careful consideration of the relative values of the two parts in the formula from which the data is derived. 311. An inspection of the range table for the 6" gun referred to in paragraph 309 above will show that for ranges from 1000 yards to 10,800 yards an increase in the weight of the projectile will cause a decrease in the range ; at 10,900 yards it will cause no change in the range; and from 11,000 yards up an increase in range will result. 312. For the standard problem 12" gun, through the entire table, from 1000 to 24,000 yards, it will be seen that increase in weight of projectile causes decrease in range. It is of interest to note, however, that from 1000 yards to about 12,000 yards this decrease in range increases with the range from a minimum of 8 yards at 1000 yards to a maximum of 42 yards at about 12,000 yards; and that from about 12,000 yards up the variation decreases until, at the highest point of the table, 24,000 yards, the decrease in range, due to an overweight of 10 pounds in the projectile, is only 12 yards. Apparently there is some theoretical point beyond the upper limit of the range table at which this quantity would change sign, and beyond which an in- crease of 10 pounds in the weight of the projectile would cause an increase in range, in a manner similar to that discussed above in regard to the 6" gun. This point is of course of no practical value in connection with the 12" gun, whereas it must neces- sarily be taken into account in dealing with the 6" gun. coi.^12, 313. Column 12. Change of Range for a Variation of Density of Air of ±10 Per Cent. — As has been stated, the range tables have been computed for a standard atmosphere of half -saturated air, for 59° F. (15° C.) and 29.53" (750 mm.) baro- metric height. Of course this exact atmospheric condition will rarely exist in actual firing, and the data in this column has been computed to enable allowance to be made for variations from standard density. It is of course easier for a projectile to travel through a less dense than through a more dense medium ; and if the air be below the standard density the range will therefore be greater than the standard range, etc. 314. For our standard problem 12" gun, range 10,000 yards, suppose the barom- eter stood at 31.00" and the thermometer at 50° F. From Table III of the Ballistic Tables, for those conditions, the value of 8 is 1.069, or the air is 6.9 per cent above 215 X 6 9 the standard density, and our actual range would be 10000— — — — ^ =9582 yards. If the barometer stood at 29.00" and the thermometer at 96° F., the value of 8 would be 0.910, that is, the air would be 9 per cent below standard density, and the actual range would be 10000+ ^1|^ =10194 yards. A^ioc EANGE TABLES; THEIK COMPUTATIOX AND USE 189 315. Or for the same problem, using Table IV of the Ballistic Tables, in the first case the actual range would be 10000 — 215x0.69 = 9852 yards, and in the second case it would be 10000 + 215x0.9 = 10194 yards; which process is shorter but cannot be understood unless the first and longer method has first been com- prehended. 316. This column is designated as referring to variations in the density of the air, this factor (8) being the one going to make up the value of the ballistic coefficient that is most apt to vary. If we remember that the formula for the value of the fw ballistic coefficient is C= J -j^ ? we can see that a variation of any given per cent in any one of the factors gives the same numerical percentage change in the value of C, remembering that an increase in / or w gives an increase in C, and an increase in 8, c or d^ gives a decrease in C. Changes in w cannot be handled in this way, owing to the resultant change in initial velocity already explained. Changes in / are not often known in such shape as to make it convenient to handle them by this method, that is, by the use of the data given in this column, although it could of course be done in this way were the percentage variation in the value of / known. Also c and d are ordi- narily constant, thus leaving 8 as the only one of the factors of the ballistic coefficient that would ordinarily be considered from this point of view. From the range table for this gun computed for c=1.00, for 3000 yards range, we have an entry of 84 yards in Column 12. If now the value of c becomes 0.95, we have a decrease of 5 per 84 X 5 cent in the value of c, and therefore our range would increase to 3000 H r-^r — =3042 yards. Xow, similarly, if we use the table for the same gun, but with c = O.Gl, we have in Column 12 of that table, for a range of 3000 yards, the data GO yards. Xow 5 if the value of c increases from 0.61 to 0.66, we have an increase of -j— =.082, or 8.2 per cent; and the decreased range resulting from this change would be 3000- ^i^A^ =2945.88. Theoretically we should be able, starting at a given range in the table for c=1.00, to reduce the range by the correction from Column 12 for a variation of 39 per cent, and thus get the range for a projectile for which c = 0.61 that would correspond to the range of 3000 yards for the projectile for which c = 1.00. Then starting with this new range in the table for which c=0.61, and applying the correction from 39 Column 12 for a variation of ^ =.64, or 64 per cent, we should get the original range from which we started as the corresponding range for the projectile for which c = 1.00. This will not work out very closely, however, because the percentage change in such a case is too large to be handled by the use of data such as that contained in Column 12, which is computed by the use of a formula based on differential incre- ments. 39 per cent and 64 per cent manifestly cannot be considered as such incre- ments. 190 EXTEEIOE BALLISTICS Col. 13, wind in range. 317. Column 13. Change of Range for Wind Component in Plane of Fire of 12 Knots. — This column is constantly used. For our standard proolem 12" gun, at 10,000 yards, a wind blowing directly from the target to the gun with a velocity of 12 knots would decrease the actual range 27 yards, and would increase it the same A^A//^ /27' ^i^ f /^'V l^/nd Figure 23. amount if blowing the other way. Suppose the line of fire were 37° true, and the wind were blowing from 210° true with a velocity of 25 knots. Then the wind com- ponent in the line of fire would be 25 cos 53°, or (by use of the traverse tables) 15 knots, and the range would be increased 27x15 12 = 3-1 yards by this component. EANGE TABLES; THEIR COMPUTATION AND USE 191 318 Column 14. Chansre of Range for Motion of Gun in Plane of Fire of 12 coi. i4, gun ■,■,-,,, motion in Knots. — Tliis column is also constantly used. For our standard problem 13 gun, at range. 10,000 yards, if the gun be moving at 12 knots directly towards the target, it will overshoot 57 yards unless the motion of the gun be allowed for in pointing; and if V < / ^ 45" > Figure 34. moving in tlie opposite direction it would undershoot by the same amount. If the line of fire be 3?° true, and the firing ship be steaming 315° true at 20 knots, the component speed in the line of fire would be 20 cos 82°, or (by the use of the traverse tables) 2.8 knots towards the target, and the gun would overshoot ■ — ^ ' ■ =13.3 yards. 192 EXTEEIOR BALLISTICS Col. 15, tar- get motion in range. 319. Column 15. Change of Range for Motion of Target in Plane of Fire of 12 Knots. — This column is also constantly used. For our standard problem 12" gun at 10,000 yards, if the target be steaming directly towards the gun at 12 knots, the gun would overshoot the mark 84 yards unless the motion were allowed for in point- ing; and if it were steaming at the same rate in the opposite direction it would under- MorfK Figure 25. shoot by the same amount. If the line of fire were 37° true, and the target were steaming 175° true at 23 knots, the component of motion in the line of fire would be 23 cos 42°, or (by the use of the traverse tables) 17.1 knots toward the gun, and the gun would overshoot 84x17.1 12 = 143.6 yards. RAXGE TABLES: THEIE COMPUTATION" AND USE 193 320. Column 16. Deviation for Lateral Wind Component of 12 Knots. — This column is also constantly used. For our standard problem 12" gun at 10,000 yards, if the wind were blowing perpendicular to the line of fire and across it from right to left, with a velocity of 12 knots, the shot would fall 14 yards to the left of the target Col. 16, wind in deflection. A&rfA. Figure 26. unless the effect of the wind were allowed for in pointing. If the line of fire be 37" true, and the wind be blowing from 310° true at 23 knots, the win^ component per- pendicular to the line of fire would be 23 sin 87°, or (by the use of the traverse tables) 23 knots, and the shot would fall — — — =27 yards to the right of the target. 13 194 EXTERIOE BALLISTICS Col. 17, gun motion in deflection. 321. Column 17. Deviation for Lateral Motion of Gun Perpendicular to Line of Fire, Speed 12 Knots. — This column is also constantly used. For our standard problem 12" gun, at 10,000 yards, if the gun be moving at 13 knots perpendicular to the line of fire, and from right to left, the shot would fall 70 yards to the left of the Figure 27. target unless the motion were allowed for in pointing. If the line of fire be 37° true, and the firing ship be steaming 100° true at 21 knots, the component of this motion perpendicular to the line of fire would be 21 sin 63°, or (by use of the traverse tables) 18.7x70 18.7 knots to the right, and the shot would fall 12 109 yards to the right. EANGE TABLES; THEIE COMPUTATION AND USE 195 322. Column 18. Deviation for Lateral Motion of Target Perpendicular to Line of Fire, Speed 12 Knots. — This eohimn is also constantly used. Note that the change of range in yards for the given speed when the target is moving in the line of fire is always the same numerically as the deviation in yards for the same speed M'hen the motion is perpendicular to the line of fire. This is manifestly correct, as the motion of the target, unlike any of the other motions considered, has no effect upon the actual motion of the projectile relative to the ground. This motion of the target simply removes the target from the point aimed at by an amount equal to the dis- /Y^r/K Col. 18, tar- get motion in deflection. Figure 28. tance traveled by it during the time of flight. For our standard problem 12" gun at 10,000 yards, if the target be moving at 12 knots perpendicular to the line of fire, from right to left, the shot would fall 84 yards astern of, that is, to the right of the target unless allowance were made for this motion in pointing. If the line of fire be 37° true, and the target be steaming 180° true at 20 knots, then the component of motion perpendicular to the line of fire would be 20 sin 37°, or (by the use of the traverse tables), 12 knots to the left, and the shot would fall the left. 84x12 12 = 84 yards to 196 EXTERIOR BALLISTICS Relation be- tween deflec- tion in yards and in knots. 323. There is another most important use to which the data contained in this column is constantly put, and that is the determination of the i^oint at which to set the deflection scale of the sight to compensate for any known deviation in yards. Deflection scales could just as properly he marked in any units, say parts of an inch motion of the sliding leaf either way from the central position, or simply in arhitrary divisions of convenient size ; and some such method was formerly employed before the present more scientific and accurate methods of pointing were introduced. What- ever the system of marking the deflection scales, the essential point is that there must be some simple and convenient means of determining quickly how many divisions change in the set of the deflection scale is necessary to correct a deviation of a known number of yards at any given range. It has therefore been found most convenient to mark the deflection scale in " knots," meaning " knots speed of target." and to make the size of the divisions such that setting the scale over by 12 knots, that is, by 12 of the divisions, will produce at any given range the number of yards deviation shown in Column 18 for that range. Our telescopic sights have their deflection scales marked in this way. To avoid the confusion that was found to arise from the necessity for using the words " right " and " left " in giving orders for sight setting, the mark of zero deflection for the sight is now commonly marked as " 50 knots," and to shift the point of fall of the shot to the left we lower the reading of the deflection scale (" left" and " lower" both begin with the letter /), and to shift the point of fall of the shot to the right we raise the reading of the deflection scale ("right" and " raise " both begin with the letter r) . For our standard problem 12" gun, at 10,000 yards, if we wish to correct a deflection of 84 yards left, we wish to shift the point of fall of the shot that distance to the right, and we accordingly set the deflection scale at 62 knots. To correct a deflection of 81 yards right, we would similarly set the scale at 38 knots. If the deflection had been 25 yards, we would have set the deflection 25x12 scale over 81 :3.6 knots, say 4 knots; and if we were correcting an error to the right (the original deflection setting having been 50), we would set the sight on 46 knots on the deflection scale; whereas had the original error been to the left the setting would be 54 knots.* 324. Column 19. Change in Height of Impact for Variation of ± 100 Yards "of pofnt'of in Sight Bar. — This column is also frequently used. For our standard problem 12" gun, at 10,000 yards, suppose the shot wer-e striking at an estimated distance of 50 feet above the target, and we want to know how much to change the setting of the Col. 19, ver- tical position Manifestly, we would lower the sight in range by 50x100 28 = 172 sight in range to hit. yards. The above is of value in shooting at objects on shore, where it is in some cases easier to estimate the vertical distance of the point of impact from the target than it is the error in range ; and also in direct flight spotting where the shot can be seen to pass over the screen. 325. We also see that, by the use of Column 19, a change, with our standard problem 12" gun of — ^^ — =72 yards in range will change the vertical position of ^8 ^ the point of impact 20 feet, at 10,000 yards range, that is, from the top to the bottom, or vice versa, of a target screen 20 feet high, which is the danger space at that range for such a target, and corresponds with the danger space as given in Column 7. * See Appendix C for a description of the arbitrary deflection scale for sights, which has recently been adopted for service use. EANGE TABLES; THEIR COMPUTATION AND USE 197 326. If we were shooting at 10,000 yards on the sight bar, with the same gun, and gave a spot of •" up 200," this would raise the position of the point of impact on 28 X 200 the target screen, or rather in the vertical plane through it, a distance of — — — =56 feet. 327. We are now in a position to proceed to the solution of some every-day Real wind practical problems by the use of the range tables, and for the first one we will take a p^obfim! ship steaming southwest at 18 knots, which wishes to fire a 12" gun (7 = 2900 f. s., w = S70 pounds, c = 0.61) at another ship that is 8000 vards distant and bears 30° off the port bow of the firing ship at the moment of firing. The target ship is steaming Figure 29, west at 22 knots, and the real wind is blowing from the south at 20 knots. The barometer is at 29.67" and the thermometer at 20° F. The temperature of the powder is 70° F. Drying out of volatiles has raised the initial velocity 25 f. s., and dampness of powder has reduced it 10 f. s. The shell weighs 875 pounds. How must the sights be set to hit? * For given atmospheric conditions 8 = 1.089, that is, the air is 8.9 per cent over standard density. Use traverse tables for all resolutions of speeds. 35 Powder is 20° below standard, reducing V by ^rr X 20 — 70 f . s. ' * - lU Drying out of volatiles increases U by + 25 f . s. Dampness of powder reduces U by — 10 f. s. Total variation of initial velocity from standard —55 f . s. * See Appendix B for a description of the Farnsworth Gun Error Computer, by the use of which these problems may be solved mechanically. 198 EXTEEIOR BALLISTICS Cause of variation Affects. Formulae. Range. Deflection. Speed of or varia- tion in — Short. Yds. Over. Yds. Right. Yds. Left. Yds. Gun - Target - Wind - Range Deflection.. Range Deflection.. Range Deflection.. Range Range Range 18 cos 30 X ^ = 15.6 X ^ 18 sin 30X^ = 9Xy| 22 cos 75 X ^ = 5.7 X -^ 22sin75x^ = 21.3X-§ 20 cos 15X^=10.3X -^ 20 sin 15 X 32" = 5.2 x ~ ^r. 229 39 S.9xf 30.9 27.3 251.9 19.5 121.0 61.1 42.0 3.5 115.4 Initial velocity . . . IV s 4.50.6 61.1 61.1 45.5 115.4 45.5 Point of fall of shot if uncorrec ted 389.5 yards short. 69.9 yards left. Heal wind and speed problem. To correct a deflection of 69.9 yards, set deflection scale to right — '--^ — =12.9 65 knots of scale. Therefore to point correctly, set sights at In range 8389.5 yards In deflection 62.9 knots or, to nearest graduations of sight scales, remembering to shoot short rather than over, In range 8350 yards In deflection 63 knots. 328. A 12" gun (7 = 2900 f. s., w = 870 pounds, c = 0.61) mounted on board a ship steaming 45° (magnetic) at 18 knots, is to be fired at a target ship on the star- board bow of the firing ship and steaming 315° (magnetic) at 14 knots, at the moment when the firing ship is 9530 yards from the point of intersection of the two courses, and the target ship is 5500 yards from the same point. The barometer is at 28.25" and the thermometer at 80° F. The temperature of the powder is 105° F. Dampness of the powder has reduced the initial velocity, at standard temperature, to 2875 f. s. The shell is 7.5 pounds over weight. There is a real wind blowing from 260° by compass (Dev. 10° E.) at 20 knots. How should the sights be set to hit? EANGE TABLES; THEIR COMPUTATION AND USE 199 By use of the traverse tables, at the moment of firing, the target will be 30° on the starboard bow of the firing ship, distant 11,000 yards. //arr^ vs^- ^rLA Figure 32. gun. In dealing with these quantities it is customary to consider deviations or errors in range separately from those in deflection ; so we would speak of the " mean deviation (or error) in range from the mean point of impact," and similarly for deflection. These quantities are also called the " mean errors " of the gun in range and in deflection. 348. The general plan of a calibration range is shown in Figure 32. A raft carrying a vertical target screen is moored in such a position that one or more observing stations, preferably two, may be established on shore, as shown at A and B. The ship is then moored at S, broadside to the target; and the screen of the latter should be as nearly as possible parallel to the keel of the ship. The angle STA should be as nearly a right angle as possible. 349. The base line AB must then be measured or determined by surveying methods; and then the positions of the ship, target, etc., must be accurately plotted, and their distance apart accurately determined. CALIBRATIOX OF SIXGLE GUXS AXD A SHIP'S BATTEEY 217 350. Having found the distance, »ST, from the gun to the target, the sights of the gun to be calibrated are set to that range, and a string (usually of four) care- fully aimed shots is fired at the target. (It is usual to set the sights a small known distance off in deflection, to prevent damage to the target and consequent frequent delays in completing practice.) Let us suppose that the first shot fell at F. By the use of sextants, plane tables or their equivalents (preferably plane tables), the angles a, /? and y should be observed, and the point of fall should be plotted on the drawing board. This process should be repeated for each shot, and the results tabulated for the gun, the errors in range and in deflection being measured from the drawing board for each shot. This process is repeated for each gun of the battery, and in doing this it is well to fire one shot from each gun in turn instead of having one gun fire its whole allowance at once, as more uniform conditions for firing the battery as a whole are obtained in this way, especially in regard to the temperature of the guns. A gun is not loaded until immediately before it is fired, for a number of reasons, among which is the fact that otherwise the temperature of the charge would be changed by contact with the heated walls of the powder chamber. 351. It is most desirable in calibration practice that the conditions of weather weather should be good, and should be uniform for the firing of all guns of the same caliber. The weather should be uniform for the whole firing, if practicable. If the weather be not uniform throughout the firing for one caliber, then it is necessary that the data for each shot be reduced to standard conditions individually before any com- bination of the results of different shots is made. The complete practice should be finished in one day if possible ; as it is bad practice to have part of the firing on the afternoon of one day and the remainder on the forenoon of the next, for instance, as the weather conditions may be entirely different on the two days, and misleading results may follow such a course. 352. The greater the number of shots fired the more reliable are the results. Number Four shots are usually fired from each gun, which is a small number ; but the cost of the ammunition so expended is not small and limits the practice to that allowed by a reasonable economy, to say nothing of the wear on the guns, especially on those of large caliber. 353. During the practice, for each shot, the observers at each shore station observations, should record: (a) The time of flash. (b) The consecutive number of the shot. (c) The angle between the point of fall and the center of the target. (d) The force and direction of the wind. 354. The observers on board ship^ in addition to the above, should record the following for each shot : (e) Time of shot (in place of time of flash). (f) Consecutive number of shot (should be same as (b)). (g) Xumber of gun from which fired. (h) Whether or not the cross wires of the telescope Mere on the center of the bull's eye at the instant the gun was fired ; and, if not, how much they were off in each direction (estimated in feet on the target screen ; lines painted on the screen should assist in making this estimate). (i) Direction and force of the wind in knots per hour. (j) Barometer. (k) Temperature of the air. (1) Temperature of the charge (assumed as the same as the temperature of the magazine, from which the charge should not be removed until it is actually needed for the firing). 218 EXTERIOR BALLISTICS Necessity for care. Plotting of observed points of fall. (m) Weight of the shell. (n) Any other information that may be desirable. ( (i), (j) and (k) need only be recorded when a change occurs, but the record must be such that the conditions at the beginning and at the end of the practice, and at the moment when any individual shot is fired may be readily and accurately obtained from it.) 355. The members of the observing parties should realize the necessity for accurate observations and records. Xothing is more disastrous than carelessness in regard to details, as inaccuracy in any one of the apparently minor points may easily result in rendering the results of the whole practice entirely worthless. Such inaccuracies may readily be of such a nature that they cannot be detected, and might lead to confident entry into battle or target practice with a battery with which it is impossible to do good work owing to the undiscovered carelessness or inaccuracy of some person charged with some of the duties in regard to the observations taken during the calibration practice. 356. Suppose we have four shots fired from a single gun, which fell as follows relative to the foot of the perpendicular from the center of the bull's eye upon the water : No. 1 a yards over a' yards to the right. No. 2 h yards short h' yards to the left. No. 3 c yards over c' yards to the right. No. 4 d yards short d' yards to the left. Figure 33 357. Then their points of fall are as shown in Figure 33, in which we have given a projection in the vertical plane through the line of fire and the center of the bull's eye, and also the corresponding projection upon the horizontal plane of the water. T is the target, the center of the bull's eye being at B, which is li feet above the water. LU is the line of sight such that the gun pointer looking along it sees the cross wires of the telescope on the center of the bull's eye. Now if the gun were in perfect CALTBEATIOX OF SIXGLE GUXS AND A SHIP'S BATTEEY 219 adjustment when fired, its sliot would travel along the trajectory X, pierce the bull's eye at B and strike the water at P. Note that the recorded errors are actually observed from the point B' on the surface of the water vertically below B. Therefore we have to reduce our observations to the point P, by subtracting for overs in range, the dis- tance B'P from the recorded range, and by adding the same distance for shorts ; and B'P may therefore be considered as a constant error affecting all shots alike. 358. The distance Sn is the danger space for a target h feet high at a range equal to the distance from the gun to the target plus the danger space. For practical purposes, however, when the range is considerable, this danger space may be taken from the range tables for the height h for a range equal to the distance from the gun to the target. The amount of correction to be applied because of the height h should also be taken from Column 19 of the range table. It sometimes happens, also, that the point of sight may not be exactly on the center of the bull's eye at the moment of firing, but may, by the check telescope, be determined to have been a certain distance above or below the proper point of aim ; in which case h would have to be modified accordingly. To work out the observed data : 1. Take a large drawing board, and plot on it to scale (scale sufficiently large to give accurate results) the positions of the observing parties, gun and center of the target. 2. Using the observed data, plot the point of fall of each shot, and measure the distance from the foot of the perpendicular through the center of the bull's eye on the water, in range and in deflection, recording the results. 3. From the results obtained by (2), find the location of the mean point of impact in range and in deflection with reference to the perpendicular noted in (2), which for range will be ~ — --^ ^ ^ — -^ yards from the foot of the perpen- dicular; a, h, c and d being taken with their proper algebraic signs, -f- for an over and — for a short; the sign + on the result showing that the point is over and — that it is short. 4. For the mean point of impact in deflection, by similar methods, the distance from the line of sight will be +o,' + i-^')+c' + {-d') . ^,^ ^,^ ^, ^^^^ ^, ^^^.^^ ^^j.^^ 4 with their proper algebraic signs, 4- for a deviation to the right and — to the left ; a 4- sign on the result will show that the mean point of impact is to the right of the line of fire and a — sign that it is to the left. It is equally as good, and sometimes more convenient, instead of using the 4- and — signs in this work, to keep to the nomen- clature of " short " and " over," etc. ; using the letters " S " and " " to represent them and " K " and '" L " to represent " rights " and " lefts "; thus a shot might be " 155 S and 25 L." 359. Having obtained from these plotted positions for the particular group of Reduction shots under consideration the mean distances in range and in defiection from the foot conditions, of the perpendicular on the water through the bull's eye, it is now necessary to reduce those distances, first to the point P as an origin, and then from firing to standard con- ditions. The method of doing this is simply an application of methods that have already been studied in this book, and it may be best understood from the solution of a problem. 220 EXTEEIOR BALLISTICS Calibration problem. 360. Let US suppose that six shots were fired on a calibration practice from a 12" gun (y = 2900 f. s., w = 870 pounds, c = 0.61) under the following conditions: Actual distance of target from gun 8000 yards. Sights set in range for 8000 yards. Sights set in deflection at 38 knots. Center of bull's eye above the water 12 feet. Bearing of target from ship -45° true. Wind blowing from 270° true. Wind blowing with a velocity of 18 knots. Barometer 30.00". Temperature of the air 75° F. Temperature of the powder 94° F. Weight of projectile 875 pounds. Measured from the foot of the perpendicular upon the water through the center of the bull's eye, the shot fell. 'No. 1. .200 yards short; 90 yards left. No. 4. .150 yards short; 85 yards left. No. 2. .150 yards short; 95 yards left. No. 5. .100 yards short; 75 yards left. No. 3. .100 yards short; 95 yards left. No. 6. . 50 yards short; 70 yards left. Find the true mean errors in range and in deflection under standard conditions, and adjust the sight scales in range and in deflection in order to have the sights properly set ; that is, under standard conditions, to have the mean point of impact at the point P when the sight is set for 8000 yards in range and for 50 knots on the deflection scale. No. of shot. Range, Short. Yds. Deflection. Left. Yds. 1 200 150 100 150 100 50 90 2 95 3 95 4 85 5 75 6 70 Mean errors on foot of perpendicular") through bulTs eye. J 6|750 125 yards short. 6|510 85 vards "left. The error in range due to the fact that the point of aim is at the bull's eye and not at the water line of the target is the correction that should be applied to the observed distance from the foot of the perpendicular on the water through the bull's eye in order to refer it to the point P as an origin. By Column 19 of the range tables, this would be 12 X -^ = 60 yards. The error in deflection intentionally introduced in order to avoid wrecking the target, by setting the sight off in deflection, would be, by Column 18 of the range table, (50-38) X "I =65 yards left. Now to bring the observed errors to their true values under standard conditions, we proceed as follows : CALIBEATION OF SINGLE GUNS AND A SHIP'S BATTERY 221 Temperature of the powder is 4° above the standard, therefore the initial 35 velocity is 4 X r-^ = 14 f . s. above standard. From Table IV, the multiplier for Column 12 is +.18. jV/nc/ = /^ /tTzoT^^ Figure 34. Therefore we have, usins: the traverse tables to resolve the wind forces: Cause of error. Affects. Wind ic Range Deflection.. Range Atmosphere Velocity Range Range Height of bull's eye Intentional deflec- tion. Range Deflection.. Formulffi. 18eos450x4=i^^ 18 sin 45^x4-=:.^^ 5X 39 10 .18 X 136 1. 229 12 X 100 20 12 X — 12 Errors on point P as an origin for standard conditions. Range. Short, Yds. 19 19.5 Over. Yds. 18.0 24.5 64.1 60.0 166.6 19.5 147.1 over Deflection. Right. Yds. 8.5 8.5 Left. Yds. 65.0 65.0 8.5 56.5 left. Observed distance from target in range 125.0 yds. short Error (where shot should have fallen) 147.1 yds. over True mean error in range under standard conditions 272.1 yds. short Observed distance from line of lire through bull's eye in deflection. . 85.0 yds. left Error (where shot should have fallen) 56.5 vds. left Tru e mean error in deflection under standard conditions 28.5 yds. left 223 EXTEEIOE BALLISTICS That is, under standard conditions, the mean point of impact of this gun is 272,1 yards short of and 28.5 yards to the left of the point of fall (P) of the perfect tra- jectory of the gun through the bull's eye. We want to so adjust the sight scales as to bring the mean actual trajectory of the gun into coincidence with the perfect tra- jectory of the gun; that is, to shift the mean point of impact of the gun to its proper theoretical position, that is, to the point P. To do this we : 1. Run up the sight in range until the pointer indicates 8273.1 yards. Then slide the scale under the pointer until the pointer is over 8000 yards on the scale. Then clamp the scale in this position. 2. From Column 18 of the range table, we see that 28.5 yards deflection at 8000 12 yards range corresponds to a movement of 28.5 x ^=5.3 knots on the deflection scale. Therefore set the sight in deflection at 55.3 knots. Then slide the deflection scale under the pointer until the pointer is over 50 knots on the scale. Then clamp the scale in that position.* When the above process has been completed, the gun should shoot, under standard conditions, so that the mean point of impact will fall at P. 361. With the sights adjusted as described above, under standard conditions, the shot should fall at the range and with the deflection given by the sight setting ; that is, the shot should all fall at the mean point of impact. And any variation from standard conditions should cause the errors indicated for such variations in the range tables; and such errors could be easily handled by the spotter. Of course this statement, if taken literally, means that all errors have been eliminated from the gun, and that all shots fired from it under the same conditions will strike in the same place, that place being the mean point of impact for those conditions. It is of course never possible to actually accomplish this, owing first to the inherent errors of the gun, and second to unavoidable inaccuracies in the work. If the work be well done, how- ever, the result will be to come as near as is humanly possible to that most desirable perfect condition. Mean 362. If the distance of the point of fall of each shot from the mean point of dispersion. jj^-,p^(^|. ]jg found for every shot fired, and the arithmetical mean of these distances be found, we have a distance which is called the " mean dispersion from the mean point of impact." This information is desirable because it gives an idea of the accviracy and of the consistency of shooting of the gun. For example, one gun of a battery may have its mean point of impact with reference to a certain target at a distance, say, of 100 yards over and 25 yards to the right, but all of its shot may fall at, say, a mean distance of only 10 yards from the mean point of impact; that is, its shot will all be well bunched and closely grouped around the mean point of impact. Its mean dispersion from mean point of impact is small, and it is a good gun ; for the spotter can readily bring its shot on the target, and when he has done this they will all fall there. If, on the contrary, with another gun, the mean point of impact be, say, only 10 yards over and 10 yards to the right of the target, but the mean dispersion from the mean point of impact be, say, 75 yards, the shot will fall scattered, the spotter will have difficulty in bringing the mean point of impact on the target and in keeping it there, and after he has done so the percentage of hits will be much smaller than * For setting the sights preparatory to adjusting the scales, given the true mean errors, we may readily figure out the following rules: „ f If the error be " short," add it to the standard range. "[if the error be " over," subtract it from the standard range. P^ „ .. fif the error be " right," subtract its equivalent in knots from 50. ■{". the error be " left," add its equivalent in knots to 50. CALIBEATION OF SINGLE GQNS AXD A SHIP'S BATTEKY 223 with the first gun. Comparing the two guns, we would say that the second gun was a poor one compared to the first. 363. All that the spotter can hope to do, with a single gun, is to bring the mean point of impact on the target and hold it there ; then if the gun shoots closely he will make the maximum possible number of hits. If, however, the gun does not shoot closely, there is nothing that can be done to increase the number of hits ; he is simply doing the best that he can with an inferior weapon. Similarly, with salvo shooting, in which the spotter tries to bring the mean point of impact of all the guns (that is, the mean point of impact of all the mean points of impact of all the individual guns) on the target (or slightly in front of it) and keep it there. After he has done this the result depends upon the mean dispersion of each gun from its own mean point of impact, and upon the accuracy with which the work of calibration has brought the mean points of impact of all the guns into coincidence for the same setting of the sights. 364. As an example of the mean dispersion from the mean point of impact, we will now determine that quantity for the 12" gun already calibrated. Power of spotter. In range. In deflection. No. of shot. Fall relative to target. Short or over. Yds. Position of mean point of impact rela- tive to target. Short or over. Yds. Variation of each shot from mean point of im- pact. Short or over. Yds. Fall relative to target. Right or left. Yds. Position of mean point of impact rela- tive to target. Right or left. Yds. Variation of each shot from mean point of im- pact. Right or left. Yds. 1 2 3 4 5 6 200 short 150 short 100 short 150 short 100 short 50 short 125 short 125 short 125 short 125 short 125 short 125 short 75 short 25 short 25 over 25 short 25 over 75 over 90 left 95 left 95 left 85 left 75 left 70 left 85 left 85 left 85 left 85 left 85 left 85 left 5 left 10 left 10 left 10 right 15 right 6|250 41.7 yards in range. 6|50 8.3 yards in de- flection. Therefore the mean dispersion of this gun from the mean point of impact is : In range 42 yards. In deflection 8 yards. 365. Note that in finding the position of the mean point of impact we com- Mean point bined the errors of the several shots with their proper algebraic signs, because we and mean were finding the mathematical center of gravity of the group of points of fall ; but in computing the mean dispersion from the mean point of impact we were simply trying to find the average distance at w^hich the shot fell from that point, without regard to direction, and so we discard the algebraic signs and simply take the arithmetical mean. It is to be noted that all corrections applied to the original data change all the shot alike, and therefore do not change their position relative to each other. We may therefore find the dispersion by using the original data before cor- rection, as was done above. Or we could correct each shot separately and then find the dispersion from the results, but the former process is of course the shorter and simpler. Corrections to the original data do of course change the positions of the shot relative to any fixed outside point, such as the target or the point P, and therefore we have the process previously employed for finding errors, etc. 224 EXTERIOR BALLISTICS EXAMPLElS. 1. For the following results of different calibration practices, compute the true mean errors under standard conditions and the mean dispersion from mean point of impact; and tell how to adjust the sight scales in each case in range and deflection to make the gun shoot as pointed when all conditions are standard. 14" gun; 7 = 2600 f. s.; w = 1400 pounds; c = 0.70. Actual distance of target from gun, yds Sights set in range for, yds Sights set in deflection for, l^nots Center of bull's eye above water, feet. . Bearing of target from ship, °true. .. , Wind blowing from, °true Wind blowing with a velocity of, knots. Barometer, inches Temperature of air, °F Temperature of powder, °F Weight of shell, pounds Number of shots fired Fall of— Shot No. 1 Shot No. 2 Shot No. 3 Shot No. 4 1. 2. 3. 4. 5. 6. 7. 13000 13500 14000 14500 13300 13700 14200 13000 13500 14000 14500 13300 13700 14200 35 40 30 42 GO 65 70 4 3 5, 6 4 5 6 45 180 80 315 270 250 345 180 225 90 180 250 165 12 15 18 20 25 15 18 28.. 50 29.00 29.50 30.00 30.50 31.00 30.25 GO 65 70 75 SO 85 90 80 85 95 100 97 82 75 1395 1390 1405 1410 1407 1393 1397 4 4 4 4 ^ 4 4 25 100 150 75 150 200 100 S. s. Ov. S. S. Ov. s. 75 70 200 20 15 30 100 L. L. L. L. R. R. R. 50 75 200 90 100 250 75 S. S. Ov. S. s. Ov s. 100 75 150 10 20 35 80 L. L. L. L. R. R. R. 100 50 175 20 125 275 50 s. S. Ov. Ov. S. Ov. s. 150 50 175 15 30 50 110 L. L. L. L. R. R. R. 75 25 130 10 110 225 S. Ov. Ov. Ov. S. Ov. 80 50 150 20 25 40 90 L. L. L. L. R. R. R. 14400 14400 63 4 270 22 29.75 83 80 1404 4 20 Ov. 50 R. 25 Ov. 55 R. 30 Ov. 60 R. 22 Ov. 50 R. ANSWERS. 1 2 3 4 5 6 7 True mean errors. Range. Yds. 11 7 0v. 8 4S. 41 2S. 449 4S. 322 9S. 479 2 0v. 89 2 0v. 134 lOv. Deflec- tion. Yds. 79. OR. 78. 5R. 20.7 R. 124.0 R. 148.7 L. 133.7 L. 146.7 L. 168.8 L. Mean dispersion from M. P. of I. Range. Yds. 25.0 37.5 18.75 48.75 16.25 25.0 31.25 3.25 Deilec- tion. Yds. 24.4 11.3 18.75 3.75 5.00 6.25 10.0 3.75 Set sights for. Range. Yds. 12988.3 13508.4 14041.2 14949.4 13632.9 13220.8 14110.8 14265.9 Deflec- tion. Knots. 42.6 43.1 48.5 40.0 63.4 61.6 62.1 63.7 Clamp scales at. Range. Yds. 13000 13500 14000 14500 13300 13700 14200 14400 Deflec- tion. Knots. 50 50 50 50 50 50 50 50 CALIBEATION OF SINGLE GUNS AND A SHIP'S BATTERY 325 2. For the following results of different calibration practices, compute the true mean errors under standard conditions and the mean dispersion from mean point of impact; and tell how to adjust the sight scales in range and deflection in each case to make the gun shoot as pointed under standard conditions. 7" gun; 7 = 2700 f. s.; w = 16b pounds; c=0.61. Actual distance of target from gun, yds Sights set in range for, yds Sights set in dellection at, knots Center of bull's eye above water, feet. . Bearing of target from ship, °true. .. , Wind blowing from, °true Wind blowing with velocity of, knots. . Barometer, inches Temperature of air, °F Temperature of powder, °F Weight of shell, pounds Number of shots fired Fall of— Shot No. 1 Shot No. 2 Shot No. 3 Shot No. 4 6000 6000 70 6 180 13 30.50 50 95 170 4 50 Ov. 150 R. 75 Ov. 165 R. 70 Ov. 155 R. 65 Ov. 150 R. 6200 6200 65 5 180 180 17 30.25 55 97 172 4 75 Ov. 100 R. 50 Ov. 95 R. 25 Ov. 70 R. 25 S. 75 R. 6500 6500 63 4 90 180 12 30.00 60 100 168 4 75 S. 95 R. 55 S. 90 R. 70 S. 85 R. 75 S. 90 R. 6700 6700 60 3 180 90 20 29.33 65 98 162 4 200 Ov. 75 R. 250 Ov. 50 R. 200 Ov. 25 R. 175 Ov. 10 L. 5. 7000 7000 55 4 350 220 25 29.67 70 93 160 4 150 S. 5 R. 125 S. 10 R. 100 s. 130 s. 5 L. 7200 7200 45 5 220 350 18 29.00 75 87 161 4 100 Ov. 5 R. 110 Ov. 10 L. 125 Ov. 25 L. 117 Ov. 20 L. 7500 7500 43 6 160 30 30 7300 7300 40 5 225 270 22 29.10 28.90 80 85 85 164 4 250 Ov. 20 L. 275 Ov. 25 L. 300 Ov. 30 L. 280 Ov. 27 L. 80 169 4 100 s. 25 L. 125 S. 30 L. 130 S. 35 L. 135 S. 40 L. ANSWERS. True mean errors. Mean Dispersion from M. P. of I. Set sights at. Clamp scales at. Range. Yds. Deflec- tion. Yds. Range. Yds. Deflec- tion. Yds. Range. Yds. Deflec- tion. Knots. Range. Yds. Deflec- tion. Knots. 1 91.8 0V. 88.7 0V. 141.7 S. 19.2 V. 312.5 S. 39.9 S. 142.10V. 63.7 S. 63.3 R. 13.7 R. 40.9 R. 39. 2L. 47.8 L. 33.6 R. 13.6 L. 45.3 R. 7.5 31.25 6.85 21.9 13.75 8.0 16.25 11.25 5.0 12.5 2.5 27.5 6.25 10.0 3.0 5.0 5908.2 6111.3 6641.7 6680.8 7312.5 7239.9 7357.9 7363.7 36.2 47.1 42.0 57.5 58.6 44.2 52.2 42.2 6000 6200 6500 6700 7000 7200 7500 7300 50 2 50 3 . . 50 4 50 5 50 6 50 7 50 8 50 15 CHAPTEE 19. Reasons for calibrating battery. Standard gun. THE CALIBRATION OF A SHIP'S BATTERY. 366. In the preceding chapter we have seen how a single gun is calibrated and the sights so adjusted that, so far as the inherent errors of the gun, etc., will permit, the gun will shoot, under standard conditions, as the sights indicate. It was stated that it is not possible to accomplish this result with absolute accuracy. If it were, we could adjust the sights of each gun of the battery separately, and then, if they were all mechanically just alike, we would have all the shot from each gun falling at its mean point of impact (within the limits of inherent errors), and the mean points of impact of all the guns would be the same. As a matter of fact, however, the mean points of impact of the several guns would not coincide, if this method were followed, and of course all the shot from any one gun would not all fall at its mean point of impact. Some remarks were made in the last chapter relative to the necessity for getting the guns so calibrated that the shot from all of them Avill fall together for the same sight setting, and, as a matter of fact, this is more important than it is to get them so that actual and sight-bar ranges coincide under standard conditions. Con- ditions are almost never standard during firing, and even if they were there are many other factors which prevent the actual and the sight-bar ranges from being the same. But if the mean points of impact of the several guns for the same sight setting can be brought very nearly into coincidence, then any variation of the resultant point from the target (that is of difference between actual and sight-bar ranges) can be readily handled by the spotter. This means that if the salvos are well bunched the spotter can control the fire successfully, but if the shots are scattered he cannot. We will now proceed with an entire battery to bring all guns to shoot together. 367. Having calibrated each gun separately, as described in the preceding chapter, we now proceed to select a gun as the " standard gun," to the shooting of which we propose to make that of all the others conform, providing the performance of any one gun is good enough to justify selecting it for the purpose. From what we have seen in the preceding chapter we would naturally select one whose mean dis- persion from mean point of impact is small, that is, one that bunches its shots ; and, other things being equal, if we have one whose sights are very nearly in adjustment, we will use that one without changing the sight adjustment. Any gun may of course be selected as the standard, and the sights of the others brought to correspond to it, but the considerations set forth above would naturally govern, as a matter of common sense. If no gun be sufficiently accurate, or if none has its sights sufficiently well adjusted to justify its selection as a standard gun, then we must correct all guns to the mean point of impact. The practical method of bringing the sights of a number of guzis to correspond is best shown by an actual problem. CALIBRATION OF SINGLE GUNS AND A SHIP'S BATTERY 227 368. The results of the individual calihration of a battery of eight 13" guns Calibration (y = 2900 f. s., ft) = 870 pounds, c=0.(.)l), at an actual range of 8000 yards, were as follows : True mean errors under standard conditions. Yards. Number of gun. In range. In deflection. Short. Yards. Over. Yards. Right. Yards. Left. Yards. 1 100 75 "so 25 5 "40 90 '125 '26 15 10 5 15 9 3 4 15 6 20 7 8 20 It is desired to calibrate the above battery. From an examination of the above results, assuming that the eight guns are equally good in the absence of any knowledge to the contrary, we will select No. 7 as the standard gun; and, as its sights are very slightly out, we will leave them unchanged and bring the sights of the other guns to correspond with them. The work, which is best expressed in tabular form, then becomes (all guns were fired with sights set at 8000 yards in range and 50 knots in deflection) : With reference to standard gun, each gun shot. To bring all sights together set them for each gun as follows: Number of gun. In range. Yds. In deflection. In range. Yds. In deflection. Yards. Knots.* Knots. 1 95 short 70 short 45 over 95 over 75 short 20 short Standard 130 over 20 left 15 right 10 right 5 right 20 left 25 left Standard 25 left 3.7 left 2.8 right 1.9 right 1.0 right 3.7 left 4.6 left Standard 4.6 left 8095 8070 7955 7905 8075 8020 Standard 7870 53 7 2 47 2 3 48 1 4 49 53 7 6 54 6 7 8 54 6 12 * From the range table, at 8000 yards, one yard in deflection corresponds to 7^^.. knots on the deflection scale. After the sights have been set as indicated in the two right-hand columns of the above table, move the sight scales under the pointers until the pointers are over the 8000-yard mark in range and the 50-knot mark in deflection in each case, and then clamp the scales in those positions. The guns are then calibrated to shoot together. It will be noted that, theoretically, we should have set the range scales for the 5 yards 228 EXTEPtlOE BALLISTICS short in range and the deflection scales for the 5 yards right in deflection of the standard gun, to be absolutely accurate ; but, as the sight scales are graduated to 50-yard increments in range only, it is impracticable to go any closer in range. It would perhaps be well to adjust each deflection scale to 51 knots instead of 50, in order to allow for the 5 yards right deflection of the standard gun. Different 369. When we wish to calibrate a ship's battery that is composed of separate batteries of different calibers, we calibrate each caliber by itself, as already described. The difference between the mean points of impact of the standard guns of the differ- ent calibers will be the difference between the centers of impact of the salvos from the several calibers, and this must be allowed for in firing all calibers together. To attempt to calibrate the sights of all calibers together by a readjustment of the sight scales would not be wise ; for if they could be brought to shoot together at one range in this way, it would necessarily ensure dispersion of the several salvos at all other ranges. Therefore the only practical way of handling this proposition is to deter- mine the error of each caliber at the range in use and apply it properly in sending the ranges to the guns; which means send different ranges to the guns of different calibers, so related that the results will bring the mean points of impact of the several calibers together at the range in use. As far as possible, these differences in ranges should be tabulated for different ranges. As the fire-control system is arranged, as a rule, to permit the control of each caliber battery independently of the others, this method presents no difficulties other than a little care on the part of the spotter group. 370. For instance, suppose that we have a ship with a mixed battery of 7", 8" and 12" guns; that each of these calibers has been calibrated at 8000 yards; and has had the mean point of impact of its salvos located with reference to the target as follows : 7" battery 100 yards over 3 knots right. 8" battery 50 yards short 3 knots left. 12" battery 150 yards over 2 knots left. Then if we wish to fire broadside salvos from this entire battery, the ranges and deflections should be sent to the guns as follows, for 8000 yards : To the 7" 7900 yards 47 knots deflection. To the 8" 8050 yards 53 knots. To the 12" 7850 yards 52 knots. CALIBEATION OF SINGLE GUNS AND A SHIP'S BATTERY 229 EXAMPLES. 1. Having determined the true mean errors of guns under standard conditions, by calibration practice, to be as given in the following table; how should the sights of each caliber be adjusted to make all the guns of that caliber shoot together? (Six separate problems.) True mean errors of guns under standard conditions. 6"— G. 7"— J. 8"— K. 12"— N, 13"— P. 14"— R. d Errors at Errors at Errors at Errors at Errors at Errors at b£ range o f 4500 range of 6500 range of 8500 range of 10000 range of 11000 range of 13000 yar Is. yards. yards. yards. yards. yards. S Eange. Defl. Range. Defl. Range. Defl. Range. Defl. Range. Defl. Range. Defl. ^ Yds. Yds. Yds. Yds. Yds. Yds. Yds. Yds. Yds. YMs. Yds. Yds. 1.. 25 20 50 30 100 5 100 15 125 25 75 15 S. R. Ov. L. R. R. Ov. L. Ov. L. R. L. 2. . 50 30 75 40 120 15 75 10 100 30 100 15 Ov. R. Ov. L. R. L. Ov. L. Ov. L. Ov. R. 3.. 75 35 100 25 90 20 50 20 75 25 75 2(1 Ov. L. Ov. R. Ov. L. Ov. R. Ov. R. Ov. R. 4.. 5 5 75 15 75 25 5 100 30 50 20 R. L. S. L. Ov. R. R. S. R. 8. L. 5.. 30 30 125 10 100 20 100 25 100 25 Ov. R. S. L. Ov. R. R. R. s. L. 6.. 50 25 10 70 SO 10 90 15 100 30 S. L. s. R. R. L. R. R. Ov. R. 7.. 100 40 100 25 70 25 75 30 > • • • to 20 s. R. Ov. R. R. R. Ov. L. Ov. R. 8.. 100 40 90 20 70 30 100 30 • • • • 70 15 Ov. L. S. L. Ov. L. R. R. S. L. AN8WERS. To bring all sights together for each caliber, set the sights for that caliber as given below, and then slide scales to standard readings and clamp. be 6"- -G. 7"— J. 8"— K. 12"- N. 13"— P. 14"— R. a 3 Range. Defl. Range. Defl. Range. Defl. Range. Defl. Range. Defl. Range. Defl. Yds. Yds. Yds. Kts. Yds. Kts. Yds. Kts. Yds. Kts. Yds. Kts. 1.. 4520 43.0 6450 56.0 8600 49.25 9900 52.8 10875 52.0 13075 51.35 2.. 4445 40.2 6425 58.0 8620 52.25 9925 52.1 10900 .52.4 12900 48.65 3.. 4420 58.4 6400 45.0 8410 53.00 9950 47.9 10925 48.0 12925 48.20 4.. Rtan dard. 6575 53.0 8425 46.25 Rtan dard. 11100 47.6 13050 51.80 5. . 4465 40.2 6625 52.0 8400 47.00 10100 47.2 13100 52.25 (5. . 4545 55.6 6510 36.0 8580 51.50 10090 48.6 12900 47.30 7. - 4595 37.4 6400 45.0 8570 46.25 9925 54.9 12925 48.20 8.. 4395 59.8 6590 54.0 8430 54.50 10100 46.5 13070 51.35 PAET VI. THE ACCURACY AND PROBABILITY OF GUN FIRE AND THE MEAN ERRORS OF GUNS. INTEODUCTIOX TO PAET VI. "We have now learned all that mathematical theory can teach us with certainty about the flight of a projectile in air, about the errors that may be introduced into such flight by known causes, and about the methods of compensating for such errors. After all this has been done there must, in the nature of things, remain certain errors that cannot be either eliminated or covered by strict mathematical theories, and such errors are known as the inherent errors of the gun. It is the purpose of Part VI to discuss the general nature of these errors, their methods of manifesting themselves, and their probable effect upon the accuracy of fire. CHAPTER 20. THE ERRORS OF GUNS AND THE MEAN POINT OF IMPACT. THE EaUATION OF PROBABILITY AS APPLIED TO GUN FIRE WHEN THE MEAN POINT OF IMPACT IS AT THE CENTER OF THE TARGET. Z. Y. (zi, !/i, etc.) . %■ n. yx- Jy yz. New Symbols Introduced. . Axis of; axis of coordinates lymg along range, for points over or short of the target. . Axis of; axis of coordinates in vertical plane through target, for points above or below the center of the target. . Axis of ; axis of coordinates in vertical plane through target, for points to right or left of center of the target. . Coordinates of points of impact in vertical plane through target. . Summation of z^, Zo, etc. . Summation of y^, t/„, etc. . Number of shots. . ]\Iean error in range. . Mean vertical error. . Mean lateral error. 371. Before proceeding to the discussion of the accuracy and probability of gun fire, it is wise to collect and consider certain definitions and descriptions of which a full understanding is necessary in order to clearly understand what is to follow. 372. There are three general classes of errors which enter into gun fire, and the classes of distinction between which must be clearly comprehended. They may be stated as ^"°^^' follows : 1. Errors Resulting- from Mistakes or Accidents. — As examples of these may be Mistakes, mentioned mistakes in estimating ranges or deliections, mistakes in sight setting, mistakes in pointing, etc. These are matters that pertain to the training of the personnel, and of course have no place in any discussion of the principles of ballistics, etc., for no theory can be developed unless all such causes of error are first eliminated. Such mistakes of course cause poor shooting, but they have no place in any theoretical investigation of the performance of the gun. 3. Preventable Errors. — These are errors arising from causes which must Preventable, necessarily exist, but in regard to which the theories are well understood, and for which it is possible to compensate by a practical application of such theories. Ex- amples of this class are the errors due to wind, to variation in the temperature of the powder, etc. One of the principal provinces of the science of exterior ballistics is to teach the principles governing such errors, and to show how they may be over- come. In general, it may be said that it is the principal duty of the spotter to discover the magnitude and direction of these errors and to give the instructions necessary to compensate for them. 3. Unpreventable Errors. — These may be generally classified as the inherent unpre^ errors of the gun. That is, they are the result of the very many elements entering into the shooting which cause variations in successive shots even when most carefully fired under as nearly as possible the same physical conditions, and which therefore ensure that any considerable number of shots from the same gun will have their ventable. 234 EXTERIOR BALLISTICS Summary. Mean point of impact. Mean trajectory. Deviation or deflection, successive points of impact more or less scattered about within a certain area. These causes are probably very numerous, it not even being certain that we have yet been able to recognize them all, and no satisfactory laws governing them have as yet been discovered, nor is it probable that such laws ever will be determined. 373. To summarize, it may be said that before entering on any theoretical investigation of the subject of gunnery we must first throw out all errors resulting from mistakes. We may then, by the study of exterior ballistics, learn certain principles governing the errors produced by certain known causes, and in conse- quence may learn how to eliminate such errors from our shooting. When all this has been done, however, we necessarily have left certain other causes of error which, although not great as compared with the others, are still sufficient to cause a scattering of the points of impact of successive shots from the same gun, even when fired under similar physical conditions. We manifestly cannot hope to eliminate these inherent errors, and therefore we must accept them as they are ; all that we can do in regard to them is to investigate their probable effect upon the results of our shooting. It is this investigation that is to be undertaken in the last two chapters of this book, and it is to be noted that here we cannot speak of anything as a certainty, even in a theoretical and mathematical way, but can only say that, mathematically, it is probable or improbable that a certain thing will happen, and in addition attempt to measure the degree of probability or improbability which attaches to a certain effort. 374. Mean Point of Impact. — Let us suppose that all errors except the inherent errors of the gun have been eliminated, and that a large number of shots be fired, under as nearly the same physical conditions as possible, at a vertical target screen of sufficient size to receive all the shot under such conditions. Manifestly, if there were no errors of any kind whatsoever, all these shot would describe the same trajectory and strike the target at the same point. Of course this result can never be attained in practice, and the many causes of inherent error tend to scatter the several shot about the target, and only a certain percentage of absolute efficiency can be secured, no matter how skillfully the gun may be handled. The point which is at the geometrical center of all the points of impact on the screen is known as the " mean point of impact," and is of course the center of gravity of the 'group of points of impact. We may also speak of the mean point of impact in the horizontal plane as well as in the vertical plane as given above. 375. Mean Trajectory. — The mean trajectory of the gun for these conditions is the trajectory from the gun to the mean point of impact. It is manifestly the trajectory over which all the shot would travel were there no errors of any kind whatsoever. 376. Deviation or Deflection. — Suppose Figure 35 to represent the vertical target screen, the point at the center being the point aimed at. Suppose the shot struck at the point P. Then the deviation or deflection of the shot from the point aimed at is the distance OP in the direction shown. So considered, however, for manifest reasons, this information is not useful, so it is usual to speak of the hori- zontal deviation or deflection, which is a, and of the vertical deviation or deflection of the shot, which is h. And algebraic signs are assigned to these deviations or deflec- tions, -t- being above or to the right and — below or to the left. Thus the deviations or deflections of the four points of impact shown in Figure 35 would be: For P +a and +&. For P" -a" and -&". For P' ....-a' and +&'. For P'" . . . . -|- a'" and -h'". In place of the signs we might speak of horizontal deviations as being to the right or to the left, and of vertical deviations as being above or below. In addition to the ACCUEACY AND PROBABILITY OF GUN FIRE 235 above we may consider the deviation (the term "deflection" i» not ordinarily used in this connection) in range, as the shot falls short of or beyond the target. These are denoted by the + sign for a shot that goes beyond and by a — sign for one that falls short, but usually the words " short " and " over " are used instead of the algebraic signs, and such shots are spoken of as " shorts " or " overs," as the case may be. - ti — — —A 1 ^-\ y 1 1 1 / \. \-6- A" / 'D 1 / 1 / ~7^'~'' St •p" -^" Figure 35. 377. Deviation or Deflection from Mean Point of Impact. — In the preceding Deviation paragraph Ave explained deviation or deflection from any given point of aim; the po^nt of results giving the actual amount by which the shot missed the point aimed at. In '™^*'^ ' theoretical consideration of the accuracy of a gun, however, it is customary to assume that the point of reference or origin of coordinates is the mean point of impact, rather than any given point of aim, and our results are then the " deviations or deflections from the mean point of impact." As the mean point of impact is, by definition, the center of gravity of the group of impacts caused by a large number of shot, it is evident that the summation of all the deviations from the mean point of impact must be equal to zero. 378. Dispersion. — Now suppose that, in Figure 35, we had disregarded the Dispersion, algebraic signs, and considered only the actual distances of the points of impact from the point aimed at. The distance OF in this case would be the " dispersion '* for the single shot; but again it is customary to separate the distance in the two directions, and we would have a " horizontal dispersion " of a and a " vertical dis- persion " of h ; although it is not customary or appropriate to speak of the "dis- persion " of a single shot, the word being collective in its nature. The " mean dis- persion " of the four shots shown in Figure 35 from the point aimed at would be: Mean lateral dispersion — — ' Mean vertical dispersion "^ ^^ 4 We may also have " dispersion in range " as well as in the vertical plane. 236 EXTERIOR BALLISTICS Mean disper- 379. Mean Dispcrsion from Mean Point of Impact. — Suppose we ajrain consider sion from . . . „ . . . . . *" mean point our clispersions from the mean point of impact as an origin. Tiien it is evident that the " mean dispersion from mean point of impact," or " mean dispersion " as it is usually called, is the average distance or arithmetical mean of the distances of the points of impact of all the shot from the mean point of impact. Now as this latter point is the one at which every theoretically perfect shot should strike, it is evident that the mean dispersion from mean point of impact gives us a measure of the accuracy of the gun, that is, of the extent to which its shooting is affected by its inherent errors. 380. It must be said in regard to the above definitions, that the terms defined are often very loosely and more or less interchangeably used in service. The term " deflection " is ordinarily used only to represent lateral displacement, in either the vertical or the horizontal planes ; and the term " deviation " is used for either vertical or lateral displacement or for displacement in range, in which case the terms " vertical deviation," " lateral deviation " or " deviation in range " are customarily used. There is also confusion in the use of the terms as to whether deviation or deflection from the point aimed at or from the mean point of impact is meant. The term " dispersion " is fairly regularly used as defined above, but even here the point as to whether dis- persion from the point aimed at or from the mean point of impact is meant is often left obscure. The context of the conversation or written matter will usually show what is meant. In this book the terms will be used strictly as defined. Figure 36 CrUTL System of coordinates. 381. For use in these last two chapters we will also introduce a special system of coordinates, as shown in Figure 36. This figure represents a perspective view of a vertical target screen of which at the center is the mean point of impact in the vertical plane. The axis of X is the line from the muzzle of the gun to 0, the mean trajectory being shown. Z is the horizontal axis and Y the vertical axis through the center of the target. It will be noted that in this system the axes of X and Z are interchanged from what they ordinarily are in geometry of three dimensions; and this is done in this particular subject to preserve the convention that has been con- sistently used throughout, that X and all functions thereof represent quantities pertaining to the range. In this system of coordinates it will be seen that, the mean ACCURACY AND PEOBABILITY OP GUN FIHE 237 point of impact at the center of the screen being considered as the origin, coordinates {z, y) will definitely locate any point of impact on the screen, while coordinates {x, z) will definitely locate any point of impact on the horizontal plane through 0. And here it may be stated that it rarely becomes necessary to consider hits in the vertical and in the horizontal plane together. Therefore for hits in the vertical plane we use as an origin the mean point of impact in the vertical plane, and for hits in the horizontal plane we use as an origin the mean point of impact on the surface of the water. In Figure 3G, being the mean point of impact in the vertical target screen, the mean point of impact in the horizontal plane of the water would lie behind the target at the point where the mean trajectory through strikes the surface of the water. 382. Having cleared up these preliminary matters, and bearing in mind that all errors have been eliminated except the inherent errors of the gun, it may now be stated that it becomes important, under these conditions, to be able to answer certain questions in regard to the probability of securing hits under given conditions. For instance, with a properly directed fire from which all avoidable errors have been removed, what are the chances of hitting a given target at a given range; what proportion of the total number of shot fired at it may reasonably be expected to hit it, etc.? 383. In other words, the preceding chapters having taught us the methods to be followed in eliminating all possible sources of error or of compensating for their effects, we now wish to conduct an investigation that will enable us to determine what are our chances of hitting under given conditions. From the results of this investiga- tion, applied to any particular case, we can tell how much of a drain it would probably be upon our total ammunition supply to make an effective attack under given con- ditions, and hence whether or not we can afford to make the attempt. To arrive at answers to such questions we must fall back upon the theory of probability. 384. It will be readily understood from what has been said that the deviations Deviations of projectiles from their mean point of impact are closely analogous to what are dental called " accidental errors " in the text books on the subjects of probability and least squares; such, for example, as errors that are made in the direct measurement of a magnitude of any kind ; and they obey the same laws. Small deviations are more frequent than large ones; positive and negative deviations are equally probable and therefore equally frequent, if the number of shot be great ; very large deviations are not to be expected at all (if one occur it must be the result of some mistake or some avoidable error). 238 EXTERIOE BALLISTICS 385. Suppose that we have as the point of aim the center, 0, of the vertical target screen shown in Figure 37, and suppose we had n points of impact as shown (18 are shown) of which the coordinates are {z^, yj, (z^, y^), {zn, yn), each with its proper algebraic sign. Then manifestly the coordinates of the mean point of impact referred to as an origin are (^^ , -^ ) , which for the 18 shot shown on the figure would place the mean point of impact somewhere near the point P. Of course the larger we can make n, the more accurately is the position of the mean point of impact determined. Then, the origin being shifted to P, we get new values of the coordinates z and y, from which we know the deviations of each shot from the mean point of impact, both horizontal and vertical. The same process is resorted to in the horizontal plane, with coordinates x and z to determine the position of the mean Y e /a t Z' 'y . e /-f- /3 /z Figure 37. point of impact in that plane, and tlience the deviations of the several shots in that plane. 386. Having found the position of the mean point of impact as described above, and the coordinates of the several points of impact in relation to it, we then get the mean dispersion from mean point of impact in the lateral and in the vertical direc- tions by taking the arithmetical mean (all signs positive) of all the z coordinates for the one and of all the y coordinates for the other, taking the mean point of impact as an origin. The mean dispersion from mean point of impact in the horizontal plane is determined in a similar way. Probability. 387. The probability of a future event is the numerical measure of our reason- able expectation that it will happen. Thus, knowing no reason to the contrary, we assign an equal probability to the turning up of each of the six different faces of a die at any throw, and we may say that the probability that an ace, for example, will turn up on any single throw is measured by the fraction ^. This does not mean that we should expect an ace to turn up once and once only in every six times, but merely that in a great number of throws, n, we may reasonably expect very nearly -— aces to be thrown, and that the greater n is the more likely it is that the result will agree with the expectation. ACCUEACY AND PEOBABILITY OF GUN FIRE 239 388. If an event may happen in a ways and fail in 6 ways, each of the a + & ways being equally likely to occur, the probability that it will happen is — ~y and the proba- bility that it will fail is r , and the sum of these two fractions, unity, represents the certainty that the event will either happen or fail. Thus, if the probability that an event will happen be P, then the probability that it will not happen must be 1 — P. For example, since of the 52 different cards which may be drawn from a pack, 13 are spades, the probability that a single card drawn from a pack will be a spade is 13 1 . 13 39 3 -^- = -- , while the chance that it will not be a spade is 1— -^^ = -^k == ^ • 52 4 52 52 4 389. If the probability that one event will happen be P, and that another independent event will happen be Q, then the probability that both events will happen will be the product of P and Q. For example, the probability that a single card drawn 12 from a pack will be a face card (king, queen or knave) is vs > and the chance that 13 it will be a spade is — ; therefore the chance that it will be either the king, queen or , . -, . 12 ^ 13 3 knave ot spades is — ^ X --^ = -z^ . '■ 52 52 52 390. It will be noted in the preceding discussion of the laws of probability that we have been dealing with cases in which one or more of a fixed number of events in question must either happen or fail; that is, with definite numbers of equally probable events. When we consider the deviations of projectiles this is no longer the case, for we are then dealing with values which may be anything whatever between certain limits. We cannot assign any finite measure to the probability that a deviation shall have a definite value because the number of values that it may have is unlimited. With a single throw of a die there are just six things that may happen and one of the six must happen. With the fire of a gun, however, within the limits which we are considering, there are an infinite number of points at which the shot may strike, and therefore no such definite fraction as ^ can be assigned, as was done with the die. We can, however, measure the probability that the deviation of a certain shot will lie within certain limits, or that it will be greater or less than an assigned quantity. Suppose, for example, that a very large number, n, of shots have been fired, and that, their lateral deviations having been measured, it is found that m of these deviations are between 2 feet and 3 feet, either plus or minus; then we could say that, in any future trial under similar circumstances, the probability that a single shot will have a lateral deviation between 2 and 3 feet is — . Or if, of the n impacts, q were less than 4 feet to one side or the other of the mean point of impact, we could say that the probability that the lateral deviation of any single shot would be less than 4 feet is ^ . The actual case given in the following paragraph will serve as an illustration, although n is not really as large as it should be. 391. On December 17, 1880, at Krupp's proving ground, at Meppen, 50 shots observed were fired from a 12-centimeter gun at 5° elevation, giving a mean range of 2894.3 meters. The points of fall were marked on the ground and the position of the mean point of impact was determined (note that this is in the horizontal plane). Measur- 240 EXTERIOR BALLISTICS ing the lateral deviations from this mean point of impact, the following results were obtained: _. . r Number of shot ^ •^*™'^^ To the right To the left Between and 1 meter 1-i 13 Between 1 and 2 meters 8 8 Between 2 and 3 meters 2 5 24 26 The mean lateral deviation was found to be 1.07 meter. Taking horizontal and vertical axes through the mean point of impact (assuming that the lateral deviations are the same in the horizontal and in the vertical plane, which is very nearly the case), laying off equal spaces to left and to right of the origin, each representing one meter, and constructing on each space a rectangle whose height represents, on any r A] A. b:/ / / 1^^- -\^' C' B' A' O A B Figure 38. convenient scale, the number of shots whose lateral deviations were within the limits corresponding to the space, we obtain Figure 38. 392. It will be seen that the distribution of the deviations is fairly symmetrical to the axis of Y , there being 26 to the left and 24 to the right; also that the maximum does not exceed three times the mean deviation ; also that the area of each rectangle divided by the whole area of the figure is the measure of the probability (as defined) that any single deviation will fall within the limits represented by its base. Thus, 14 the area OAA^ — \^, divided by the total area, 50, is the probability, -^ ? that any single deviation will lie between and +1 meter, the area OAA-^A^' A' = 27, divided 27 by the total area is the probability, -^ , that any single deviation will lie between + 1 50 meter and —1 meter; and the total area divided by itself, -^ = 1 = certainty, is the probability that no deviation will exceed 3 meters. ACCUEACY AND PEOBABILITY OF GUN FIEE 241 393. Now if the number of shot be increased, while the width of the horizontal spaces be diminished in the same proportion, the area of each rectangle divided by the whole area of the figure will continue to measure, with increasing accuracy, the probability that any one deviation will fall within the limits represented by its base. At the limit, when the number of shot is infinite and the width of the horizontal spaces has been reduced to the infinitesimal dz, the height, y, of each rectangle will still be finite ; the upper contour of the figure will become a curve approximately like that shown in the figure ; the area of each rectangle now becomes the elementary area Mathematical theory. ydz and the whole area under the curve now becomes ydz, and the quotient of the first by the second will still measure the probability that any one shot will fall between the limits represented by the base of the rectangle, that is, between z and z + dz. The area between any two ordinates of the curve, that is, ydz, divided by the whole area, will still measure the probability that any one deviation will lie between a and h. 394. The curve just described is the probability curve for the lateral deviations of the projectiles from the particular gun considered, under the given conditions, and while tlie probability curve for the vertical deviations for the same case, or for either lateral or vertical deviations or for deviations in range in the case of other guns or other conditions would difi^er from the particular curve shown in Figure 38, they would all present the following general features : 1. Since plus and minus deviations are equally likely to occur, the curve must be symmetrical to the right and to the left of the origin, which is the mean point of impact. 2. Since the deviations are made up of elemental deviations which, as they may have either direction, tend to cancel one another, small deviations are more frequent than large ones, so the maximum ordinate occurs at the origin. 3. Since large deviations can only result when most of the elemental deviations have the same directions and their greatest magnitudes, such large deviations must be rare, and deviations beyond a certain limit do not occur at all. Therefore the curve must rapidly approach the horizontal axis, both to the right and to the left of the origin, so that the ordinate, which can never be negative, practically vanishes at a certain distance from the origin. 395. If y = F(z) be the equation to the probability curve, the general features stated in the preceding paragraph require that: 1. F(z) shall be an even function; that is, a function of z'. 2. F{0) shall be its maximum value. 3. It shall be a decreasing function of z- and shall practically vanish when z is large. Since it is impracticable to so select the function F that F(z) shall be con- stantly equal to zero when z exceeds a certain limit, this last condition requires that the curve shall have the axis of Z for an asymptote; in other words, we must have F{±cc)=0. 396. The foregoing characteristics being thus established, and taking as a basis Probability curve* the axiom that the arithmetical mean of the observed values (made under similar circumstances and with equal care) of any quantity is its most probable value, the theory of accidental errors deduces as the equation to the probability curve 1 -A Try in which y is the mean error, or in our case the mean deviation from mean point of Conditions existing. y= (222) 16 243 EXTERIOR BALLISTICS impact, 7r = 3.1416, and e = 2.7183, and the factor — has been introduced to make the whole area under the curve equal to unity '' ■-f-oo ___z£_ e Try- dz = Try thus obviating the necessity for dividing the partial area by the whole area whenever a probability is to be computed. 397. Figure 39 represents the probability curve for the Krupp 12-centimeter siege gun, taking its mean error to be 1.07 meter, as given by the 50 shots previously described. There is also shown in dotted lines, for comparison, the probability curve ^' Figure 39. for a gun whose mean error is three-quarters that of the 12-centimeter gun. In both cases the ordinates are exaggerated ten times as compared with the abscissa?. 398. The maximum ordinate being the value of y when 2 = 0, is therefore in- versely proportional to the mean deviation, that is, y = — ; the probability that any one deviation will be less than OB = OA is the numerical value of the area AA'CB'B in the one case, and of the area AA'C'B'B in the other; the probability that any one deviation will exceed OB = OA is the area under that part of the curve which is to the left of AA' and to the right of BB' ; the whole area under the curve has the numerical value of unity. It will be seen how very small is the probability that any deviation will exceed three times the mean deviation. ACCURACY AND PROBABILITY OF GUX FIRE 243 399. The probability, P, that the deviation of any single shot will be numerically less than a given quantity, a, being measured by the area between the ordinates of the probability curve at 2= dza^ and that curve being symmetrical to the axis of Y, we have Try Jo dz (223) 400. In order to avoid repeated integrations, the following table gives the value of P, calculated from the above equation, but arranged for convenient use with the ratio — as an argument. Knowing the mean deviation of a gun, y, to find the probability of a shot striking within a given distance of the mean point of impact, it is only necessary to take from the table the value of P which corresponds to — . It is to be noted that if a and y relate to lateral errors on a vertical screen, we get, by the use of this table, the probability that any one shot will strike between the two vertical lines on the screen distant a to the right and left, respectively, of the mean point of impact on the vertical screen ; that if a and y relate to vertical errors on a vertical screen, we get the probability that any one shot will strike between two horizontal lines distant a above or below the mean point of impact, respectively ; if a and y relate to the point of impact in the horizontal plane and to lateral deflections, we get the probability that any single shot will fall between two lines drawn on the surface of the water parallel to the horizontal trace of the vertical plane of the mean trajectory and distant a to the right and left, respectively, from the mean point of impact ; and if a and y relate to the point of impact in the horizontal plane and to deviations in range, we get the probability that any single shot will fall between two lines drawn on the surface of the water perpendicular to the horizontal trace of the vertical plane of the mean trajectory and a short of or beyond the mean point of impact. Each one of these four cases is of use under proper conditions. PROBABILITY OF A DEVIATION LESS THAN a IN TERMS OF THE RATIO — . a a a a P. P. P. P. 7 7 7 7 O.I .004 " 1.1 .620 2.1 .906 3.1 .987 0.2 .127 1.2 .662 2.2 .921 3.2 .990 0.3 .189 1.3 .700 2.3 .934 3.3 .992 0.4 .2.50 1.4 .735 2.4 .945 3.4 .994 0.5 .310 1.5 .768 2.5 .954 3.5 .995 0.6 .368 1.5 .798 2.6 .962 3.6 .996 0.7 .424 1.7 .825 2.7 .969 3./ .997 0.8 .477 1.8 .849 2.8 .974 3.8 .998 0.9 , .527 1.9 .870 2.9 .979 3.9 .998 I.O .575 2.0 .889 3.0 .983 4.0 .999 401. x\s an illustration of the use of the above table, we will find the probability of a deviation not exceeding 1 meter and 2 meters in the case of a gun whose mean lateral deviation is 1.07 meter, and will compare our results with those given by the actual firing of 50 shots from the Krupp 12-centimeter gun. Taking a = l meter, we have — = — -— ; =,935, and from the table P=.544. The probability that the lateral deviation will not exceed 1 meter is therefore .544; therefore of 50 shots 27 should fall within 1 meter on either side of the mean point of impact, and actually 27 did so fall. Taking a = 2 meters, we have -— — =1.87, whence P = .864, which is the 244 EXTERIOR BALLISTICS probability that the lateral deviation of any one shot will not exceed 2 meters. Therefore of 50 shots 43 should fall within 2 meters on either side of the mean point of impact, and actually 43 did so fall. 402. If P be the probability that the deviation of any single shot will not be greater than a, then evidently lOOP will be the probable number of shots out of 100 which will fall within the limits ±a; in other words, lOOP is the percentage of hits to be expected upon a band 2a wide with its center at the mean point of impact. Thus we see from the table that the half width of the band which will probably receive 25 per cent of the shot is 0.4y, while the half width of the band that will prob- ably receive 50 per cent of the shot is 0.846y. These facts are usually expressed by saying that the width of the 25 per cent rectangle is 0.80 and of the 50 per cent rectangle is 1.69 times the mean error. 403. The half width of the 50 per cent rectangle is known as the " probable error," or in our case the " probable deviation," since it is the error or deviation which is just as liable to be exceeded as it is not to be exceeded. 404. If we wish to find the probability of hitting an area whose width is 2& and whose height is 2h, since the lateral and vertical deviations are independent of each other, the probability is the product of the two values of P taken from the table with the arguments — and — - , where y« and vy are the mean lateral and mean vertical deviations, respectively. Thus, supposing jz to be 4 feet and jy to be 5 feet, the probability of hitting with a single shot a 20-foot square with its center at the mean point of impact is PiP2 = .954x .889 = .848, Pi = .954 being the value of P for -^ = ^ =2.5 and P. = -889 being the value of P for A = ^ =2. EXAMPLES. 1. The coordinates {z, y) of 10 hits made by a 6-pounder gun on a vertical target at 2000 yards range, axes at center of target, were as follows, in feet : (-10, +13) ( + 11, +9) (+4, -2) (-1, +1) (- 4, + 2) (+ 2, +1) (-1, -2) ( 0, -4) (- 1, - 3) (- 4, -4) Find the mean point of impact and the mean vertical and lateral deviations. Answers. Zq=—QA; ^^=+1.1; yj/ = 4.14; y~ = 3.72. 2. The coordinates of 8 hits made by a 28-centimeter gun on a vertical target at 4019 meters range, axes at center of target, were as follows, in centimeters : (-80, -90) (- 10, +210) ( + 30, -70) (-70, +355) ( + 30, +40) (-220, -150) (-40, +40) (-65, + 90) Find the mean j^oint of impact and the mean vertical and lateral deviations. Answers. z^i=—bd; ^^=+53; y^ = 123.9; yj = 55.7 ACCURACY AND PEOBABILITY OF GUN FIRE 245 3. The following ranges and lateral deflections from the plane of fire, in meters, were given by 10 shots from a 28-centinieter gun at 8° 30' elevation: Range. Deflection left. Range. Deflection left. 6285 18 6204 16 6228 21 6141 17 6187 15 6200 19 6187 13 6256 15 6192 17 6205 18 Find the mean point of impact, the mean lateral deviation and the mean deviation in range. Ansivers. Mean range 6208.5 meters ya; = 28.7 meters. Mean lateral deflection 16.8 meters y-= 1.8 meters. 4. A and B shoot alternately at a mark. If A can hit once in n trials and B once in n — 1 trials, show that their chances are equal for making the first hit. What are the odds in favor of B after A has missed the first shot? Answer, n to n — 2. 5. What is the probability of throwing an ace with a single die in two trials? Answer. 777^ . 3o 6. Taking the mean vertical error given from Example 1, and supposing the mean point of impact to be at the center of a vertical target, what would be the per- centage of hits on targets of unlimited width and of heights, respectively, of 8 feet, 12 feet, 16 feet, 20 feet and 24 feet? Anstvers. 55.9;i;; 75.1^; 87.6^; 94.6^; 97.9^. 7. Taking the mean errors given from Example 1, what percentage of shot would enter a gun port 4 feet square, supposing the mean point of impact to be at the center of the port? What would be the percentage if the port were 3 feet high by 5 feet wide? Answers. 9.9^; 9.2^. 8. What would be the probability of a single shot from the 28-centimeter gun of Example 2 hitting a turret 2 meters high and 8 meters in diameter at the range for which the mean errors are given, supposing the fire to be accurately regulated ? Answer. 0.48. 9. If a zone of a certain width receives 20^ of hits, how many times as wide is the zone which receives 80;^ of hits? Answer. 5.05 times. 10. At Bucharest, in 1886, 94 shots were fired from a Krupp 21-centimeter rifled mortar at a Gruson turret, distant 2510 meters, without hitting it. The mean devia- tions were 33.27 meters in range and 9.90 meters laterally, and the mean point of impact practically coincided with the center of the turret. What was the probability of hitting, supposing the target to have been a 6-meter square (it was really a circle of 6 meters diameter) ? Answer. 0.011. 11. How many of the 94 shots of Example 10 would probably have struck a rectangle 80 meters by 16 meters, with the longer axis in the plane of fire? Answer. '31.8i. CHAPTER 21. THE PROBABILITY OF HITTING WHEN THE MEAN POINT OF IMPACT IS NOT AT THE CENTER OF THE TARGET. THE MEAN ERRORS OF GUNS. THE EFFECT UPON THE TOTAL AMMUNITION SUPPLY OF EFFORTS TO SECURE A GIVEN NUMBER OF HITS UPON A GIVEN TARGET UNDER GIVEN CONDITIONS. SPOTTING SALVOS BY KEEPING A CERTAIN PRO- PORTIONATE NUMBER OF SHOTS AS " SHORTS." Mean point 405. In the preceding chapter we considered only the chance of hitting when not°It'?^nter the mean point of impact is at the center of the target, but this is far from being an of target. ^|.^^-j^g^|^jg Condition in the service use of guns, especially of naval guns. In fact to bring the mean point of impact upon the target is the main object to be attained in gunnery, for, from what has already been said, if the mean point of impact be brought into coincidence with the center of the target and kept there, we will get the maximum number of hits possible, and it is to the accomplishment of this that the spotter gives his efforts. Even with a stationary target, at a known range, however, it is difficult to so regulate the fire as to bring about and maintain this coincidence of center of target and mean point of impact; and when the target is moving with a speed and in a direction that are only approximately known ; when the range is not accurately known; when there is a wind blowing which may vary in force and direc- tion at different points between the gun and the target; when the density of the air may vary at different points between the gun and the target; and when the firing ship is also in motion, etc. ; even the most expert regulation of the fire by the observa- tion of successive points of fall can do no more than keep the mean point of impact in the neighborhood of the object attacked. All this applies to a single gun, and in salvo firing we have the additional trouble that the mean points of impact of the several guns cannot be brought into coincidence. This makes it necessary for the spotter to estimate the position of the mean point of impact of the whole salvo, that is, the mean position of the mean points of impact of all the guns, and it is this com- bined mean point of impact of all the guns that the spotter must determine in his own mind and endeavor to bring upon the target and keep there. The difficulties attending this process are manifest. 406. In Figure 40, let be the mean point of impact of a single gun, and let ABCD be the target at any moment, and let the coordinates of the center of ABCD, with reference to the horizontal and vertical axes through be z^ and ?/o ; also let the mean lateral and vertical deviations of the gun be y^ and jy, respectively, and let the dimensions of the target be 2& and 21i. Then the probability that a shot will fall between the vertical lines Cc and C'c' is the tabular value of P for the argument ?^i^ , which we will call Pz{z^ + h) ; and the probability that a shot will fall between ^^' z -b Dd and D'd' is the tabular value of P for the argument -^ , which we will call 7z P^(zo-h). Therefore the probability that a shot will fall between Cc- and Dd is one-half the difference of the two preceding probabilities, or ^[P,(z, + h)-P.iZo-h)] (224) Similarly, the probability that a shot will fall between the horizontal lines C'C and ^'i^is i[Pyiyo+h)-PAyo-h)] (225) ACCUEACY AND PEOBABILITY OF GUN FIEE 247 Hence the probability of hitting ABCD is the product of the two expressions given in (224) and (225), or i[P^{^^o + ^)-PAz,-h)-\x[PAu, + h)-Py{ij,-h)] (226) T c' 7>' 3' ^' I- ^. A' O \a.' -f d' Figure 40. 1 I ^ 7c I jr. Zh. B 407. To illustrate, suppose we wish to find the probable percentage of hits on a gun port 4 feet square, if the mean point of impact be 3 feet to one side of and 4 feet below the center of the port, the value of yz being 3.72 feet and of yj/ being 4.14 feet. Here we have : P,(2,+ &)=F,(5)=.717 P,(;,/„ + /0=P,(G)r:=.751 F,{z,-l) =P,il) =.Vm Pyiyo-h) =P,j{2) =: .298 Pz{o) -P.(l) =;547 P,(0) -P,(2) =.453 From which we have P = ^x .547 x .453 = .062 Therefore the percentage of hits under the given conditions would be 6.2 per cent. Under the same conditions, but with the mean point of impact at the center of the port, the percentage of hits would be 9.9 per cent. 408. From what has been said it is evident that the less the mean errors of the gun, the more important it becomes to accurately regulate the fire; for if the dis- tance of the mean point of impact from the target be more than three times the mean error of the gun we would get practically no hits at all. Therefore a reduction in the mean error of the gun renders imperative a corresponding reduction in the distance within which the spotter must keep the mean point of impact from the target if hits are to be made. Therefore, unless good control of the fire be secured,- a gun with a small mean error will make fewer hits than one with a larger mean error, and this has sometimes been used as an argument in favor of guns that do not Bearing: of mean errors upon fixe control. 248 EXTERIOR BALLISTICS shoot too closely. Conversely, however, if good control be secured — that is, if the spotter be competent and careful — the close-shooting gun will secure more hits than the other. Therefore the scientific method of securing hits is to have a competent spotter and a close-shooting gun; the other process is a discarding of science and knowledge and a falling back upon luck, which cannot but meet with disaster in the face of an enemy using proper and scientific methods. 409. To illustrate the statements contained in the preceding paragraph, we will take the case of a 6" gun firing at a turret 25 feet high by 32 feet in diameter, and 3000 yards distant. Suppose the»mean vertical and lateral deviations each to be 10 feet; if the mean point of impact coincides with the center of the target, the probable percentage of hits will be 54.3 per cent; but if the sights be set for a range of 10 per cent more or less than the true distance the mean point of impact will be raised or lowered about 43 feet (this is one of the older 6" guns; not the one given in the accompanying range tables), and the percentage of hits will be reduced to 0.7 per cent. If, on the other hand, the mean errors of the gun were each 20 feet, or double the first assumption, while the percentage of hits with perfect regulation of fire (that is, with the mean point of impact at the center of the target) would be reduced to 18.2 per cent, that with sight setting for a range 10 per cent in error would be 4.7 per cent. Thus we see that if the fire be not accurately regulated a gun will be severely handicapped by its own accuracy if the range be not known within 10 per cent. 410. The work for the problem in the preceding paragraph is as follows : Figure 41. Case 1. Mean deviation 10 feet. Mean point of impact at (Figure 41), a^ 16 10 L2J 10 = 1.6 :.798 2 =1^=1.25 P,, = .681 P,xPy = . 543438 Therefore the percentage of hits is 54.3 per cent. ACCUEACY AND PEOBABILITY OF GUN FIEE 249 Case 2. Mean deviation 10 feet. Mean point of impact at (Figure 42), Chances of hitting between A'D and B'C. P,(2o + 16)-P.(.-o-16)=F^(lG)-P.(-16) =1.596 P,(16)=.T98 .0 = ^- — =1.6 Chances of hitting between .45 and CD. Py{y, + 12.h) -P,(,y,-12.5) =P,(55.5) -P,(30.5) ^-43-^ =-^=5.55 P,(55.5) =1.0000 -^^ y^ 10 = .0168 i I .026208 .0065 ■^3 _ 30.5 10 = 3.05 P2,(30.5)= .9832 .0168 Percentage of hits is 0.7 of 1 per cent. Case 3. Mean deviation 20 feet. Mean point of impact at (Figure 41). P, = .477 Pj, = .382 «1 = 16 20 = 0.8 'yy = 12.5 20 = .62 5 P«xPj,= . 182214 Percentage of hits is 18.2 per cent. Case 4. Mean deviation 20 feet. Mean point of impact at (Figure 42). Y <-r-3Z 1 1 1 > 1 /N ZS' ' i i < /6' > 3' f3' A O A' Figure 42. Chances of hitting between A'D and B'C. P,(2o + 16)-P.(2o-16)=P^(lC)-P.(-16) = gj _ 1^ — 8 y. 20 • .954 250 EXTEEIOR BALLISTICS Chances of hitting between AB and CD. Py{y, + 12.5) -Py(y^-12.5) =Py(55.D) -P.,(30.5) = .1972 5 o^ 55.5 .. .^. .,..-_. ._.. 4 I .1881765 .^Q -2.775 Pj,(55.5) =.97275 ^-^ :^ = ^ = 1.525 P^(30.5) =.77550 Pj,(55.5) -Fj,(30.5) =.19725 Percentage of hits is 4.7 per cent. 411. If we know the percentage of hits at a given range on a target of given size, we can make a rough estimate of the mean errors of the gun by assuming that the mean point of impact was at the center of the target, and the greater the number of rounds fired the more nearly correct will this determination probably be. For example, on a certain occasion, the eighty 6" gun of certain British ships, firing separately, made 295 hits out of 650 rounds fired; that is, 45.4 per cent of hits ; on a target 15 feet high by 20 feet wide, at a mean range of 1500 yards. Here we have given that the product -^(f)x-»a=-- Assuming that yz = yy, we may solve the above by a process of trial and error, that is, by assuming successive integral values of y, and by this process we see that when y = 7 we have ^) X Py (Jy) = -^45 X .581 = .432 and as .432 is very nearly .454, we can say that the mean deviations are slightly less than 7 feet; and we could go on and determine the solution of the equation more accurately by trying 6.9 feet instead of 7 feet as the value of y. This would probably not make the result any nearer the truth, however, as any correction resulting there- from would probably be less than the error caused by the assumption that the mean point of impact was at the center of the target. 412. The number of roimds necessary to make at least one hit may be determined by the following method: Let p be the probability of hitting with a single shot; then 1 — ;^ is the probability that a single shot will miss; and (1 — ;j)" is the prob- ability tliat all of n shots will miss. Therefore the probability of hitting at least once with n shots is P=l— (1 — p)". Solving this equation for n, we get log(l— jP) =n log(l — p) log(l-p) and by giving P a value near unity we can find the value of n which will make one hit as nearly certain as we wish. 413. As an example, taking a case in which 94 shots were fired from a mortar at a turret, and in which the calculated probability of a hit with a single shot was .011, let us see how many rounds would have to be fired to make the probability of at least one hit .95. In that case, /j = .011, and so we have, from (227), _ log(l-.95) _ log .05 _ 8^69897_^10 _ -1.3Q1Q3 _^„. ^ log (1-. Oil) log .989 9.99520-10 -0.00480 Therefore 271 shots must be fired to make the odds 19 to 1 that there will be at least one hit. The probability of at least one hit with the 94 shots fired was P=l-(l-/;)'*^ = l-.354 = .646 ACCUEACY AXD PEOBABILITY OF GUN FIRE 251 414. Tlie deviations of the projectiles fired from a gun on a steady platform, the mean of which we will call the mean error of the gun, lateral or vertical as the case may be, are principally caused by : 1. Errors of the gun pointer in sighting the gun, but bear in mind that this does not include " mistakes," which are supposed to have been eliminated, but only accidental errors that must necessarily ensue even when the pointer is working as accurately as it is humanly possible to do. 2. An initial angular deviation of the projectile; that is, when the projectile does not leave the muzzle in a path in line with the axis of the gun. 3. Variations in initial velocity between successive rounds. 4. Differences between projectiles existing even after all possible differences have been eliminated. 415. With open sights the most expert gun pointers make a considerable angular error (this is an angular error in sighting, not to be confused with the angular deviation described in 2 of the preceding paragraph), which varies from round to round, when the gun is pointed by directing the line of sight at a target. With telescopic sights this error is greatly reduced but still exists. There is also always an error in setting the sights (we suppose the range to be unchanged from shot to shot, but that the sights are reset for each round). The mean angular error of sighting can only be estimated. 416. The initial angular deviation results from the projectile not leaving the gun in the exact line of the axis of the latter. This deviation, which occurs indifferently in all directions, was quite large with smooth-bore guns and with some of the earlier rifles, but with modern guns, using projectiles rotated by forced bands, it is un- doubtedly much less. 417. With all powders the muzzle velocity varies somewhat from round to round, no matter what care be taken to insure uniformity in the charges. With the nitro- cellulose powder now used in our navy, if the charges have been made up with proper care, and if the projectiles are all of the same weight, the average difference between the velocities given by successive rounds and the mean velocity of all the rounds fired on any one occasion will prol)ably not be great. If a large number of rounds be fired, and the velocity for no one round differs a f . s. from the average, then somewhat more than half the velocities will be within -;- f . s. of the average, a not o being large. 418. The projectiles of any gun differ among themselves, but when they all have the same form of head and are not of greatly different lengths, the resulting devia- tions are not so important as those caused by variations in the weight. 252 EXTERIOE BALLISTICS 419. Of course the only correct way of determining the mean errors of a given gun is by actually firing a large number of rounds at a target and measuring the deviations. That the errors are very small under favorable circumstances is illus- trated in Figure 43, vi^hich represents a target made at Meppen on June 1, 1882, with a 28-centimeter gun, the distance of the target from the gun being 2026 meters (2215 yards). The dotted cross is the mean point of impact, whose coordinates referred to the horizontal and vertical axes at the center of the target are 2 = 32.4 inches and y=—11.6 inches. The mean lateral deviation is 9.5 inches, and the mean vertical deviation is 11.6 inches. These actual deviations are considerably less than those encountered in service, which may be plausibly ascribed to the fact that in proving I I I I I I I 1 I I I I I I I ^:s^ >j 1 Figure 43. ground firings greater care can be taken in pointing than is usually practicable under service conditions. Effect of 420. The three angular motions of a ship's deck, caused by the rolling, pitching pitching and and yawing, greatly increase the actual mean errors of naval guns in service, but y&wing, , o -^ their effects depend so much upon the skill of the gun pointer, as well as upon the state of the sea and the characteristics of the particular ship and gun mounting, that only the roughest estimates of their values can be made. Many naval guns are mounted^ in broadside and only train from bow to quarter, and even those mounted on the midships line are likely to be most used in broadside; thus the roll, which is the greatest and most rapid of a ship's motions, has its largest component in the plane of fire, and acts principally to increase the vertical deviations. The principal effect of pitching, on the other hand, is to increase the lateral deviations by causing the plane of the sights to be more or less inclined, now to one side and now to the other of the plane of fire. Motion in azimuth, yawing, mostly due to unsteady steering, affects the lateral deviations only. 421. If the target be in motion, the person controlling the fire of the gun must of course estimate its speed and direction in order to direct the fire at the point where the target will be when the projectile strikes, and his corrections must always vary in accuracy from round to round, thus increasing both the lateral and ihe vertical deviations. Furthermore, variations in the accuracy of the estimated cor- rections for the speed of the firing ship and for the effect of the wind must occur Uotion of target. errors. ACCURACY AND PROBABILITY OF GUN" FIRE 253 from round to round, which will also affect the lateral and the vertical deviations. Moreover, the changing direction of the target will cause the angle between the direc- tion of the wind and the plane of fire to vary, thus necessitating a variable allowance for wind effect and again increasing the deviations of projectiles. 422. Taking everything into account, probably a fair estimate of the mean vertical deviation of modern naval guns of medium and large caliber, at 2000 yards range, with skilled fire-control personnel and gun pointers, and under average con- ditions, would be 5 feet. The mean lateral deviation, which for guns on steady platforms, is from two-thirds to three-fourths the mean vertical deviation, may be taken to be the same as the mean vertical deviation in the case of naval guns, without any great error. Both vertical and lateral deviations may be taken to be proportional to the range, at least up to 4000 or 5000 yards range, though the former really in- creases somewhat more rapidly than the range. For ranges greater than those given, the increase in the deviations will be at a greater rate. 423. It will be noticed that, in the earlier part of the discussion of the subject, inherent we referred to the " accidental deviations " of a gun as being due to the " inherent errors " of the gun, but we have now seen that there are " accidental errors " that are not really inherent in the gun itself, although their results are similar. The angular deviation resulting from the fact that the shell does not leave the gun exactly in the axis of the gun is strictly an inherent error of the gun ; but the angular error in pointing due to the fact that even the most perfectly trained and most skillful pointer cannot point twice exactly alike is an error that does not pertain to the gun itself but to its manipulation. The results, as stated, are similar, however, and may therefore be considered together, as making up the sum of the accidental errors that cause the deviations. To recapitulate, we have first deviations due to mistakes, which we eliminate from consideration. Then M^e have deviations resulting from known causes, which we also eliminate by the methods of exterior ballistics. When these two sources of error have been eliminated, we have remaining two sources of error, those pertaining to the imperfections of the gun itself and those pertaining to the inherent imperfections of even the most perfect personnel handling the gun. It is these last two only that may be considered under the theory of probability. And bear in mind the difference between a mistake and an accidental error. A mistake is the result of bad judgment, and may cause a large error, as, say, a mistaken estimate of two points in the direction of the wind; and an accidental error is that small error which must necessarily be made even by a thoroughly trained judgment. Accidental errors are necessarily small, and are necessarily as likely to occur on one side as on the other. 424. Although the targets of naval guns are generally vertical, the fire of such guns must, as a rule, be regulated by the observation of points of fall in the horizontal plane. The lateral deviations are practically the same whether measured on the vertical plane perpendicular to the line of fire, or on the horizontal plane, provided the error of the shot in range be not too great. The deviations in range, however, differ very greatly from the vertical deviations, the ratio between them being the cotangent of the angle of fall. 425. Since the mean deviation in range, jx, is related to the mean vertical devia- tion, yy, by the formula y3-=yyCotM, and since the angle of fall increases and its cotangent correspondingly decreases with increase of range at about the same rate as the mean vertical deviation, it will be seen that the mean deviation in range remains nearly the same for widely different ranges. Thus, for example, while the estimated mean vertical deviation of the 12" gun of 2800 f. s. initial velocity increases from 2.5 feet at 1000 yards range to 10.5 feet at 4000 yards range, the corresponding 254 EXTERIOR BALLISTICS deviation in range only changes from 119 yards to 104 yards; and while in the case of smaller guns the mean deviation in range decreases more rapidly, still the change is always very much less proportionately than the change in range itself. 426. The principal use of knowledge of the mean deviation in range is in the regulation of gun fire by observation of the points of fall. Suppose the axis of Z, in Figure 44, represents the water-line of the target, the axis of Z being the hori- zontal trace of the vertical plane of the mean trajectory, and let the distance from the axis of Z to the dotted lines aa, a' a', hh, h'b', cc, c'c', dd, d'd', etc., represent the mean deviation in range, y^, of the gun. Then if the point of impact be on the axis of Z, that is, on the water-line, half of all the shot will fall short ; if it be on a' a' the percent- X J a \--^ a c l-i c b \.Jl i, ^ yi- a' ^--y ^' ,' i.fi „' C \-ft c' d! ^_Zl a' Figure 44. age of shot that will fall short will be increased by the number which fall between the axis of Z and ala! , or, from the table of probabilities, it will be 50 H ^ = 79 per cent. If the mean range be still further short, so that the mean point of impact falls on h'h' , SS 9 the percentage of shorts will be 50 -f —^ — \- 94 per cent; and, finally, if the mean point of impact be three or more times the mean deviation short, then practically all the shot will fall short. The same reasoning shows that if no shot strike short of the axis of Z, the mean point of impact is three or more times the mean deviation in range beyond the axis of Z ; if about 6 per cent strike short, the mean point of impact is about twice the mean deviation in range beyond the axis of Z ; and if about 21 per cent are short, it is about the mean deviation in range beyond the axis of Z. Thus, by observation of the percentage of shot which strike short it is possible to determine with some degree of accuracy how much the setting of the sight in range should be increased or decreased to bring the mean point of imj^aet on the target. ACCURACY AND PROBABILITY OF GUN FIRE 255 427. Let us now suppose that we are going to fire salvos from a battery of 12" o-uns, for which 7 = 2900 f. s., w = S70 pounds and c = 0.61. Let us also assume that we have a vertical target 30 feet high and wide enough to eliminate the necessity for considering lateral deviations due to accidental errors. Let us take the mean errors of the gun in range, first as 40 yards, next as GO yards, and then again as 80 yards; and also that they are approximately the same at all ranges. Let us also assume that the mean point of impact is at the center of the water-line of the target, in which case, as we have already seen, 50 per cent of the shot will fall short. Let us also assume that the three mean deviations correspond to total deviations of 150, 200 and 300 yards, respectively. Now let us see what percentage of the shot in each salvo will probably hit, at a range of 7000 yards, at which range the danger space for a target 3U feet high is, by the range table, 180 yards. ■'l^- JO ^ I I Figure 45. 1. Mean dispersion in range 40 yards; maximum dispersion in range 150 yards (Figure 45), , 30 _ 1 ^^''"^ 540 -18 450 18 y = 25 feet That is, for a maximum dispersion of 150 yards, or 450 feet, no shot would pass more than 25 feet above the water-line of the target, and all shots that do not fall short would hit. Therefore, by our assumption, we would have 50 per cent of shorts and 50 per cent of hits. 3oy Figure 46. 2. ^lean dispersion in range of 60 yards; maximum dispersion in range of 200 yards (Figure 46). 600 18 y = 33^ feet 256 EXTEKIOR BALLISTICS Also the mean dispersion in range is 180 feet, therefore the mean vertical dis- persion is y, = 180tanto= 1^=10 feet ' 18 Our problem therefore becomes to find how many shot will pass between the top of the target and a line parallel to it and 3^ feet above it, knowing that 50 per cent of the shot fired will fall short, and the other 50 per cent will pass between the water- line of the target and a horizontal line 33^ feet above it. As we have already taken out the 50 per cent of the shot that fall short, the ^ disappears from the formula, and we have Pj,(33)-Pj,(30)=.009 33 y 7 = ^=3.3 Pj,(33)=.992 10 30 10 = 3.0 Py{30)=.d83 .009 That is, .9 of 1 per cent of the shot that do not fall short will pass over the top of the target, leaving 99.1 per cent of them as hits. Therefore, of the 100 per cent of shot fired, 50 per cent will fall short; 99.1 per cent of 50, or 49 per cent of them, will hit; and 1 per cent of them all will go over. 3. Mean dispersion in range 80 yards ; maximum dispersion in range 300 yards. 900 18^' 50 feet yy- ■240 x^^ 40 feet and in the same manner as in 2, to find the number of shot that will pass between the top of the target and a horizontal line 50 feet above the water-line, we have Pj,(50)-Pj,(30)=.035 fli _ 50x3 y y 40 30x3 40 = 3.75 = 2.25 Pj,(50)=.9975 P3,(30)=.9275 .07 Therefore 7 per cent will go over, and 93 per cent will hit out of the 50 per cent that do not fall short. Therefore we have that there will be 50 per cent of shorts, 46 per cent of hits and 4 per cent of overs. Proceeding similarly for other ranges, we can make up a table like the following : Mean point of impact at water-line. Mean dispersion in range of — Danger Range. Yds. 40 yards 60 yards. 80 yards. space. Yds. Percentage of — Percentage of — Percentage of — Shorts. Hits. Overs. Shorts. Hits. Overs. Shorts. Hits. Overs. 7000 50 50 50 49 1 50 46 4 180 10000 nO 49 1 50 42 8 50 35 15 108 13000 50 42 8 50 32 18 50 25 25 70 15000 50 36 14 50 27 23 50 21 29 55 18000 50 28 22 50 20 30 50 15 35 39 ACCTJEACY AND PEOBABILITY OF GUN FIRE 257 The above mean dispersions are less than have been experienced at recent target practices. The above chances of hitting are based only on vertical errors ; if the target be short they will be materially reduced by the lateral errors. 428. For the above problem let us now suppose that the mean point of impact had been at the center of the danger space, instead of at the water-line, and we had desired to tabulate the same data as before. Let us start with the range of 7000 yards, for which the danger space is 180 yards, and compute the results for a mean dispersion in range of 40 yards, corresponding to a total dispersion of 150 yards. A /^ Mea.Tr. Po/rxr 0f Jmpctcf Figure 47. All shot that fall less than 270 feet short of the mean point of impact are hits. For space between mean point of impact and target, A = |I^ = A:^2.25 P=.9275 Therefore, for those short of the mean point of impact, 92.75 per cent will hit and 7.25 per cent will fall short. But only 50 per cent of the total number of shot fired fall shot of the mean point of impact, therefore the above percentages become, of the total, Hits 46.375 per cent Shorts 3.625 per cent For the space between A and C, which is 270 feet, we know that any shot that falls between A and C hits, and any that falls beyond C is over. Therefore the work is the same as the above, and we have Hits 46.375 per cent Overs 3.625 per cent Therefore the total is Shorts 3.625 per cent Hits 92.750 per cent Overs 3.625 per cent 17 258 EXTERIOE BALLISTICS Working out similar data for the other ranges and dispersions gives us the following table : Mean point of impact at center of danger space. Mean dispersion in range of — Range. Danger Yds. 40 yards. 60 yards. 80 yards. space. Yds. Percentage of — Percentage of — Percentage of — Shorts. Hits. Overs, Shorts. Hits, Overs, Shorts. Hits. Overs. 7000 4 93 3 12 77 11 19 63 18 180 10000 14 72 14 24 53 23 30 40 30 108 13000 25 49 26 32 36 32 37 27 36 70 15000 29 42 29 36 29 35 40 21 39 55 18000 35 31 34 40 21 39 42 16 42 39 429. From the tables given in the two preceding paragraphs, if we assume the mean point of impact on the target, we see that, as the mean dispersion increases, the percentage of hits decreases very rapidly. 430. It will also be seen that, to get the greatest possible number of hits, a greater percentage of shorts is necessary at long ranges than at short ranges. 431. It will also be seen that, where the mean point of impact in range is at some distance from the target, an increase in dispersion gives an increase in the number of hits, which is in accord with the principles previously enunciated. It may be shown mathematically that the mean dispersion for maximum efficiency is equal to 80 per cent of the distance from the mean point of impact in range to the center of the danger space. 432. From what has been said, it will readily be seen that, in controlling the firing of salvos from a battery of similar guns, we desire to keep a certain proportion of the shot striking short of the target in order to get the maximum number of hits. There are also other good reasons for so keeping a number of the shot striking short. From what we have seen, we may determine certain general rules which will govern the spotter in thus controlling salvo firing. This question, however, is one that may more appropriately be considered at length in the study of another branch of gunnery, so there will be no further discussion of it in this book. 433. It will be interesting to compare the results of actual firing with the com- puted results in some one case, to see how closely the two agree, and to get some idea of the correctness for service purposes of percentages determined mathematically. Such results are given in the following table taken from Helie's well-known Traite de Balistique. It represents the results of about 500 shots fired at Gavre from a 16.5-centimeter rifle at various ang-les of elevation : Probability that the lateral deviations will not exceed — 7z 4 ■ 72 2 ■ 73- 27«. 37.. By table Bv firinc 0.158 0.176 0.310 0.300 0.575 0.592 0.889 0.885 0.9R3 0.988 ACCURACY AXD PEOBABILITY OF GUN FIEE 259 EXAMPLES. 1. Supposing a row of gun ports, each 4 feet high by 6 feet wide, are spaced 18 feet between centers; show by comparing the percentages of shot which would enter a port that it woukl be better to aim a gun whose mean vertical and lateral errors are each 5 feet at the center of a port than half way between two ports. Answe7\ 9.5 per cent to 7.0 per cent. 2. Compare the percentages of hits on ports in the two cases of Example 1 if the mean errors of the gun were 7.5 feet instead of 5 feet. Ansiver. 5.8 per cent to 5.8 per cent. 3. The 12" guns of a certain ship made 69 per cent of hits on a target 15 feet high by 20 feet wide at 1700 yards range. Supposing the mean vertical and lateral devia- tions to have been equal, what was their approximate value? Supposing through ignorance of the range the sights had been set for 1870 yards, thus raising the mean point of impact 9 feet, what would the percentage of hits have been? Anstvers. 5 feet; 35.7 per cent. 4. The mean errors, laterally and in range, of a rifled mortar are 3.5 yards and 53 yards, respectively, at a mean range of 3357 yards. What is the chance of hitting a ship's deck (taking its equivalent area to be a rectangle 300 feet by 60 feet) when she is end on; (1) if the mean point of impact be at one corner of the rectangle; (2) if it be at the center of the rectangle ? Answers. .217 ; .555. 5. What are the chances of hitting under the same circumstances as in Example 4, excepting that the ship is broadside on? Answers ..059; .120. 6. At a range at which the mean vertical error of the guns equals the freeboard of the enemy, what is the ratio of the respective probabilities of hitting her when you aim at her water-line and when you aim at her middle height, suppose the fire to be accurately regulated in each case ? What is the same ratio at a range at which the mean vertical error is only half the freeboard ? Ansivers. .288 to .310; .445 to .575. 7. A torpedo-boat steaming directly for a ship at 24 knots is discovered and fire is opened on her at 1500 yards range. If the probability of a single 3" shot striking her is .02, and there are eight 3" guns each firing 12 rounds a minute at her, what is the chance that she will be struck at least once before she is within 500 yards? What is the chance of her being hit at least twice? Answers. .911 ; .694. 8. At 4000 yards, a ship with 30 feet freeboard gives a danger space of 90 yards for the 3" gun (F = 2800 f. s.) and the mean error in range of the same gun at 4000 yards is 30 yards (estimated). How closely must the range of such a ship be known to make the probable percentage of hits as great as 0.5 per cent, supposing the guns to be pointed at the middle of the freeboard? Answer. 141 yards. 9. The mean error in range of the 12" gun (F = 2800 f, s.) at 4000 yards range is 100 yards (estimated), and its danger space for 30 feet freeboard is 300 yards. How closely must the range be known to make the probable percentage of hits as great as 0.5 per cent, supposing the guns to be pointed at the middle height of the freeboard? Answer. 470 yards. 260 EXTERIOE BALLISTICS 10. Fire is opened with eight 3" guns on a torpedo-boat coming head on when she is at 1500 yards range. She covers 100 yards every 7.5 seconds, and each gun fires once every 7.5 seconds. The mean lateral and vertical deviations are each 6 feet, and the target offered is 6 feet high by 15 feet wide. If an error of 100 yards in the sight setting displaces the mean point of impact 3 feet vertically, and the sights are all set for 1000 yards range, what is the probable number of hits while the boat advances to 500 yards range? Aiiswer. 10 + . 11. The turrets of a monitor steaming obliquely to the line of fire present a vertical target consisting of two rectangles, each 24 feet wide by 12 feet high, and 36 feet from center to center. If the mean errors of a gun be 12 yards laterally and 8 yards vertically, would it be better to aim at a turret or half way between them ? Answers. 1st case, P=.057; 2d case, F=:.0G1. 12. A gun has 30 shell, one of which, if landed in a certain gun position, would silence the gun contained therein. The gun pit is 10 yards in diameter, and the probability of hitting it with the gun in question is .05. What would be the prob- ability of silencing the gun, using all the ammunition? Ansiver. P = .7S5. 13. The 12" guns of a ship made 68 per cent of hits on a target 15 feet high by 20 feet wide at 1700 yards range. What was the probable value of the mean devia- tions, vertical and lateral ? Supposing the mean deviation to be proportional to the range, what percentage of hits would the same guns make on the same target at 3400 and at 5100 yards? A^isivers. 5 feet; 25.9 per cent; 12.6 per cent. 14. If the probability of hitting a target with a single shot is .05, what will be the probability of making at least two hits with 50 shots? Answer. .721. 15. What is the greatest value of the mean deviation of a gun consistent with a probability equal to .90 of its making at least one hit in a hundred shots on a gun port 2 feet wide by 4 feet high? Ansiver. 5.95 feet. 16. Compute the data for 10,000 yards contained in paragraph 427. 17. Compute the data for 13,000 yards contained in paragraph 427. 18. Compute the data for 15,000 yards contained in paragraph 427. 19. Compute the data for 18,000 yards contained in paragraph 427. 20. Compute the data for 10,000 yards contained in paragraph 428. 21. Compute the data for 13,000 yards contained in paragraph 428. 22. Compute the data for 15,000 yards contained in paragraph 428. 23. Compute the data for 18,000 yards contained in paragraph 428. APPENDIX A. FORMS TO BE EMPLOYED IN THE SOLUTION OF ;rHE PRINCIPAL EXAMPLES GIVEN IN THIS TEXT BOOK. NOTES. 1. In preparing these forms the problem taken has been the 8" gim (gun K in the tables) for which 7 = 2750 f. s., m; = 260 pounds, c = 0.61, generally for a range of 19,000 A-ards. More specific data is given at the head of each form, 2. In the problems under standard conditions, which should give the exact results contained in the range tables, it should be borne in mind that the latter are given, for the angle of departure to the nearest tenth of a minute, for the angle of fall to the nearest minute, for the time of flight to the nearest hundredth of a second, etc., only. Also that results given in the range tables are entered after the results of the computations have been plotted as a curve, and the faired results are those contained in the tables. Small discrepancies between the computed results and those given in the tables may therefore sometimes be expected. 3. Also, results obtained by direct computation are of course more accurate than those obtained by taking multiples of quantities given in the range tables, and small discrepancies may be expected in some such cases between computed results and those taken from the range tables. Form No. Chapter Example No. No. 1 8 7 INDEX TO FORMS IN" APPENDIX A. Nature of Example. For computing 0, etc., for a given range, by Ingalls' method of successive approximations. 2 8 8 For computing (p, etc., for a given range, by Ingalls' method, knowing /. 3 8 9 For computing ((>, etc., for a given range, by Alger's method of successive approximations. 4 9 1 For computing the elements of the vertex for a given R and 4>, by successive approximations. 5 9 2 For computing the elements of the vertex for a given R and 0, knowing /. 6 9 3 For deriving special formulae for y and tan 6 for a given tra- jectory. . 7 9 4 For computing values of y and 6 for given abscissae in a given trajectory. 8 10 1 For computing R, etc., for a given 4>, by successive approxima- tions. 9 10 2 For computing R, etc., for a given <}>, knowing /. I 11 1 For computing 0, ^, 0, t and v for an elevated or depressed target. 11 12 2 For computing change in range resulting from a variation in initial velocity. 12 12 3 For computing change in range resulting from a variation in atmospheric density. 13 12 4 For computing change in range resulting from a variation in weight of projectile; method by direct computation 14 12 5 For computing change in range resulting from a variation in weight of projectile; method by using Columns 10 and 12 of range table. 15 12 6 For computing change in position of point of impact in vertical plane through target resulting from a variation in the setting of the sight in range. For computing the drift at a given range. For computing sight-bar height and set of sliding leaf for a given range and deflection. For computing the effect of wind. For computing the effect of motion of the gun. For computing the effect of motion of the target. For computing the penetration of armor. For range table comnutations, 0, etc. For range table computations. For range table computations. For wind and speed problems. Real wind. For wind and speed problems. Apparent wind. 1 and 2 For the calibration of a single gun. For the calibration of a ship's battery. 16 13 1 17 13 2 18 14 1 19 14 2 20 14 3 21 - 16 — 22 16 2 23 16 3 24 16 4 25A'I 25B J 17 8 26 17 9 27 18 1 28 19 1 APPEN^DICES 263 Form No. 1. CHAPTER S— EXAMPLE 7. FORM FOR THE COMPUTATION OF THE DATA CONTAINED IN COLUMNS 2, 3, 4 AND 5 OF THE RANGE TABLES; THAT IS, FOR THE VALUES OF THE ANGLE OF DEPARTURE (4>), ANGLE OF FALL (o.), TIME OF FLIGHT (7") AND STRIKING VELOCITY (v^,) FOR A GIVEN RANGE, CORRECTING FOR ALTITUDE BY A SERIES OF SUCCESSIVE APPROXIMATIONS, THE ATMOSPHERE BEING CONSIDERED AS OF STANDARD DENSITY— INGALLS' METHOD. FORMULAE. C,=K= ^ ,Z= ^; sin 2cf> = AC; Y-A"C tan (^; loglog /=log F + 5.01765-10; tan ti = B' tan <{>; T = CT' sec4>; Vo] = Uo, cos <^ sec w PEOBLEM. Cal. = 8"; 7 = 2750 f. s.; «' = 260 pounds; c = 0.61; Eange = 19,000 yds. = 57,000 ft. C, = K= (from Table VI) colog 9.17654-10 A' = 57000 log 4.75587 Zi = 8558.75 log 3.93241 ^, = .08673+ -QQ^^^^y^-^^ - -'''^l^'' -.084673 (from Table II) 4i = .084673 log 8.92775-10 . C^= log 0.8234G 2<^i = 34° 19' 36" sin 9.75121-10 01 = 17° 09' 48" (first approximation, disregarding /) A "->948- ^ X (--006 4) X71 50x0 .000273x71 .g^^. .^ , , jj. A^ _^J48- ^^^ X ^Q3^ + ^^^ + _QQ33^ _d0.7.5 (iable 11) ^/' = 3027.5 log 3.48109 C^= log 0.82346 c/), = 17° 09' 48" tan 9.48974-10 J\= log 3.79429 Constant log 5.01765-10 /i= log 0.06485 loglog 8.81194-10 C^= log 0.82346 Co= log 0.88831 colog 9.11169-10 X = 57U00 ■ log 4.75587 ^2 = 7371.5 log 3.86756 i, = .06520+^^0m^5 - -00^^9^^X50 ^03333,3 yl2 = . 0638876 log 8.80542-10 C,= locr 0.88831 202 = 29° 36' 14" sin 9.69373-10 02 = 14° 48' 07" (second approximation) A" o.jQu 50 ,, (-.0046) x67 , 50x0 , .001488x67 ^.^^^ A, =2398- ^0 X -^^^ + "loo- + .0035 =^^^^-^ 264 APPEXDICES 42" = 2499.5 log 3.39786 0^= log 0.88831 , = U° 48' 07" tan 9.42201-10 Y.= log 3.70818 Constant log 5.01765-10 /,= log 0.05319 loglog 8.72583-10 C^= log 0.82346 C^= log 0.87665 colog 9.12335-10 Z = 57000 ■ log 4.75587 ^3 = 7572.15 log 3.87922 A3 = .06851+ '^^^'^^^J^-^^ _ -.YY^f^-^^ =.067137 .001 70x72.15 _ .00520X 50 100 100 43 = .067137 log 8.82696-10 C,= log 0.87665 2(/)3 = 30° 21' 22" sin 9.70361-10 (^3 = 15° 10' 41" (third approximation) A" 9i«- 50 ,, (-.0048) x68 , 50x0 ^ .002237x68 _.;,^Q-, ., As =2460- ^ X ^0025 + -Tor + ,0025 "^^^^-^ ^3" = 2591.1 .log 3.41349 C^= log 0.87665 (^3 = 15° 10' 41" tan 9.43342-10 Y^= log 3.72356 Constant log 5.01765-10 f^= log 0.05511 loglog 8.74121-10 C^= log 0.82346 C^= log 0.87857 colog 9.12143-10 X = 57000 log 4.75587 Z, = 7538.75 log 3.87730 A - 06851+ 'Q'^^QX^^-^^ - •QQ^^.^.X^Q:^ .066569 ^*--"^° ^ 100 100 ^, = .066569 log 8.82327-10 C^= log 0.87857 2<^, = 30° 13' 10" sin 9.70184-10 ^4 = 15° 06' 35" (fourth approximation) .„ ^,_ 50 ,, (-.0048) X68 , 50X0 .001669x68 _ 4. =3465-— X ^0025 + "lor + .0025 —575.6768 A/' = 2575.7 log 3.41090 C^= log 0.87857 (^,'=15° 06' 35" tan 9.43137-10 Y^= log 3.72084 Constant log 5.01765^1 f^= log 0.05476 loglog 8.73849-10 C,= log 0.82346 ■ ^5= log 0.87822 colog 9.12178-10 Z = 57000 log 4.75587 ^5 = 7544.85 log 3.87765 A n^e^i L .00170x44.85 .00520x50 nrrr-?^^ ^, = .06851+. ^^ ^^^^-~ =.0bbb,245 APPEJ^DICES 2G5 A, = MG672-i5 log 8.82395-10 C.= W 0.87822 2, = 30° 14' 42" sin 9.70217-10 , = lo° 07' 27" (fifth approximation) J "_.2ifi=i 50 ^ ( -.0048) X 68 50x0 ^ .001773x68 ...^ ^ ^^ -^^^'- Too >< M25 + -loT + — ^025— =^^^^-^ ^." = 2578.5 log 3.41137 C'5= log 0.87822 , = 15° 07' 27" tan 9.43175-10 i\= '. log 3.72134 Constant los 5.01765-10 h= log 0.05483 loglog 8.73899-10 C^= log 0.82346 C^= log 0.87829 colog 9.12171-10 Z = 57000 • loff 4.75587 Ze = 7543.65 log 3.87758 i n^Q-i , .00170x43.65 .00520x50 n^rr-o /l, = .068ol+ — -^-^^— =.0666o2 ^Q = .066652 log 8.82381-10 C«= loff 0.87829 208 = 30° 14' 21" sin 9.70210-10 (/)g = 15° 07' 10" (sixth approximation) i " oiaK 50 ^ (-.0048) x 68 , 50x0 . .001752x68 ^.^^^ 4e" = 2577.9 .log 3.41126 C^= log 0.87829 <^6 = 15° 07' 10" tan 9.43166-10 1\= log 3.72121 Constant los 5.01765 - 10 f,= log 0.05481 loglog 8.73886-10 C\= log 0.82346 C,= log 0.87827 colog 9.12173-10 X = 57000 loff 4.75587 Z, = 7544.0 log 3.87760 J nrQ-1^ -00170 X44 .00520x50 nrrfi=;Q A, = M^ol + —^^ --^00 = -0^66o8 ^, = .066658 log 8.82386-10 C,= log 0.87837 2(^, = .10° 14' 30" sin 9.70213-10 <;(). = 15° 07' 15" (seventh approximation) J" OAc- 50 ^ (-.0048) X 68 ^ 50x0 ^ .001758x68 „.^q , ^^ =^^^'-Wq'' :0025 + -T00- + :0025— =^^^^-^ 266 APPENDICES A/' = 2578.1 log 3.41130 C,= log 0.87827 l = 15° 07' 15" tan 9.43170-10 Y,= log 3.72127 Constant log 5.01765 - 10 /,= log 0.05481 loglog 8.73892-10 Ci= log 0.82346 Cs = log 0.87827 As C^ = Ct, the limit of accuracy has been reached and the work of approxima- tion can be carried no further. By the preceding work we have derived the following data for the remainder of the problem : = 15° 07' 15" Z = 7544.0 log (7 = 0.87827 From Table II, with the above value of Z, log5' = .2652+ :^0^4 ^ .002^6^x50 ^.g,^^ r = 4.600+ :^^ _ -1^3X50 ^^g,^, «^=1086-^^ + ^^^ =1097.0 100 100 B'= log 0.26751 C= log 0.87827 (^ = 15° 07' 15" tan 9.43170-10. .sec 0.01530 cos 9.98470-10 r' = 4.554 log 0.65839 Wa, = 1097 log 3.04021 (0 = 26° 34' 40" tan 9.69921-10 sec 0.04850 7 = 35.642 lo2f 1.55196 t;„ = 1184.2 log 3.07341 EESULTS. By above computations. As given in range table. <^ 15° 07' 15" 15° 07' 00" oy 26° 34' 50" 26° 35' 00" T 35.642 seconds 35.64 seconds Vui 1184.2 foot-seconds 1184 foot-seconds Note to Form No. 1. — The number of approximations necessary to secure correct results increases with the range, therefore problems for shorter ranges will not involve so much labor as the one worked out on this form. APPENDICES 2Gr Form No. 2. CHAPTEE 8— EXAMPLE 8. FORM FOR THE COMPUTATION OF THE VALUES OF THE ANGLE OF DEPART- URE (0), ANGLE OF FALL (o.), TIME OF FLIGHT (T) AND STRIKING VELOCITY (i^^) FOR A GIVEN RANGE, MAXIMUM ORDINATE AND ATMOSPHERIC CONDITION. FORMULA. C= -^^^' ^—~(T'> sin2<^ = AC; tan w = 5' tan ^ ; T = CT' sec (f>; r^^w^ cos c^ sec w PEOBLEM. Cal. = 8"; F = 2750 f. s.; w = 2G0 pounds; c = O.CA; Eange = 19,000 yards = 57,000 feet; Barometer = 28.33"; Thermometer = 82.7° F.; Maximum ordinate = 5261 feet. From Table III, 8 = .9139G; |F = 3507', hence /= 1.0962 from Table V. K= (from Table VI) log 0.82346 /=1.0962 '.. log 0.03989 8 = .91396 log 9.96093-10. .colog 0.03907 C= log 0.90242 colog 9.09758-10 Z = 57000 log 4.75587 Z = 7136.0 log 3.85345 From Table XL ^^^^^^^^_^ ^00158x36 _ .00475x50 ^^^^^^^^ log 5' = .2551+ ^^^^ + --^0^X50 ^_,.,,^ rp, , OOP , .089x36 .163x50 _, ^qq^ T =4.238+ -^^^ ^^^— -4.1885 110 1 10x36 , 34x50 ..^r/A 268 APPENDICES 4 = .060204 log 8.77963-10 B'= log 0.25677 r = 4.1885 log 0.62206 w^= 1137.4 log 3.05591 C= W 0.90242 loir 0.90242 2(^ = 28° 44' 38" sin 9.68205-10 3 = 36° 04' 46" sin 8.77004-10 <^3 = 18° 02' 23" (third approximation) sec 0.02190 Ts= log 1.59694 Ts-= log 3.19388 g = 32.2 log 1.50786 8 colog 9.09691 - 10 ^3 = 6290.0 log 3.79865 fF3 = 4193.3, hence /a^ 1.1168 /3 = 1.1168 log 0.04797 Ci= log 0.76503 C\= log 0.81300 colog 9.18700-10 Z = 57000 log 4.75587 AS^= 8767.4 log 3.94287 Sv= 2565.2 4„^ = 1315.96 r„^=:6.592 ^„^ = l"S^ ^.==J00^ Ty = 0M9_ i{l= 997.6 A.44 = 1215.73 Ar, = 5.773 A.44 = 1215.73 log 3.08483 A*S', = 8767.4 log 3.94287 f Subtractive. 4-^^i =.13866 log 9.14196-10 A04 /f = . 04832 -Mi-/ =.09034 log 8.95588-10 Ar, = 5.773 log 0.76140 C^= log 0.81300 log 0.81300 2<^, = 35° 58' 04" sin 9.76888-10 4 = 17° 59' 32" (fourth approximation) sec 0.02175 T^= log 1.59615 T,^= log 3.19230 ^ = 32.2 log 1.50786 8 colog 9.09691 - 10 F, = 6267.2 log 3.79707 §7^ = 4178.1, hence /4 = 1.1163 372 APPENDICES /4 = 1.1163 log 0.04778 C^= log 0.76503 ^5= log 0.81281 colog 9.18719-10 Z:=57000 log 4.75587 ^8,= 8771.2 log 3.94306 ^F= 2565.2 A«^=1316.97 T„^ = 6.596 ^„„ = 1133674 Ar=J00^3 Tf^O-819 u^-= 997.4 AA5 = 1216.74 AT, = 5.777 A45 = 1216.7 log 3.08518 ^ A^, = 8771.2 log 3.94306 jSubtractive. ^^ = .13871 log 9.14212-10 /f = . 04832 -^^-/^ = .09039 log 8.95612-10 Ar5 = 5.777 log 0.76170 C^= log 0.81281 log 0.81281 2.^5 = 35° 58' 20" sin 9.76893-10 <^. = 17° 59' 10" (fifth approximation) sec 0.02176 T^= log 1.59627 T^-= log 3.19254 ^ = 32.2 log 1.50786 8 colog 9.09691-10 75 = 6270.6 log 3.79731 §75 = 4180.4, hence /3 = 1.1164 /5 = 1.1164 log 0.04782 Ci= log 0.76503 C^- log 0.81285 colog 9.18715-10 Z= 57000 W 4.75587 A»S6= 8770.4 log 3.94302 Sv= 2565.2 A„^ = 1316.97 r„^ = 6.596 ^„„=Ii^^6 AU^m^ r;=o^ wl= 997.4 A/l6 = 1316.74 AT8 = 5.777 APPENDICES 273 A.4« = 1216.7 loff 3.08518 A.4. a5« = 8769.5 log 3.94302 rSubtractive. —=^=.13873 log 9.14216-10 /r = . 04832 M6._7^^.09041 log 8.95622-10 A/bg Ar6 = 5.777 log 0.76170 Cr= loff 0.81285 log 0.81285 2<^6 = 35° 59' 10" sin 9.76907-10 (^6 = 17° 59' 35" (sixth approximation) sec 0.02177 Tg= log 1.59632 T^-= log 3.19264 g = 32.2 log 1.50786 8 colog 9.09691-10 F6 = 6272.0 log 3.79741 fF6 = 4181.3, hence /e^ 1.1164 /e = 1.1164 log 0.04782 Ci= log 0.76503 C,= log 0.81285 We see that Ct = Cq. Therefore further work will be simply a repetition of the last two approximations, and the limit of accuracy has been reached. IS 274 APPENDICES From the preceding work, therefore, we have the following data for the remainder of the problem : Wa,=997.4 ^=:.13873 Ar = 5.777 log (7 = 0.81285 -=^=.13873 Z„, =.31477 From Table I. 77 = . 04832 -^=.13873 ■|^-/. = .09041 7„„- ^=.17604 -^ -/;. = . 09 041 log 8.95622-10 /„^- ^ = .17604 log 9.24561-10 Ar=5.777 log 0.76170 C= log 0.81285 log 0.81285 log 0.81285 2 colog 9.69897-10 Wc. = 997.45 W 2.99887 2<^ = 35° 59'10" ..sin 9.76907-10 = 17° 59' 35" 2 sec 0.04354 sec 0.02177.. cos 9.97823-l( (0 = 32° 18' 28" tan 9.80097-10 sec 0.07303 r=39.4745 log 1.59632 i;„ = 1122.4 log 3.05015 EESULTS. <^ 17° 59' 35". (o 32° 18' 28". T 39.4745 seconds. Vo) 1122.4 foot-seconds. Note to Form No. 3. — The number of approximations necessary to secure correct results increases with the range, therefore problems for shorter ranges will not involve as much labor as the one worked out on this form. APPENDICES 275 Form No. 4. CHAPTER 9— EXAMPLE 1. FORM FOR THE COMPUTATION OF THE ELEMENTS OF THE VERTEX FOR A GIVEN ANGLE OF DEPARTURE (c/>) AND GIVEN ATMOSPHERIC DENSITY, CORRECTING FOR ALTITUDE BY A SERIES OF SUCCESSIVE APPROXIMATIONS. FORMULA. loglog/ = log F + 5.01T65 — 10; Xo = Czq; 1^^ = 01^' sec <^; 1^0 = ^0 cos ^ PROBLEM. Cal. = S"; F = 2750 f. s.; it; = 260 pounds; c=0.61; Range = 19,000 yards; = 30° 14' 00" sin 9.70202-10 00/ = .0641385 . log 8.80712 - 10 ^ "-o.^oQ 50 ^ (-.0046) X 67 , 50x0 , .0017385x67 ^..-^^ o ^3 --Jb-— X -^^^ ^-^OT"^ :0025 ^-^^^-^ 276 APPENDICES A," = 2506.2 log 3.39901 C,= log 0.89490 <;!> = 15° 07' 00" tan 9.43158-10 Y,= log 3.72549 Constant loo- 5.01765 - 10 f.= log 0.05535 loglog 8.74314-10 C,= 10^0.83963 C^= log 0.89493 colog 9.10502-10 2(^ = 30° 14' 00" sin 9.70202-10 f7o/=.06417 log 8.80704-10 A "-oqqs 50 ^ (-.0046)x67 , 50x0 , .0Q1727X 67 _o^^. ^ A, -^39«-^x -^^^ +-ior+ :oo25~-^^"''-^ A4"=:2505.9 log 3.39896 C^= log 0.89498 <;i, = 15° 07' 00" tan 9.43158-10 r, = 5315.25 log 3.72552 Constant log 5.01765 — 10 f,= log 0.05536 loglog 8.74317-10 C,= log 0.83963 Cr= log 0.89499 colog 9.10501-10 2.^ = 30° 14' 00" sin 9.70202-10 aj= .0641256 log 8.80703 - 10 A" o-^QQ 50 ,, (-.0046)x67 , 50x0 .001725 6 X 67 _o^ A, -2398- ^^- X -^^ + -j^ + —^0025 -^..0o.9 A^" = Ai", therefore the limit of accuracy has been reached, and we have for the data for the remainder of the problem : a ' = .0641256 102 = 0.89499 ,^ = 4100+ i^, r.0017256- 50x_(^,0046l = 4261.0 .0025 L 100 ,, onnp , .063X61 .079x50 _^^..,t, t, =2.0o6+-j^^ ^-—— 2.0o49 ,.Q_ 21x61 , 66x50 ..-,_ ^0 = 1^90 ^^ + -^^ = 1GU.2 C= log 0.89499 log 0.89499 <;i. = 15° 07' 00" sec 0.01529 cos 9.98471-10 2o = 4261 log 3.62951 V = 2.0349 log 0.30854 M, = 1615.2 log 3.20822 a:, = 33458 log 4.52450 L = 16.551 loir 1.21882 i'o = 1559.3 log 3.19293 EESULTS. Xo 11152.7 yards. tj^ = Y 5315.25 feet. t^ 16.551 seconds. r„ 1559.3 foot-seconds. Note to Form No. 4. — The number of approximations necessary to secure correct results increases with the range, therefore problems for shorter ranges will not involve as much labor as the one worked out on this form. APPENDICES 277 Form No. 5. CHAPTER 9— EXAMPLE 2. FORM FOR THE COMPUTATION OF THE ELEMENTS OF THE VERTEX FOR A GIVEN ANGLE OF DEPARTURE AND GIVEN ATMOSPHERIC DENSITY, GIVEN ALSO THE MEAN HEIGHT OF FLIGHT FROM WHICH TO CORRECT FOR ALTITUDE. FOEMUL.E. C = ^ K; %' =A = ^^^^ ; yo = Y = A"C tan ; Xo = Czo ; t^ = CtJ sec cf> ; v^ = u^ cos PEOBLEM. Cal. = 8",; F = 2750 1 s.; iv = 260 pounds; c = 0.61; Eange=:19,000 yards; = lo° 07' 00" tan 9.43158-10 ^-^=4.0535 log 0.60783 RESULTS. ?/ = 4.0535 (.066643- a) .-c tan ^ = 4.0535(.066643-a') Note to Form No. 6. — To determine the above equations with accuracy for any given trajectory in air, the value of log C must be determined by the process of approximation given on Form No. 1, for the range for which the special equations are desired. This value of log C must then be used as was done in the above problem. An approximate result may be obtained by determining the value of / by means of the maximum ordinate given in the range table, from which the value of E may be approximately corrected for altitude. APPENDICES 279 Form No. 7. CHAPTER 9— EXAMPLE 4. FORM FOR THE COMPUTATION, FOR ANY GIVEN TRAJECTORY, OF THE ABSCISSA AND ORDINATE OF THE VERTEX AND OF THE ORDINATE AND OF THE ANGLE OF INCLINATION OF THE CURVE TO THE HORIZONTAL AT ANY POINT OF THE TRAJECTORY WHOSE ABSCISSA IS KNOWN, HAVING GIVEN THE SPECIAL EQUATIONS FOR / AND TAN 6 FOR THE GIVEN TRAJECTORY. FOEMUL.^. y, = Y = A"Ctan; x, = Cz,; y= ^ (A-a)x; tan ^= *^ (A-a') PEOBLEM. Cal. = 8"; 7 = 2750 f. s.; w = 260 pounds; c = 0.61; Eange = 19,000 yards = 57,000 feet; log (7 = 0.87827; «/) = 15° 07' 00"; ?/ = 4.0535(.0G6643-a)a:; tan ^ = 4.0535 (.066643 -a') ; a-^^SOOO yards = 24,000 feet; a;2=: 16,000 yards = 42,000 feet. For Vertex: From data, /I = .066643, and for vertex ao=A, therefore, from Table II for ao' = -066643. A" oiPK 50 ,, ( -.0048) X 68 , 50x0 , .001743 X 68 _o^^^ ^ ^ =^^*^^- Too ^ .0025 + -T00~ + .0025 '^^^^'^ Zo = 4200 + 001713 (--0018) X50 ■^^^^^'^ ■ 100 = 4365.7 .0025 A" = 2577.7 log 3.41123 (7= log 0.87827 log 0.87827 <^ = 15° 07' 00" tan 9.43158-10 2n = 4365.7 log 3.64005 a;o = 32,985 feet= 10,995 yards log 4.51832 ^„ = F = 5261.1 feet log 3.72108 For x^ = 8000 yards = 24,000 feet : C= colog 9.12173-10 a;i = 24000 log 4.38021 2 = 3176.4 log 3.50194 .,^^, , .00075x76.4 .00131x50 ni779Q a=.01.81+ ^^ j^3^^— =.017728 ,.^.0,08+. -0019X76.4 _ :003 1^x50 ^_,^,,02 A = .066643 A = .066643 a=. 017728 a' = .040702 A-a=.048915 log S. 68945-10 yi-a' = . 025941 log 8.41399-10 ^^^ =4.0535 log 0.60783 log 0.60783 A O G a:i = 24000 log 4.38021 2/1 = 4758.7 log 3.67749 '=6° 02' 10" tan 9.02182-10 280 APPENDICES For jr. = 16,000 yards = 48,000 feet : C= colog 9.12173-10 a;, = 48000 log 4.68134 2 = 6352.8 log 3.80297 ^ ^^^^^ , .00137X52.8 a = .05030+ ^^^ _. 00386 X 50 ^Q^9Qg3 , iQ^Q, .0044x52.8 a =.1358+ ^^^ - .0105X50 ^.^3,g.3 ^ = .066643 A = .066643 a=. 049093 a' = .132873 A-a=.017550 log 8.24428-10 A-a'=(-).066230 (-)log 8.82105-10 ^^^ =4.0535 log 0.60783 log 0.60783 a;, = 48000 loir 4.68124 ?/o = 3414.7 log 3.53335 ^2=(-)15° or 39" (-)taii 9.42888-10 For point of fall, a: = Z= 19,000 yards = 57,000 feet: C= colog 9.12173-10 a;= 57000 log 4.75587 2 = 7544.0 log 3.87760 a= .06851+ -Q^Yo^^^ - ^^^^^^ = .066658 a'= .1946+ :^255X44 _ .0140 x50^^^,,,,q A= .066643 A= .066643 a= .066658 a'= .190020 l-a=(-).000015 (-)log 5.17609-10 A -a'=(-). 123377 ". (-)log 9.09125-10 ^~^ =4.0535 log 0.60783 .' . . .log 0.60783 a;=57000 W 4.75587 ?/„=(-)3.4657 feet (-)log 0.53979 ^„=(-)26° 34' 15" (-)tan 9.69908-10 RESULTS. For vertex. For a?i=:8000 yards. For a-;Z= 16,000 yards. For point of fall. 0:0 = 10,995 yards 7/^ = 4758.7 feet ^, = 3414.7 feet ?/u,= (- )3.4657 feet ?/o=F= 5261.1 feet ^i = 6°02'10" ^2= ( - )15° 01' 39" ^a,= (- )26° 34' 15" Note to Form No. 7. — In the above problem of course the ordinate at the point of fall should be zero. The angle 6 at the point of fall should equal — w; for which the work gives 6= ( — )26° 34' 15", and the range table gives w = 26° 35' 00". These comparisons give an idea of the degree of accuracy of the above method. APPENDICES 281 Form No. S. CHAPTER 10— EXAMPLE 1. FORM FOR THE COMPUTATION OF THE VALUES OF THE RANGE (R), ANGLE OF FALL (io), TIME OF FLIGHT (T) AND STRIKING VELOCITY M FOR A GIVEN ANGLE OF DEPARTURE () AND ATMOSPHERIC CONDITION, CORRECTING FOR ALTITUDE BY A SERIES OF SUCCESSIVE APPROXI- MATIONS. FOEMUL^. C= -t K; ao' = /l = ^HL^ ; X = CZ; Y=A"C tan cf> ; \og\og f = \og r + 5.01765 -10; tan (x) = B' tan cf>; T = C T' sec ; v^ = Wc^ cos <^ sec w PROBLEM. Cal. = S"; 7 = 2750 f. s.; w = 260 pounds; c = 0.61; <^ = 15° 07' 00"; Barometer =29.00"; Thermometer = 82° F. K= log 0.82346 S = .937 loir 9.97174-10. .coloff 0.02826 C\= log 0.85172 colog 9.14828-10 2'-orm 50 ^ (-.0051) X 69 , 50x0 , .0008435 X 69 _..Q^;, A, -.601- ^ X -^^ + -j^ + -^^ -2687.7 4/' = 2687.7 log 3.42938 C^= log 0.85172 d> = 15° 07' 00" tan 9.43158-10 1\= log 3.71268 Constant locv 5.01765-10 /,= log 0.05374 loglog 8.73033-10 C\= log 0.85172 C„= log 0.90546 colog 9.09454-10 2 = 15° 07' 00" tan 9.43158-10 ^3= log 3.72968 Constant log 5.01765-10 /3= log 0.05589 loglog 8.74733-10 Ci= log 0.85172 C^= log 0.90761 colog 9.09239-10 24> = 30° 14' 00" sin 9.70202-10 ao/=.0622885 log 8.79441-10 A "_oQQi 50 ^ (-.0045) X67 , 50x0 , .0022885x67 _„.-^ ^ "^^ —^"^1- 100 ^ ^024 + ^^r + :0024 — ^^^-^ .4/' = 2457.7 log 3.39053 C^= log 0.90761 = 15° 07' 00" tan 9.43158-10 1\= log 3.72972 Constant log 5.01765-10 /,= log 0.05589 loglog 8.74737-10 Ci= log 0.85172 C,= log 0.90761 which equals log C^ ; therefore the limit of approximation has been reached, and we have the following data : A = .0622885 log (7 = 0.90761 <^ = 15° 07' 00" ■ ^ Z = 7100+ -^[.002785- 50x (-J M75)j ^^,g^ ^ log5' = .2578+ :0025X67^ ^ -0017x50 ^,,^3, rr'_A Q9.V I .090X67.9 .165x50 _, ^^>^ ^ -^'^^^^ — 100 — ioo~ -^-30^^ ,,,, 10x67.9 , 33x50_^^9O7 C= log 0.90761 log 0.90761 Z = 7267.9 log 3.86141 = 15° 07' 00" tan 9.43158-10. .sec 0.01529. .cos 9.98471-10 B'= log 0,26035 r = 4.3056 log 0.63403 w„ = 1123.7 log 3.05065 Z = 58752 loff 4.76902 w = 26° 11' 44" tan 9.69193-10 sec 0.04707 r = 36.052 log 1.55693 i'^= 1208.9 log 3.08243 EESULTS. R 19,584 yards. T .'. 36.052 seconds. to 26° ir 44". Vo; 1208.9 foot-seconds. Note to Fokm No. 8. — The number of approximations necessary to secure correct results increases with the angle of departure, therefore problems for a smaller angle of departure will not involve so much labor as the one worked out on the form. APPENDICES 283 Form No. 9. CHAPTER 10— EXAMPLE 2. FORM FOR THE COMPUTATION OF THE VALUES OF THE HORIZONTAL RANGE (R), ANGLE OF FALL (o.), TIME OF FLIGHT (T) AND STRIKING VELOCITY (i'^) FOR A GIVEN ANGLE OF DEPARTURE (c/>), ATMOS- PHERIC CONDITION AND MAXIMUM ORDINATE. FORMULA. C=^K; A = ^^^^^ ; X=CZ; tan w = 5' tan «^ ; T = CT'sec; C Va, = Uco COS sec 0) PROBLEM. Cal. = 8"; 7 = 2750 f. s.; w = 260 pounds; c=0.61; <^ = 15° 07' 00"; Barometer = 29.00"; Thermometer = 82° F.; Maximum ordinate = 5400 feet; fF = 3G00 feet. R= log 0.82346 /= 1.099 log 0.04100 8 = .937 log 9.9717-i-lO. .colog 0.02826 C= log 0.89272 colog 9.10728-10 2<^ = 30° 14' 00" sin 9.70202-10 A = .064461 log 8.80930-10 ^=''''+wk 100 1 P' oron I -0024x5 , .0023x50 ^..^^ log B = .2620 + -3^^^ + 100 = .26407 r = 4.508 + ^^^1^ - ''^^^^ ^^ =4.4276 ,,.1095-^5 + ^0=1110.1 C= log 0.89272 log 0.89273 Z = 7405 log 3.86953 (^ = 15° 07' 00" tan 9.43158-10. .sec 0.01529.. cos 9.98471-10 n'= log 0.26407 r = 4.4276 log 0.64617 2^^ = 1110.1 log 3.04536 Z = 57843 log 4.76225 (0 = 26° 23' 24" tan 9.69565-10 sec 0.04780 r = 35.824 los 1.55418 ra, = 1196.4 log 3.07787 RESULTS. From work with From above work. same data on Form No. 8. R 19,281 yards 19,584 yards oj 26° 23' 24" 26° 11' 44" T 35.824 seconds 36.052 seconds Vco 1196.4 foot-seconds 1208.9 foot-seconds Note to Form No. 9. — To solve the above problem with strict accuracy it must be done as shown on Form No. 8. In order to get a series of shorter problems for section room work, an approximately correct value of the maximum ordinate is given and employed as above. The comparison of results by the two methods given at the bottom of the above work gives an idea of the degree of inaccuracy resulting from the employment of the method given on this form. 281 APPEJ^DICES Form No. lOA. CHAPTER 11— EXAMPLE 1 (WHEN y IS POSITIVE). FORM FOR THE COMPUTATION OF THE VALUES OF THE ANGLE OF ELEVA- TION {xp) (THE JUMP BEING CONSIDERED AS ZERO), ANGLE OF INCLI- NATION TO THE HORIZONTAL AT THE POINT OF IMPACT {6), AND TIME OF FLIGHT TO (0 AND REMAINING VELOCITY AT (i' ) THE POINT OF IMPACT WHEN FIRING AT A TARGET AT A KNOWN HORIZONTAL DIS- TANCE FROM THE GUN AND AT A KNOWN VERTICAL DISTANCE ABOVE THE HORIZONTAL PLANE OF THE GUN, FOR GIVEN ATMOSPHERIC CONDITIONS. EOEMUL^. C=^K; tan p = ~; z = ^; sin 2(^x = aC ; shi(2(f) — p) =sin p(l + cot p sin 2(f)x) ; A= — -^p^ ; tan 6= ^ (A—a'); t = Ct' sec cp; v = ucoscj> sec 6; \l/ = cf> — p PEOBLEM. Cal. = 8"; F = 27o0 f. s.; w = 260 pounds; c = 0.61; Gun below target 900 feet; Horizontal distance = 18,000 yards = 54,000 feet; Maximum ordinate = 4170 feet; Barometer = 29.00"; Thermometer = 40° F.; §7 = 2980 feet. K= .' log 0.82346 /=1.0804 log 0.03358 8=1.021 W 0.00903 colour 9.99097-10 C= log 0.84801 colog 9.15199-10 y = 900 log 2.95424 Subtractive. a;=54000 loff 4.73239 log 4.73239 p = 0° 57' 18" tan 8.22185-10 2 = 7662.6 log 3.88438 n- A^noi, -00172X62.6 .00532x50 _ nrqro^ a_.070.1+ ^ — -.068627 a'=.2001+ -^'056X62.6 _ .0143^x50 ^ -^^^^g ^:.1077-^^ + ^^=1086.5 i' ^4.692 + ^09^^ - :i:?^= 4.6627 APPEXDICES 285 C= log 0.84801 a = .068627 log 8.83649-10 2cf>^= sin 9.68450-10 L p = 0° 57' 18" cot 1.77815 cot 77 sin 2(/)^. = 29.016 log 1.46265 1 + cot/; sin 2-p = 30° 01' 02" sin 9.69920-10 p= 0° 57' 18" 2.^ = 30° 58' 20" sin 9.71149-10 <^ = 15° 29' 10" /;= 0° 57' 18" ^^ = 14° 31' 52" C= colog 9.15199-10 ^ = .07303 log 8.86348-10 A= .07303 a= .19646 ^i_fl=(_). 12343 (-)log 9.09143-10 <;!> = 15° 29'10" tan 9.44258-10.. sec 0.01606. .cos 9.98394-10 A = .07303 log 8.86348-10. .colog 1.13652 r = 4.6627 ^ log 0.66864 « = 1086.5 log 3.03603 C= log 0.84801 ^=(-)25° 05' 38" (-)tan 9.67053-10 sec 0.04306 ^ = 34.097 log 1.53271 t' = 1156.2 log 3.06303 The range table gives for 7^ = 18,500 yards (/) = 14° 26.9' for 7? = 18,600 yards = 14° 34.9' Therefore, for an angle of elevation of 1/^ = 14° 31.9', the sight setting in range would be 2^=18500+ ^2S^ =18562.5 yards. o EESULTS. i}, 14° 31' 52". 6 (-)25° 05' 38". t 34.097 seconds. V .1156.2 foot-seconds. Setting of sight in range. . . .18,550 yards. Note to Form No. lOA. — Note that the work on this form, for a target higher than the gun, is the same as that on Form No. lOB for the same problem with the gun higher than the target, down to and including the determination of the value of cot p sin 20j., except that the sign of that quantity and of the position angle (p) is positive in Form No. lOA, and negative in Form No. lOB. Compare the results obtained on these two forms, having in mind the remarks made in paragraphs 191, 192 and 193 of Chapter 11 of the text. 286 APPENDICES Form No. lOB. CHAPTER 11— EXAMPLE 1 (WHEN y IS NEGATIVE). FOEM FOR THE COMPUTATION OF THE VALUES OF THE ANGLE OF ELEVA- TION (.//) (THE JUMP BEING CONSIDERED AS ZERO), ANGLE OF INCLI- NATION TO THE HORIZONTAL AT THE POINT OF IMPACT {6), AND THE TIME OF FLIGHT TO {i) AND REMAINING VELOCITY AT (i^) THE POINT OF IMPACT WHEN FIRING AT A TARGET AT A KNOWN HORI- ZONTAL DISTANCE FROM THE GUN AND AT A KNOWN VERTICAL DIS- TANCE BELOW THE HORIZONTAL PLANE OF THE GUN, FOR GIVEN ATMOSPHERIC CONDITIONS. FOEMULJ^. C= '^K; tan p — ^; z=-^ \ sin 2(f>j: = aC; sin(2(f> — p) =sin p(l + coip sin 2(^j) ; ^_ sm_^ . ^^^ ^_ an (f> ^j^_^'^ . f — Qf gee <^; v = u cos ^ sec 6; ij/ = (j> — p PEOBLEM. CaL = 8"; 7 = 2750 f. s.; iv = 260 pounds; c = 0.61; Gun above target 900 feet; . Horizontal distance = 18,000 yards = 54,000 feet; Maximum ordinate = 4470 feet; Barometer =29.00"; Thermometer = 40 ° E.; fF = 2980 feet. E= log 0.82346 /= 1.0804 log 0.03358 S = 1.021 W 0.00903.. coloff 9.99097-10 C= log 0.84801 colog 9.15199-10 rr = 54000 log 4.73239J log 4.73239 p={-)0° 57' 18"...(-)tan 8.22185-10 2 = 7662.6 log 3.38438 . n-noi , .00172x62.6 .00532x50 ^^0^0^ a=.0.021+ — j^^^— =.068627 „> ooni J -0056x62.6 .0143x50 -.nrir ^=-^^^ + — 100 locf-^-^^^^^ ^=1077--^^^|- +^^^:^1086.5 ^ = 4.692+^^93^^6 _ -175 X 50 ^^ ^^^^ APPENDICES 287 C= log 0.84801 a = . 068627 W 8.83649-10 2jc= sin 9.68450-10 p={-)Q° 57' 18" (-)cot 1.77815 cot/jsm2<^^=(-)29.016 (-)log 1.46265 l + cot/Jsm20.r=(-)28.O16 (-)log 1.44741 ;;=(-)0° 57' 18" (-)sin 8.22185-10 2-p= 27° 50' 10" ( + )sin 9.66926-10 p={-) 0° 57' 18" 2= 26° 52' 52" sin 9.65528-10 (f>= 13° 26' 26" I p=(-) 0° 57' 18" " iA= 14° 23' 44" C= coloff 9.15199-10 A = .06416 log 8.80727-10 A= .06416 a'= .19646 A-rt'= (-).13230 (-)log 9.12156-10 cf> = 13° 26' 26" tan 9.37836-10. .sec 0.01206. .cos 9.98794-10 A = .06416 .log 8.80726- 10.. colog 1.19273 f = 4.6627 log 0.66864 w = 1086.5 log 3.03603 C= log 0.84801 ^zr(-)26° 13' 58" (-)tan 9.69265-10 sec 0.04720 ^ = 33.784 W 1.52871 t; = 1178.0 log 3.07117 The range table gives for i2 = 18,400 yards (^ = 14° 19.0' for i2 = 18,500 yards <^ = 14° 26.9' Therefore, for an angle of elevation of ;/' = 14° 23.7', the sight setting m range would be E = 18400+ '^''^^^^^ =18459.5 yards. EESULTS. ip 14° 23' 44". (I (-)26° 13' 58". t 33.784 seconds. V 1178.0 foot-seconds. Setting of sight in range. .. .18,450 yards. Note to Form No. lOB. — Note that the work on this form, for a target lower than the gun, is the same as that on Form No. lOA for the same problem with the target higher than the gun, down to and including the determination of the value of cot p sin 2(;6x, except that the sign of that quantity and of the angle of position (j)) is minus in Form No. lOB and plus in Form No. lOA. Compare the results obtained in the two cases, having in mind the remarks made in paragraphs 191, 192 and 193 of Chapter 11 of the text. 288 APPENDICES Form No. 11. CHAPTER 12— EXAMPLE 2. FORM FOR THE COMPUTATION OF THE CHANGE IN RANGE RESULTING FROM A VARIATION FROM STANDARD IN THE INITIAL VELOCITY, OTHER CONDITIONS BEING STANDARD. FORMULA. PROBLEM. Case 1. — Correcting for Altitude by Table V. Cal. = 8"; 7 = 2750; w = 260 pounds; c = 0.61; Range = 19,000 yards = 57,000 feet; Maximum ordinate = 5261 feet; Variation from standard of 7= +75 f. s.; §7=3507 feet. K= log 0.82346 /=1.0962 log 0.03989 C= log 0.86335 colog 9.13665-10 Z = 57000 log 4.75587 Z = 7807.6 log 3.89252 A nn^^c^r , -00012X7.6 .00049x50 nnf;Q9di Af4 = .00556-^ — ^j^ =.0053241 „ ^o^^ , .0040x7.6 .0093x50 ,000^ ^=-^^^^+ 100 lor— =-^^^^^ Afa = .0053241 log 7.72625-10 7^ = 19000 log 4.27875 5 = .13335 log 9.12500-10 colog 0.87500 87=( + )75 log 1.87506 A7=100 io? 2.00000 coloff 8.00000-10 AEf=( + )568.93 yards log 2.75506 PROBLEM. Case 2. — Using Corrected Value of Obtained by Successive Approximations on Form Xo. 1. Cal. = 8"; 7 = 2750 f. s.; w = 260 pounds; c = 0.61; Range = 19,000 yards = 57,000 feet; log (7 = 0.87827 ; Variation from standard of 7= +75 f. s. C= colog 9.12173-10 Z = 57000 log 4.75587 Z = 7544.0 log 3.87760 A,, ^.00520+ :00012X44 _ .00046^X50 ^.0050228 £ = .1261+ -^^^^ - -003^8^X50^,^33^ APPENDICES 289 Ar^ = .0050328 : log 7.7009-i-lO 7? = 19000 log 4.27875 ^=.12337 log 9.09121-10 colog 0.90879 8V=( + ) 75 log 1.8750G A7=100 log 2.00000 colos 8.00000-10 Ai2F=( + )580.15 log 2.76354 Note to Form No. 11. — The method of Case 2 is of course the more accurate, and gives the range table result. The method shown in Case 1 is introduced to give practice in the use of this formula without the necessity of taking up the successive approximation method in order to determine the value of C accurately. 19 290 APPENDICES Form No. 12. CHAPTEE 12— EXAMPLE 3. FORM FOR THE COMPUTATION OF THE CHANGE IN RANGE RESULTING FROM A VARIATION FROM STANDARD IN THE DENSITY OF THE ATMOS- PHERE, OTHER CONDITIONS BEING STANDARD. FOEMUL^. C- fw . y_ X -p _ (B-A)R ^ AO PEOBLEM. Case 1. — Correcting for Altitude by Table V. Cal. = 8"; 7=2750 f. s.; w = 260 pounds; c = 0.61; Eange = 19,000 yards = 57,000 feet; Maximum ordinate = 5261 feet; Variation in density =+15 per cent; fF = 3507 feet. K= ' log 0.82346 /= 1.0962 log 0.03989 C= log 0.86335 colog 9.13665-10 Z = 57000 log 4.75587 Z = 7807.6 log 3.89252 A = .07368+ -QQ^^^Q^^-^ - :00^<50 =.071035 5^.1377+^^^^^ _ -009^3^X50 ^133354 ^ = .133354 4 = .071035 S-A = .062319 log 8.79462-10 i2 = 19000 log 4.27875 ^=.13335 log 9.12500-10 colog 0.87500 \n ~- = .15 , log 9.17609 - 10 Ai2fi= ( - ) 1331.9 log 3.12446 PEOBLEM. Case 2. — Using Corrected Value of C Obtained by Successive Approximations on Form No. 1. Cal. = 8"; 7 = 2750 f. s.; w = 260 pounds; c=0.61; Eange = 19,000 yards = 57,000 feet; log C = 0. 87827; Variation in density =+15 per cent. C= colog 9.12173-10 Z = 57000 loff 4.75587 Z = 7544.0 log 3.87760 A AfiQr^i, -001^0X44 .00520x50 ^acRKQ A = .06851+— j^^ 100— =-^^^^^^ R 19^1 , .0038x44 .0088x50 ,.,0...., £ = .1261+ —^^ ^^^^^ =.Uooa APPENDICES 291 5=. 123373 ^ = .066658 5-4 = .056714 log 8.75369-10 i?= 19000 log 4.27875 5=. 12337 log 9.09121-10 colog 0.90879 4^ = .15 log 9.17609 - 10 ARs= ( - ) 1310.1 log 3.11732 Note to Form No. 12. — The method of Case 2 is of course the more accurate, and gives the range table result. The method shown in Case 1 is introduced to give practice in the use of this formula without the necessity for taking up the successive approximation method in order to determine the value of C accurately. 292 APPENDICES Form No. 13. CHAPTER 12— EXAMPLE 4. FORM FOR THE COMPUTATION OF THE CHANGE IN RANGE RESULTING FROM A VARIATION FROM STANDARD IN THE WEIGHT OF THE PRO- JECTILE, OTHER CONDITIONS BEING STANDARD. DIRECT METHOD WITHOUT USING COLUMNS 10 AND 12 OF THE RANGE TABLES. FORMULA. C= i^ ; 87= -0.36 — V; cd- w Ai?, = ifi,' + AK,"=^-4 X 4^ XB+ (^=^ X ^ ^^ ^ *^ B AK B w PROBLEM. Case 1. — Correcting for Altitude by Table V. Cal. = 8"; 7 = 2750 f. s.; m; = 260 pounds; c = 0.61; Range = 19,000 yards = 57,000 feet; Maximum ordinate = 5261 feet; Variation in weight= +10 pounds; §7 = 3507 feet. K= log 0.82346 /=1.0962 log 0.03989 C= log 0.86335 colog 9.13665-10 Z = 57000 log 4.75587 Z = 7807.6 log 3.89253 A.. = .00556+:«M|X1,6 _ .00049x50 ^.^^^3,^^, ^ = .07368+ -Q^^^X^-^ --00556X50 ^^^^^3^ ■ B = .1377+-Q7^f-^ - ^^^^^0 =.133354 Aw;= +10 log 1.00000 w = 260 colog 7.58503-10 7 = 2750 log 3.43933 .36 log 9.55630-10 8V={-) log 1.58066 Afa = . 0053241 log 7.72625-10 72 = 19000 log 4.27875 5=. 13335 log 9.12500-10 colog 0.87500 S7=(-) (-)log 1.58066 A7' = 100 log 2.00000 colog 8.00000-10 AB«,'=(-)288.84 log 2.46066 5 = . 133354 ^ = .071035 5-A = .062319 log 8.79462-10 22 = 19000 log 4.27875 £ = .13335 colog 0.87500 Aw= +10 log 1.00000 w = 260 colog 7.58503-10 AE«;"= ( + ) 341.51 log 2.53340 ARw= + 52.67 yards, APPENDICES 293 hence an increase in weight gives an increase in range for this gun at this range, therefore this quantity would carry a negative sign in Column 11 of the range table for this range. PROBLEM. Case 2. — Using Corrected Value of C Obtained by Successive Approximations on Form No. 1. Cal. = 8"; 7 = 2750 f. s.; w = 260 pounds; c = 0.61; Eange = 19,000 yards = 57,000 feet; log (7 = 0.87827; Variation in weight = +10 pounds. C= colog 9.12173 - 10 Z = 57000 log 4.75587 Z = 7544.0 log 3.87760 A,, = .00520 + ^^^^^^ - :000i^0 ^ ^^.^Q^.g A = .06851 + :00mx44 _ -00520^X50 ^ ..g^^g P ,^^,^.0038x44 .0088x50 -.ooory. 5=.1261 + — ^^^— ^^^— =.123o72 Aw= +10 log 1.00000 tt; = 260 colog 7.58503-10 7 = 2750 log 3.43933 .36 log 9.55630-10 hV= ( - ) log 1.58066 Afa = .0050228 log 7.70094-10 E = 19000 log 4.27875 £ = .12337 log 9.09121-10 colog 0.90879 hV={-) log 1.58066 A7 = 100 locr 2.00000 colog 8.00000-10 Ai?M,'=(-)294.53 log 2.46914 5 = .123372 4 = .066658 £-.4 = .056714 log 8.75369-10 i^ = 19000 log 4.27875 5 = . 12337 colog 0.90879 A«=+10 log 1.00000 m; = 260 colog 7.58503-10 Ai?,„"=( + )335.94 log 2.52626 Ai?,,= ( + ) 41.44 hence an increase in weight gives an increase in range for this gun at this range, and this quantity would carry a negative sign in Column 11 of the range table for this range. Note to Form No. 13. — The method of Case 2 is of course the more accurate, and gives practically the range table results. The method shown in Case 1 is introduced to give practice in the use of these formulae without the necessity for taking up the successive approximation method in order to determine the value of G accurately. 294 APPENDICES Form No. 14. CHAPTEE 12— EXAMPLE 5. FORM FOR THE COMPUTATION OF THE CHANGE IN RANGE RESULTING FROM A VARIATION FROM STANDARD IN THE WEIGHT OF THE PRO- JECTILE, OTHER CONDITIONS BEING STANDARD. SHORT METHOD, USING DATA CONTAINED IN COLUMNS 10 AND 12 OF THE RANGE TABLES. FOEMUL^. 87 = 0.36 ^^ Y; ^E^ = ^EJ + ^EJ' = ^RYX S +^ XAScXAS w oV w PEOBLEM. Cal. = 8"; 7 = 2750 f. s.; w = 260 pounds; c = 0.61; Eange = 19,000 yards; Erom Column 10 of range table, Ai25oy = 387 yards; From Column 12 of range table, AEioO=874 yards; Variation in weight= +10 pounds. Aw= +10 log 1.00000 w; = 260 colog 7.58503-10 7 = 2750 log 3.43933 .36 log 9.55630-10 hV={-) log 1.58066 AJ?5of = 387 log 2.58771 87=(-) (-)log 1.58066 87' = 50 colog 8.30103-10 A7?^'= (-)294.71 ( - )log 2.46940 Ai?^oC= 874 log 2.94151 Aw= +10 log 1.00000 w = 260 colog 7.58503-10 A8= 10 lo^ 1.00000 Ai2«;"=( + )336.15 log 2.52654 Ai^i<,= ( + ) 41.44 yards, hence an increase in weight gives an increase in range for this gun at this range, therefore this quantity would carry a negative sign in Column 11 of the range table at this range. Note to Form No. 14. — The method gives practically the range table result. APPENDICES 295 Form No. 15. CHAPTER 12— EXAMPLE 6. FORM POR THE COMPUTATION OF THE CHANGE IN THE VERTICAL POSI- TION OF THE POINT OF IMPACT IN THE VERTICAL PLANE THROUGH THE TARGET RESULTING FROM A VARIATION IN THE SETTING OF THE SIGHT IN RANGE, ALL OTHER CONDITIONS BEING STANDARD. , FORMULA. H = AX tan 0) PROBLEM. Cal. = S"; y = 2750 f. s.; iv = 260 pounds; c = 0.61; Range = 19,000 yards; u) = 26° 35' 00" (from range table) ; Variation in setting of sight= +150 yards. AZ = 450 log 2.65321 ' = 466.67 log 2.66901 Multiplier = 1.5 log 0.17609 ^=.53 log 9.72428-10 n = 25 log 1.39794 colog 8.60206-10 A =.32 log 9.50515-10 h C= log 0.86335 2 log 1.72670 <^ = 15° 07' 00" sec 0.01529 3 sec 0.04587 2> = 281.29 yards log 2.44916 PEOBLEM. Case 2. — Using Corrected Value of C and (f> Obtained by Successive Approximations on Form No. 1. Cal. = 8"; 7 = 2750 f. s.; w = 260 pounds; c=0.61; Eange = 19,000 yards = 57,000 feet; log (7 = 0.87827; «^ = 15° 07' 15" (from Form No. 1). C= colog 9.12173-10 Z = 57000 log 4.75587 Z = 7544.0 log 3.87760 r>,_4oo 4. 20>^ _ 33X50 _ ^-^^"^+"^00" "100^-^^^-'^ APPENDICES 297 D' = 4U.3 log 2.61731 Multipliers 1.5 log 0.17609 fi = .53 log 9.72428-10 n = 25 log 1.39794 colog 8.60206-10 ^=.32 log 9.50515-10 h ° C= log 0.87827 2 log 1.75654 <^=15° 07' 15" sec 0.01530 3 sec 0.04590 Z) = 267.50 yards log 2.42733 Note to Form No. 16. — The method of Case 2 is of course the more accurate, and gives practically the range table result. The method shown in Case 1 is introduced to give practice in the use of this formula without the necessity for taking up the successive approximation method in order to determine the exact values of C and . Form No. 17. CHAPTER 13— EXAMPLE 2. FORM FOR COMPUTATION OF SIGHT BAR HEIGHTS AND SETTING OF SLIDING LEAF. (Permanent Angle = 0°.) FOEMUL.^. h = ltan d=^-^^D K PROBLEM. Cal. = 8"; 7 = 2750 f. s.; w = 260 pounds; c = 0.61; Range = 19,000 yards; and T from range table) ; Wind com- ponent along line of fire = 15 knots an hour with the flight; Wind component perpendicular to the line of fire = 10 knots an hour to the right. 15 log 1.17609 10 ' log 1.00000 6080 log 3.78390 log 3.78390 60x60x3 = 10800 colog 5.96658-10. .colog 5.96658-10 Wr= log 0.92657 Wz= log 0.75048 7 = 2750 2 log 6.87866 2(^ = 30° 14' 00" sin 9.70202-10 Z = 57000 log 4.75587 colog 5.24413-10 ^ = 32.2 colog 8.49214-10 n = 2.0746 log 0.31695 2w = 4.1492 2n-l = 3.1492 log 0.49820 colog 9.50180-10 71 = 2.0746 log 0.31695 ^ 60 X 60 X 3 ' " gX ' n ^ Z cos <^ ^ . ^ _ V cos <^ 60 X 60 X 3 ' gX ap — ^ V, -ST cos <^ r • n — ^ n PROBLEM. Cal. = 8"; 7 = 2750 f. s.; w; = 260 pounds; c = 0.61; Range = 19,000 yards = 57,000 feet; ^ = 15° 07' 00" (from range table) ; Speed component in line of fire = 9 knots an hour against the flight; Speed component perpendicular to the line of fire = 18 knots an hour to the left. 9 log 0.95424 18 log 1.25527 6080 log 3.78390 log 3.78390 60x60x3 = 10800 colog 5.96658-10. .colog 5.96658-10 G:.= log 0.70472 Gz- log 1.00575 7 = 2750 2 log 6.87866 2(^ = 30° 14' 00" sin 9.70202-10 g = 32.2 colog 8.49214-10 Z = 57000 loff 4.75587 colog 5.24413-10 n = 2.0746 log 0.31695 2n = 4.1492 2n-l = 3.1492 log 0.49820 colog 9.50180-10 n = 2.0746 log 0.31695 Z = 57000 log 4.75587 = 15° 07' 00" cos 9.98471 - 10 7 = 2750 colog 6.56067-10 G^= log 0.70472 A/?G = 66.791 yards short log 1.82472 Z = 57000 log 4.75587 7 = 2750 colog 6.56067-10 <^ = 15° 07' 00" sec 0.01529 Gz= log 1.00575 Z)o = 217.56 yards left log 2.33758 300 APPENDICES Form No. 20. CHAPTEE 14— EXAMPLE 3. FORM FOR THE COMPUTATION OF THE EFFECT OF MOTION OF THE TARGET. FORMULA. ^^=Sf-3'^^^-^^^^-^^ = ^^^ PROBLEM. Cal. = 8"; y = 2750 f. s.; w = 2Q0 pounds; c=0.61; Range = 19,000 yards: Time of flight^ 35.64 seconds (from range table) ; ^a; = Speed component in line of fire in knots per hour =17 knots with flight; >S'« = Speed component perpendicular to line of fire in knots per hour = 19 knots to left. r = 35.64 log 1.55194 log 1.55194 S:, = 17 log 1.23045 Sz = 19 log 1.27875 6080 log 3.78390 log 3.78390 60x60x3 = 10800 colog 5.96658-10. .colog 5.96658-10 Ai2r= 341.09 yards over log 2.53287 Dt = 381.22 yards right log 2.58117 Note to Form No. 20. — Note that this example is simply the arithmetical problem of determining how far the target will move in the given direction at the given speed during the time of flight; the speeds being given in knots per hour, and the results required in yards for the time of flight. APPENDICES 301 Form No. 21. CHAPTER 16— EXAMPLE 1. FORM FOR THE COMPUTATION OF THE PENETRATION OF HARVEYIZED (E^] AND OF FACE-HARDENED {£,) ARMOR BY CAPPED PROJECTILES. EORMUL.^. Harveyized Armor (Davis). Face-Hardened Armor (De Marre). p 0.8 _ vw^-^ -p o.7_ vw"-"- "" "^1 ~ E-'JO.S -^2 — T' J0.75 '^ K'd''-^ - A'cZ"'-^^ De Marie's Coefficient log A" = 3.25313 log A = 3.00945 PEOBLEM. Cal. = 8"; F = 2750 f. s.; w = 260 pounds; c = 0.61; Eange = 19,000 yards; r„ = 1184 f. s. (from range table) ; De Marre's coefficient=1.5. w = 260 log 2.41497 0.5 log 1.20748 i;^ = llS4 log 3.07335 v„w°-^= log 4.28083 log 4.28083 K'= colog 6.74688-10 K= colog 8.99055-10 d = S W 0.90309.. 0.5 colog 9.54846-10. .0.75 colog 9.32268-10 i;i°-«= log 0.57617 10 8 I lo g 5.76170 ^1 = 5.2506" log 0.72021 log 0.59406 10 y I log 5.94060 log 0.84866 De Marre's coefficient = 1.5 coloff 9.82391-10 .£^2 = 4.7051" log 0.67257 Form No. 22. CHAPTER 16— EXAMPLE 2. FORM FOR THE COMPUTATION OF <^, «, f, /„, D, Y, AND THE PENETRATION, GIVEN R AND /; ATMOSPHERIC CONDITIONS STANDARD.* FOEMUL.^. C=-^-^ =fK; ^ = ~p J sin 2c;6 = x4C; tan m = B' tan 0; t)£j = w£j cos sec w w/i cos'' C -^1^'*— 7^/^0.5 (Harveyized — Davis) r = A"C tan 0; £'2°-'= /^-^o.^r ;^ 5 (Face hardened— De Marre) Cal. = 8"; 7 = 2750 f. s.; tv = 2Q0 feet; /= 1.1345. log A' 0.82346 log/ 0.05481 logC 0.87827 cologC 9.12173- logJC 4.75587 PROBLEM. pounds; c = 0.61; Range = 19,000 yards = 57,000 10 logZ 3.87760 Z = 7544.0 (2) 9^ = 15° 07' 15" logC 0.87827 log A 8.82386 log sin 20 9.70213 2 = 30° 14' 30" (3) cu = 26° 34' 40" logB' 0.26751 log tan 9.43170 log tan cu 9.69921 (4) T = 35.642 seconds logC 0.87827 logT' 0.65839 log sec 0.01530 log T 1.55196 (5) t;„ = 1184.2 logM^, 3.04021 log cos 9.98470 log sec o) ' 0.04850 ^0 10 To 10 logv,,, 3.07341 (6) 23 = 267.50 log At (.53) 9.7242S cologn(25) 8.60206 log ^-(.32) 9.50515 log constant 7.83149 logC^ 1.75654 log sec' 0.04590 logD' 2.61731 2.25124 log 1.5 (if used) 0.17609 logD 2.42733 10 10 10 "10 A = .06851 + .00170 X 44 100 00520 X 50 100 = 2465-^X = .066658 (—.0048) X 68 50 X "^ 100 "^ = 2578.1 .0025 001758 X 68 0025 log B' = . 2652 + .0023 X 44 100 .0026 X 50 T' = 4.600 + 100 .092 X 44 = .267512 173 X 50 100 100 = 4.55398 u,„ = 1086 — 9 X 44 , 30 X 50 D' = 422 + 100 = 1097.04 20 X 44 + 100 33 X 50 100 100 = 414.30 (8) Y = 5263.4 feet log A" 3.41130 logC 0.87827 log tan 9.43170 — 10 logY 3.72127 (9) Harveyized armor. £1 = 5.2515 in. logio"-'* 1.20748 colog^' 6.74688 — 10 colog d""' 9.54846 — 10 logi?^ 3.07341 logE°-^ 0.57623 logE, 0.72028 (9) Face hardened, i:^ = 4.706 in. logw"-'* 1.20748 colog ^ 6.99055 colog fZ"-" 9.32268 — 10 logVy 3.07341 log£?2»' 0.59412 log(B, X1.5) 0.84874 colog 1.5 9.82391 10 10 logE, 0.67265 RESULTS. = 15° 07' 15" D = 267.50 yards. w = 26° 34' 40". Y = 5263.4 feet. T = 35.642 seconds. E^ = 5.252 inches. r„, = 1184.2 f. s. E„ = 4.706 inches. * If we have a problem in vv^hich / is not known, then we must first determine the value of for the given range by the use of Form No. 1 in paragraph 273, Chapter 16. APPENDICES 303 Form No. 23. CHAPTER 16— EXAMPLE 3. FORM FOR THE COMPUTATION OF S, ARy, ^Rc, A/?^, FOR A GIVEN R AND f. FORMULA. ^= i^ =/^'- ^= # (g-^)B \ ^; 87 = 0.36^7; 8 w l+-^cot >^^y=-B^^Yv^^' AR^ = ARu,' + ^Rw" = ^RvX f^^ + — XA7?5XA8 o K it' Cal. = 8"; 7 = 2750 1 s.; w = 260 feet; = 26° log£: 0.82346 log/ 0^05*^ logC 0787827 cologC 9.12173 logX 4.75587 PROBLEM. pounds; c = 0.61; Range = 19,000 yards = 57,000 3-i' 40"; /= 1.1345 ; /i = 20 feet; Aw=±5 pounds. .00012 X44 10 logZ 3.87760 Z = 7544.0 (7) /S = 13.335 yards logy (6.6667) 0.82391 log cot a. 0.30079 log (6.6667 cot a>) 1.12470 cologT? 5.72125 log ^'^^^^ cot u. 6.84595 — 10 R — 10 6.6667 R cot 03 = .00070 , 6.6667 , 1 + — J.— cot w = 1.0007 log 0.00030 6.6667 cot w log 1.12470 logs 1.12500 (10) A/?, =386.77 yards logAr^ 7.70094 cologS 0.90879 log5F(50) 1.69897 cologAy(lOO) 8.00000- logR 4.27875 10 10 log ^R,,r 2.58745 RESULTS. /S = 13.335 yards. ^R^y= 386.77 yards. Ai?,o5 = 873.43 yards. Ai?jo = ± 20.70 yards. Af^ = .00520 + 100 00046 X 50 A = .06851 + 100 .00170 X 44 = .0050228 100 00520 X 50 B = .1261 + 100 .0038 X 44 = .066658 0088 X 50 100 100 = .123372 (12) A7?5 =873.43 yards log (B — A ) 8.75369 — 10 log 7? 4.27875 colog B 0.90879 log (0.1) 9.00000 — 10 logA/?5 2.94123 (11) A«,„ = ± 20.70 yards log Aw 0.69897 colog w 7.58503 — 10 log V 3.43933 log 0.36 9.55630 — 10 logSF 1.27963 logA/?,„v 2.58745 logSr 1.27963 colog 5F'(50) 8.30103 — 10 log AT?,/ 2.16811 logA7?i„5 2.94123 log Aiy 0.69897 colog 10 7.58503 — 10 logA5(10) 1.00000 log ARu" 2'!22523 AR^' = If: 147.27 ARy," = ± 167.97 A7?,t, = ±: 20.70 An Increase in weight gives an increase in range for this gun at this range, there- fore this quantity would carry a negative sign in Column 11 of the range table for this range. 504 APPEXDICES Form No. 24. CHAPTER 16— EXAMPLE 4. FORM FOR THE COMPUTATION OF EFFECTS OF WIND AND OF MOTION OF GUN AND TARGET; ALSO CHANGE IN HEIGHT OF POINT OF IMPACT FOR VARIATION IN SETTING OF SIGHT IN RANGE FOR A GIVEN R AND /. FORMULA. gX \ 2n — l VI \ 7cos^/' \2n — l V / Vcostfi Dt=T,T; ]Fx = Etc.= FX6080 3x60x60 PROBLEM. Cal. = 8"; 7 = 2750 f. s.; ?(7 = 260 pounds; c = 0.61; Range = 19,000 yards = 57,000 feet; <^ = 15° 07' 15"; r = 35.642 seconds; (o = 26° 34' 40". Value of n logV^ 6.87866 log sin 20 9.70213 — 10 cologgf(32.2) 8.49214 — 10 cologZ 5.24413 — 10 logn 0.31706 2n n = 2.0752 2n = 4.1504 -1 = 3.1504 (13) ARw =151.74 yards logn 0.31706 logZ 4.75587 log cos 9.98470 — 10 colog(2n — 1) 9.50163 — 10 colog V 6.56067 — 10 ■"s^l^^w "^^^ T = 35.642 log 22.462 1.35145 logWa; 0.82966 logARw 2.18111 (16) Dw =95.740 yards logZ 4.75587 colog y 6.56067 — 10 log sec 0.01530 X .1.33184 V cos

logG^ 0.82966 log Da 2.16150 (15) Ai2T = D2 = 240.78 yards log Ta, = T« 0.82966 logT 1.55196 log AR T =D T 2.38162 (19) H= 150.08 feet log Z = 300 2.47712 log tan to 9.69921 — 10 log if 100 2.17633 RESULTS. n = 2.0752. ARw = 151.74 yards. Dw = 95.740 yards. ARa = 89.040 yards. Do = 145.04 yards. ARt = Dt = 240.78 yards. fl' = 150.08 feet. APPENDICES 305 Form No. 25A. CHAPTER 17— EXAMPLE 8. FORM FOR THE SOLUTION OF REAL WIND AND SPEED PROBLEMS. PEOBLEM. Cal. = 8"; F = -2750 f. s.; iv = 2C,0 pounds; c = O.Gl; Eange = 7500 yards; Peal wind, direction from, 225° true; velocity, 15 knots an hour; Motion of gun, course, 355° true; speed, 20 knots an hour; Motion of target, course, 330° true; speed, 25 knots an hour; Target at moment of firing, 75° true; distant, 7500 yards; Barometer = 30.00"; Thermometer = 10° F.; Temperature of powder =75° F.; Weight of projectile = 263 pounds. — 35 Temperature of powder, — j-^ X 15 : 52.5 foot-seconds. Use Table IV to correct for density. Use traverse tables for resolution of wind and speeds. 1/77^ o/" F/r^ Wind 20 306 APPENDICES Cause of error. Speed of or varia- Affects. tion in. Gun Tarsret Wind Initial velocity . . Density. Range .... Deflection. Range .... Deflection. Range .... Deflection. Range .... Range .... Range . 12 Formulae. 44 3.5 X 44 12 ~ 12 20 sin 80 X 55 12 ~ in.7 X 55 12 25 cos 65 X 68 10.6 X 68 12 ~ 12 25 sin 65 X 68 22.7 X 68 12 12 15 cos 30 X 24 13 X 24 12 12 15 sin 30 X 13 7.5 X 13 12 12 207 52.5 X^ 3X 43 1.25 X 180 30.2 yards X ~?ro=5-3 knots on deflection scale. 68 Errors in — Range. Short. Yds. 217.4 25.8 225.0 468.2 98.9 369.3 Over. Yds. 12.8 60 26 98.9 Deflection. Right. Yds. 128.6 128.6 98.4 30.2 Left. Yds. 90 Set sights at: Exactly in range, 7869.3 yards; in deflection, 44.7 knots. Actually in range, 7850.0 yards; in deflection, 45.0 knots. Remember to shoot short rather than over. APPENDICES 307 Form No. 25B. CHAPTER 17— EX A:\rPLE 8. FORM FOR THE SOLUTION OF REAL WIND AND SPEED PROBLEMS. PROBLEM. Cal. = 8"; 7 = 2750 f. s.; w = 260 pounds; c = 0.61; Range = 7000 yards; Real wind, direction from, 160° true; velocity, 20 knots an hour; Motion of gun, course, 2()0° true; speed, 18 knots an hour; Motion of target, course, 170° true; speed, 23 knots an hour; Target at moment of firing bearing, 115° true; distant, 7000 yards; Barometer = 29.00"; Thermometer = 85° E.; Temperature of powder = 95° F. ; Weight of projectile = 258 pounds. Temperature of powder. + 35 10 X 5 = + 17.5 foot-seconds. Use Table IV for correction for density, speeds. Use traverse tables for resolving ^n 308 APPENDICES Cause of error. Speed of or vari- ation in. Gun. Target. Wind, Initial velocity Density. Affects. Range Deflection.. Range Deflection.. Range Deflection.. Range Range , Range . 12 FormulsB. 4-> 14 7 V 4'> 18 cos 35 x^ = 12 12 -^ . „. ^ 52 10.3X52 18 sin do X -vv = 12 12 63 13.2 X 03 23 cos 55 X ^ = ^ — 23 sin 55 X 63 _ 18.8 X 63 12 12 20 cos 45 X 21 U 14.1 X 21 12 20 sin 45 X 11 14.1 X 11 12 12 17.5 X 197 "50 43 .69 X 157 67 yards X "^= 12. 8 knots on deflection scale. Errors in — Range. Short. Yds. 51.5 69.3 24.7 145.5 Over. Yds. 69.0 17.2 108.3 194.5 145.5 49.0 Deflection. Right. Yds. Left. Yds. 44 44. G 98.7 12.9 111.6 44.6 67.0 Set sights at : Exactly in range, 6951 yards ; in deflection, 62.8 knots. Actually in range, 6950 yards; in deflection, 62.0 knots. Remember to shoot short rather than over. APPENDICES 309 Form No. 26. CHAPTER 17— EXA]\rPLE 9. FORM FOR THE SOLUTION OF APPARENT WIND AND SPEED PROBLEMS. PEOBLEM. Cal. = 8"; 7=2750 f. s.; w = 2G0 pounds; c = 0.61; Range = 7300 yards; Apparent wind, direction from, 45° true; velocity, 30 knots an hour; Motion of gun, course, 80° true; speed, 21 knots an hour; Motion of target, course, 100° true; speed, 28 knots an hour; Target at moment of firing bearing, 300° true; distant, 7300 yards; Barometer = 28.50"; Thermometer = 10° F.; Temperature of powder = 60° F. ; Weight of projectile = 255 pounds. Temperature of powder, - x 30= — 105 foot-seconds. Use Table IV for correction for density. Use traverse tables for resolution of speeds. Ji^'ziC^ l/'ve of f}'re 310 APPENDICES Cause of error. Speed of or vari- ation in. Gun. Target . Wind Initial velocity. Density. Affects. Range Deflection.. Range Deflection. . Range Deflection.. Range Range Formulae. 21 cos 40 X 66 16.1 X 66 12 12 66 13.5 X 66 21 sin 40 X -j^ ^ 12 66 26.3 X 66 28 cos 20 X j^~ = ^2 66 9.6 X 66 ilO sill -V /\ 12 12 30 cos 75 X 23 7.8 X 23 12 12 30 sin 75 X 12 _ 29 X 12 12 12 Range . 105 X 203 50" 43 X -5 .69 X 171 12 7.5 yards deflection X -7t^= 1-4 knots on deflection scale. 66 Errors in — Range. Short. Yds. 88.6 426.3 118.0 632.9 202.6 430.3 Over. Yds. 144.7 14.9 Deflection. Right. Yds. 74 43.0 202.6 74 Set sights at: Exactly in range, 7730.3 yards; in deflection, 51.4 knots. Actually in range, 7700.0 yards; in deflection, 51.0 knots. Remember to shoot short rather than over. Left. Yds. 52.8 29.0 81.8 74.3 7.5 APPENDICES 311 Form No. 27. CHAPTER 18— EXAMPLES 1 AXD 2. FORM FOR THE COMPUTATIONS FOR THE CALIBRATION OF A SINGLE GUN AND FOR DETERMINING THE MEAN DISPERSION. PROBLEM. Cal. = 8"; 7 = 2750 f. s.;w = 260 pounds; c= 0.61 ; Actual range = 8500 yards; Sights set at, in range, 8500 yards; in deflection, 40 knots; Center of bull's eye above water, 12 feet; Bearing of target from gun, 37° true; Wind blowing from 350° true, with velocity of 8 knots an hour; Barometer= 29.85"; Thermometer = 80° P.; Temperature of powder=100° P.; Weight of shell = 268 pounds; iSTumber of shot = 4, falling as follows: Range. Deflection. Shot. Short. Yds. Over. Yds. Right. Yds. Left. Yds. 1 100 125 85 90 85 9 80 3 60 4 55 Mean errors on foot of perpendicular through bull's eye . . . 41400.0 100.0 short 4|280 70.0 left Correction in range due to height of bull's eye : ^X 12 = 38.7 yards over 31 •' Correction in deflection due to intentional displacement: -||x 10 = 66.7 yards left Temperature of powder: ±M X 10 = + 35 foot-seconds Use Table IV for density correction and traverse tables for resolution of wind forces. 313 APPENDICES ^md Lme of F/'re TA/ZQ£rT Short. Yds. Over. Yds. Right. Yds. Left. Yds. Wind - Range Deflection. . Range Range Range Range Deflection.. 31 5.5 X 31 8cos47Xi2- 12 8sin47x;^--f^ «x^ .318 X 229 227 ='X50 -x^» '°xS 14.2 65.6 .... 72.8 158.9 38.7 8.4 .... Density Initial velocity . . . Height of bull's eve .... Intentional deflec- tion 66.7 79.8 270.4 79.8 190.6 8.4 66.7 8.4 Errors on point P as an origin i or standard conditions 58.3 Observed distance from target in range 100.0 yards Error (where shot should have fallen) 190.6 yards short, over. True mean error in range under standard conditions 290.6 yards short. Observed distance from target in deflection 70.7 yards left. Error (where shot should have fallen) 58.3 yards left. True mean error in deflection under standard conditions. . 11.7 yards left. APPENDICES 313 That is, under standard conditions the mean point of impact of the gun is 290.6 yards short of and 11.7 yards to the left of the point P. We wish to adjust the sight scales so that the actual mean point of impact of the gun shall be at P. To do this we : 1. Eun up the sight until the pointer indicates 8790.6 yards in range, then slide the scale under the pointer until the latter is over the 850"0-yard mark on the former, and then clamp the scale. 12 2. 11.7 yards in deflection equals -x^ X 11.7 = 1.8 knots on the deflection scale at the given range. Set the sight for a deflection of 51.8 knots, then slide the scale until the 50-knot mark is under the pointer, and then clamp the scale. MEAN DISPERSION FROM MEAN POINT OF IMPACT. In range. In deflection. Number of shot. Fall relative to target. Short or over. Yds. Position of mean point of impact relative to target. Short or over. Yds. Variation of each shot from mean point of im- pact. Short or over. Yds. Fall relative to target. Right or left. Yds. Position of mean point of impact relative to target. Right or left. Yds. Variation of each shot from mean point of impact. Right or left. Yds. 1 100 short 125 short 8.5 short 90 short 100 short 100 short 100 short 100 short 25 15 10 85 left 80 left 60 left 55 left 70 left 70 left 70 left 70 left 15 2 10 3 10 4 15 4|50 12.5 4|50 12.5 Mean dispersion from mean point of impact: In range 12.5 yards. In deflection 12 . 5 yards. 314 APPENDICES Form No. 28. CHAPTER 19— EXAMPLE 1. FORM FOR THE COMPUTATIONS FOR THE CALIBRATION OF A SHIP'S BATTERY. PEOBLEM. Cal. = 8"; 7 = 2750; w = 260 pounds; c = 0.61; Eange = 8500 yards. For a battery of eight of the above guns, having determined the true mean errors to be as given below (by previous calibration practice), how should the sights of the guns be adjusted to make all the guns shoot together? Note that no one of the guns shoots closely enough to be taken as a standard gun. 12 At 8500 yards, one yard in deflection equals -^^r :.15 knot on deflection scale. Number of gun. Errors. With reference to point P, each gun shot. To bring all sights to- gether set them for each gun as follows : Range. Yds. Deflection. Yds. In range. Yds. In deflection. In range. Yds." In deflec- tion. Kts. Yards. Knots. 1 25 short 100 over 100 short 75 short 150 over 75 short 80 short 90 over 25 left 50 left 75 right 50 right 100 left 75 left 80 right 25 right 25 short 100 over 100 short 75 short 150 over 75 short 80 short 90 over 25 left 50 left 75 right 50 right 100 left 75 left 80 right 25 right 3.75 left 7.5 left 11.25 right 7.5 right 15.0 left 11.25 left 12.0 right 3.75 right 8525 8400 8600 8575 8350 8575 8580 8410 53 75 2 57.50 3 38 75 4 42.50 5 65.00 6 61 25 7 38.00 8 46.25 After the sights have been set as indicated above, move the sight scales under the pointers until the latter are over the 8500-yard marks in range and over the 50-kuot marks in deflection, and then clamp the scales. APPEXDIX B. THE FARNSWORTH GUN ERROR COMPUTER. PURPOSE AND USE. 1. This instrument was devised by Midshipman J. S. Famsworth, U. S. N., class of 1915, during his first class year at the Naval Academy. 2. It is intended for the purpose of determining quickly and accurately, by mechanical means and without computations, the errors in range and in deflection introduced into gun fire by : * (a) Wind. (b) ]\Iotion of firing ship. (c) Motion of target ship. (d) Variation from standard in the temperature of the powder, (e) Variation from standard in the density of the atmosphere. Plate I shows the device on an enlarged scale, so that the graduations can be clearly seen. The radial arm shown at the right of the drawing is secured to the same axis as the discs. 3. The uses and advantages of the instrument are readily apparent. It can be used by both spotting and plotting groups if desired, but presumably it would be used in the plotting room. Its use will enable the initial errors to be allowed for in firing ranging shots to be accurately and quickly determined, so that with it a spotter has a vastly greater chance of having the ranging shot strike within good spotting distance of the target than by any " judgment " or " rule of thumb " methods. This should enable a ship to begin to place her salvos properly in a shorter time and with less waste of ammunition than could be done without the device. 4. Errors due to changes in courses, speeds, wind, or other conditions during firing can be similarly quickly obtained by the use of the computer. 5. The accompanying drawing (Plate I) shows the device arranged for working with apparent wind, and for determining deflection errors directly in knots of the deflection scale of the sight, and not in yards. The device could be equally well arranged for real wind, for deflections in yards, or for any other desired system, by simply drawing the proper spiral curves on the smaller disc; but the arrangement shown here is believed to be the most useful one for service conditions. The drawing * Throughout this description the " errors " have been considered and not the " cor- rections." In the practical use of the computer it must be remembered that, having determined an " error " tlie " correction " to compensate for it is numerically equal but of opposite sign. Thus, an " error " of 100 short calls for a correction of " up 100," etc. A very clear and concise statement of the purpose and principle of the gun error com- puter is contained in the following extract from a report thereon submitted to the com- mander-in-chief of the United States Atlantic fleet by Ensign H. L. Abbott, U. S. N. : " The gun error computer is a combination of a set of curves showing the correction to be applied at various ranges to range and deflection for unit variation from normal of the conditions considered, such as wind ; and of a specially graduated numerical or circular slide rule for modifying the correction for unit variation to give the correction for the actual variation. This instrument can be made to take the place of the range tables, and with its aid the corrections for any particular set of conditions can be picked out with much greater ease and facility than with the present cumbrous range tables and accom- panying necessary calculations." 316 APPENDICES does not show the three curves in colors, as they should be drawn on a working device, each curve being of a radically different color from the others; and the powder temperature error and density error curves are not shown on the drawing. In the following descriptions it is assumed that the several cui-ves would be drawn as follows : Wind range curve in red. Wind deflection curve in green. Target and gun range curve in black. Powder temperature error curve in blue. Density error curve in yellow. METHOD OF CONSTRUCTION. 6. In external appearance and in some principles of construction, the device is similar to an omnimeter. It consists, as shown on the plate, of two circular discs, an outer or larger, and an inner or smaller one, concentrically secured on the same axis and capable of independent rotary motion around that axis; and, in addition, of a radial arm secured on the same axis and capable of free rotary motion around that axis. These parts should be so arranged that the radial arm can be clamped to the inner disc without clamping the two discs together, and so that the two discs can be clamped together without clamping the radial arm to the inner disc. The radial arm should be made of some transparent material, with the range scale line scribed radially from the center of the axis down the middle of the arm. 7. The salient features of the device are : (a) The Range Circle. — The graduations on the outside of the larger circle on the larger disc. (b) The Deflection Circle. — The graduations on the inside of the larger circle on the larger disc. (c) The Speed Circle. — The graduations on the outside of the smaller circle on the larger disc. This circle is in coincidence with the periphery of the smaller or inner disc. (d) The Correction Circle. — The graduations on the periphery of the smaller disc. (e) The Range and Deflection Curves. — Drawn on the face of the smaller disc, spirally, from the center of the disc outward. They are the : (1) Wind range curve. (2) Wind deflection curve. (3) Target and gun range curve. (4) Powder temperature curve. (5) Density curve. (f ) The Radial Arm. — Bearing the range scale. Of the above, a, b and c are all on the larger disc, and their positions relative to one another are therefore fixed. Also d and e are both on the smaller disc, and their positions relative to each other are therefore fixed. However, a, h and c can be rotated relative to d and e. The range scale, being on /_, can be rotated relative to either or to both of the discs. 8. Of the above, only the curves vary for different guns. It would therefore be necessary to construct the apparatus and then have the curves scribed on it for the particular type of gun with which the individual instrument is to be used. Thus, there would be one computer for each caliber of gun on board. Plate I shows the device as arranged for the 12" gun for which 7 = 2900 f. s., w = 870 pounds and c = 0.61; the necessary data for its construction having been obtained from the range . table for that srun. APPENDICES 317 9. The mathematical principles involved in the construction of the several ele- ments of the device are described herein (the description being based on the assump- tion that the reader is not familiar with the omnimeter). (a) Range Circle. — The entire circumference is divided into parts representing logarithmic increments in the secant of the angle, from zero degrees to the angle whose logarithmic secant is unity (84° + ). These increments are laid down on the circle in a counter-clockwise direction according to the logarithmic secants, and the scale is marked with the angles corresponding to the given logarithmic secants. For example, the point marked 23° lies in a counter-clockwise direction from the zero of the scale, and at a distance from it equal to .03597 of the circumference (log sec 23° = 0.03597). (b) Deflection Circle. — The entire circumference is divided into parts repre- senting logarithmic increments in the sine of the angle, from the angle whose logarithmic sine is 9.00000 — 10(5° + ) to 90°. These increments are laid down on the circle in a clockwise direction according to the logarithmic sines, and the scale is marked with the angles corresponding to the given logarithmic sines. For example, the point marked 23° lies in a clockwise direction from the zero of the scale, and at a distance from it equal to .59188 of the circumference (log sine 23° = 9.59188 — 10). The zero of the scale coincides with the zero of the scale of the range circle. (c) Speed Circle. — The entire circumference is divided into parts representing logarithmic increments in the natural numbers from 1.0 to 10 (the decimal point may be placed wherever necessary, and the point marked " 10 " may be considered as the " zero " of this scale, and will hereafter be referred to as such in this description). The increments are laid down on the circle in a counter-clockwise direction from zero, and the divisions of the scale are marked with the numbers corresponding to the given logarithms. For example, the number 2.3 lies in a counter-clockwise direc- tion from the zero, and at a distance from it equal to .36173 of the circumference (log 2.3 = 0.36173). The zero of this scale coincides with the zeros of the range and deflection circles. (d) Correction Circle. — The construction of the correction circle is exactly the same as that of the speed circle, except that the scale is laid down in a clockwise direction from the zero.''' (e) Range and Deflection Curves. — Each of these is based on the data in the appropriate column of the range table for the given gim, and these curves are there- fore different for different guns. The method of plotting them is described below. (f ) Range Scale. — A radius of appropriate length to fit the discs is subdivided as a range scale. These divisions are purely arbitrary, and Plate I shows the increments in range as decreasing in relative magnitude on the scale as the range increases ; so that the divisions are larger and more easily and accurately read at the ranges that will most likely be used ; becoming smaller as the range becomes very great. The size of these divisions, either actual or relative to one another, does not affect the work of the instrument, provided this range scale be prepared first and then used in plotting the error curves in the manner described below. f * Those familiar with the omnimeter will perceive that up to this point the prin- ciples of that instrument have been followed: but that the scales of the range circle (logarithmic secants) and speed circle (logarithms of numbers) have been laid down in the opposite direction from those on the omnimeter. f If the device be made of a good working size, these divisions may all be made of the same size and still be clearly read, and this is the best way to construct it. 318 APPENDICES 10. To plot the range and deflection curves for wind and for motion of firing or target ship, the data is obtained from the proper column in the range table for the error and gun under consideration (Columns 13, 14, 15, 16, 17 and 18). Thus, for instance, to locate the point of the wind range curve for a range of 10,000 yards for the given 13" gun. Column 13 of the range table (Bureau Ordnance Pamphlet No. 298) shows that the error in range caused by a 12-knot wind blowing along the line of fire is 27 yards, and it would therefore be 2.25 yards for a 1-knot wind. Therefore the desired point of the curve is plotted on the inner disc on a radius passing through the 2.25 mark on the correction circle, and at a distance from the center correspond- ing to 10,000 yards on the range scale. Enough points are plotted in this manner to enable an accurate curve to be scribed through them. The other curves are plotted in a similar manner, but instead of plotting deflection curves in " yards error," they are plotted in " knots error of the deflection scale of the sight," thus enabling the deflection error to be determined directly in knots for application to the sight drums. For example, for the wind deflection curve, the data for plotting would be found by dividing the data in Column 16 of the range table for the given range by the corre- sponding data in Column 18. Approximate values of the correction scale reading are marked at intervals along the curves to aid the operator in placing the decimal point correctly and in getting the result in correct units. 11. To plot the powder temperature curve, it will be seen from Column 10 of the range table that, for a range of 15,000 yards, 50 f. s. variation from standard in the initial velocity causes 377 yards error in range, therefore 1 foot-second variation in initial velocity would cause 7.54 yards error in range. From the notes to the range table, it will be seen that, for this gun, 10° variation from standard in the temperature of the powder (90° being the standard) causes a change of 35 f. s. in initial velocity, therefore a variation of one degree in temperature would cause a change of 3.5 f. s. in initial velocity. Consequently, a variation of one degree from standard in the temperature of the powder would cause an error of 3.5 x 7.54 = 26.39 yards in range. Therefore, to locate the point on the curve corresponding to 15,000 yards range, place the desired point on the face of the smaller disc on a radius passing through the 26.39 mark on the correction circle, and at a distance from the center corresponding to 15,000 yards on the range scale. 12. Before proceeding to a description of the method of plotting the density curve, a brief preliminary discussion of another point is necessary. As the density of the air depends upon two different variables, one being the barometric reading and the other the temperature of the air (assuming, as is done in present methods, that the air is always half saturated), it is not practicable from a point of view of easy operation to lay down a single curve for use in determining the density corresponding to given readings of barometer and thermometer. Therefore a sheet of auxiliary curves is necessary for this purpose, for use in connection with the computer. Such a set of curves is shown on Plate II, and is good for any gun. It is really a graphic representation of Table IV of the Ballistic Tables (the table of multipliers for Column 12), and values of the multiplier can be taken from these curves much more quickly than they can be obtained from the table by interpolation. These curves have been designated atmospheric condition curves, and on Plate II show as straight lines, giving values for the multipliers ten times as great as those given in the table. This has been done in order to have the computer retain the principle on which it is constructed for all other errors; namely, that the first reading taken from the correction circle by bringing the range mark on the range scale into coincidence with the proper curve shall show the error due to unit variation in the quantity under consideration. (The same thing could be done iu this case by plotting the curve for APPENDICES 319 the full errors due to 10 per cent variation in the density, and then using the values of the multipliers as given in the table; but this would make the principle of con- struction different in this case from what it is in all the others, and it was deemed best to adhere to the same principle throughout.) It will also be seen that the curves in question are plotted as straight lines, whereas they would not be quite that if accurately plotted from the table. The straight lines have been plotted as repre- senting as nearly as possible the mean value of the curve that would be obtained by plotting accurately from the table. At the end of this description will be found a mathematical demonstration of the fact tliat these curves must be straight lines, whence it follows that the table is theoretically slightly in error in so far as it departs from this requirement. 13. To plot the density curve, it will be seen from Column 12 of the range table that, for a range of 15,000 yards, a variation from standard in the density of the atmosphere of 10 per cent will cause an error in range of 451 yards; therefore, a variation of 1 per cent in density will cause an error of 45.1 yards. Therefore to plot the point on the curve corresponding to 15,000 yards range, place the desired point on the face of the disc on a radius passing through the 45.1 mark on the cor- rection circle, and at a distance from the center corresponding to the 15,000-yard mark on the range scale. Proceed in a similar manner to plot points corresponding to enough other ranges to enable the curve to be accurately scribed. METHOD OF USE. 14. Before proceeding to describe the use of the computer, the following rule must be laid down : (a) Always use the angle that is less than 90° that any of the directions makes with the line of fire, in order that we may always : (b) Determine all range errors involving angles by multiplying results in the line of fire by the cosine of the angle; that is, by dividing by the secant. (c) Determine all deflection errors involving angles by multiplying results perpendicular to the line of fire by the sine of the angle. 15. As an illustration of the method of using the computer, its manipulation will now be described in finding the error in range that would be caused by an apparent wind of 45 knots an hour blowing at an angle of 50° to the line of fire, for a range of 15,000 yards. The gun is the same 13" gun. 16. Move the radial arm until the 15,000-yard mark on the range scale cuts the wind range curve (red curve). The value on the correction circle where it is now cut by the range scale line is the error for 1-knot wind in the line of fire, and will show as 5 yards. Now swing the inner disc and radial arm together until the range scale line and 5 on the correction circle are in coincidence with the 45 mark of the speed circle and clamp the two discs together. The reading now showing on the correction circle in coincidence with the zero of the speed circle is 235 yards, or the product of 5 X 45, and this would be the error caused by a 45-knot wind blowing along the line of fire. This is not what is wanted in this case, however, so with the two discs still clamped together, swing the radial arm until the range scale line is in coincidence with the 50° mark on the range circle, and then read across by the range scale line to the correction circle, where the coincident mark will be 144 yards, which is the desired result, and will be found to check with the results of work with the range table by ordinary methods. 17. The above process may be more fully explained as follows, for the benefit of those who are not familiar with the omnimeter. As a result of the manner in which the wind range curve (red) was plotted, when the 15,000-yard mark on the range 320 APPENDICES scale was brought to cut the wind range curve, and the reading was noted where the range scale line cut the correction circle, that reading was 5 yards, or the error due to a 1-knot wind blowing along the line of fire. JSTow had the zeros of the correction and speed circles been in coincidence when this was done, when the inner disc was moved around in a counter-clockwise direction until the five of the correction circle coin- cided with the zero of the speed circle, the zero of the correction circle moved a dis- tance equal to log 5. Now when the motion of the inner disc was continued in the same direction until the 5 of the correction circle coincided with the 45 of the speed circle, the zero of the correction circle moved a further distance in the same direction equal to log 45. The total travel of the zero of the correction circle must therefore have been log 5 4- log 45 = log 225 ; and the reading on the speed circle now coincident with the zero of the correction circle (the measure of the total travel of that zero) must be 225 yards, which is the error in range due to a 45-knot wind in the line of fire. The decimal point has moved one digit to the right because of the fact that the zero of the correction circle traveled between one and two complete circumferences during the operation. Now clamp the two discs together as they stand. If the range scale on the radial arm be first placed at the zero of the speed circle (where we read 225 on the correction circle), which is also the zero of the range or log secant circle; and then be moved in a counter-clockwise direction until the range scale line is coin- cident with the 50° mark on the range circle, the range scale line will have traveled a distance from the 225 mark on the correction circle equal to log sec 50°, and if the range scale line be then followed across from the 50° mark on the range circle to the correction circle, the reading on the latter will be log 225 — log sec 50°, or log 225 + logcos50°; that is, the logarithm of 225 X cos 50°, which is the desired result; and reading off the anti-logarithm on the correction circle corresponding to the above result, the reading will be 144 yards, which is the desired error ; that is, the error in range caused by an apparent wind of 45 knots blowing at an angle of 50° with the line of fire. The sign of the error, that is, whether it is a " short " or an " over," will at once be apparent from a glance at the plotting board, on which the direction of the apparent wind should be indicated relative to the line of fire. 18. To determine the deflection due to the wind, proceed in a similar manner, using the wind deflection curve (green), and taking the angle from the deflection circle. Setting the radial arm with the 15,000-yard mark of the range scale in coincidence with the wind deflection curve gives, from the correction circle, that a 1-knot wind perpendicular to the line of flre causes an error of 0.245 knots (on the deflection scale of the sight) in deflection. Moving the .245 mark of the correction circle around to coincide with the 45-knot mark on the speed circle and reading the zero of the correction (or speed) circle will give 11.0 knots as the error due to a 45-knot wind perpendicular to the line of flre (this would not be noted in actual practice unless the wind were actually blowing perpendicularly to the line of fire, in which case it would be the desired result) ; and reading across from the 50° mark on the deflection circle to the correction circle would give 8.5 knots as the amount of error in deflection. As before, the sign of the error must be determined from the plotting board. What was really done here, after determining the value 0.245, was to perform the addition log 0.245 -flog 45 = log 11, and then the addition log 11 -flog sin 50°= log 8.5. That is, the value 0.245 was first found mechanically, and then the compound operation 0.245 x 45 x sin 50° = 8.5 was mechanically performed. 19. As the apparent wind was used in the preceding operations, the errors for the motion of the gun would be taken from the same curve as those for the motion of the target. For the error in range the method is exactly the same for both gun motion and target motion as for the wind error in range, using the target and gun APPENDICES 321 range curve (black). For the error in deflection the work might be done in either one of two ways, as follows : (a) Use the target and gun range curve (black) for deflection errors as well as for range errors (Columns 15 and 18 of the range table are numerically the same) proceeding as before, which would give the deflection error in yards, which would then have to be transformed into knots of the sight deflection scale. (b) Solve by the principles laid down in paragraph 24, subparagraph (b), below, for the solution of right triangles. This can be done because what we desire is the resolution of the speed into a line at right angles to the line of fire, which is the speed in knots multiplied by the sine of the angle, the result being in knots of the deflection scale. This is the shortest method, requires no curve on the computer, and is the one actually used in practice. Suppose the firing ship were steaming at 15 knots on a course making an angle of 36° with the line of fire. Bring the 15 on the correction circle into coincidence with the zero of the speed circle. Then read across from the 36° mark on the deflection circle to the correction circle, where the 8.8 mark shows that the required error is 8.8 knots of the deflection scale of the sight. The operation here performed was 15 X sin 36°. 20. For the error in range due to the motion of the target, proceed exactly as was done in the case of motion of the gun, using the same curve ; the target and gun range curve (black). The process for the deflection error is also the same as before. Suppose the target were moving at 18 knots at an angle of 40° with the line of fire. Put 18 on the correction circle in coincidence with the zero of the speed circle. Then read across from 40° on the deflection circle to the correction circle, and 11.6 knots will there be shown as the required error. 21. To use the powder temperature curve, bring the range scale into coincidence with the powder temperature curve at the given range mark, and clamp the radial arm and smaller disc together. Determine the variation from standard (90° F.) in the temperature of the powder (90° '^t° = T° ; where ^° is the temperature of the charge, and T° is the variation from standard). Turn the smaller disc and radial arm together until the range scale line cuts the speed circle at the T° mark. Then read the mark on the correction circle that is coincident with the zero mark on the speed circle (or the mark on the speed circle that is coincident with the zero mark on the correction circle), and this reading will be the desired error in yards resulting from a variation of T° from standard (90° F.) in the temperature of the powder. A powder temperature higher than standard always gives an increase in range, and the reverse. 22. To use the density curve, bring the range scale into coincidence with the density error curve for the given range, and clamp the radial arm and smaller disc together. Determine the value of the multiplier for the given barometer and thermometer readings from the atmospheric condition curves. Turn the smaller disc and radial arm together until the range scale line cuts the speed circle at the mark indicating the value of the multiplier thus determined. Then read the mark on the correction circle that is coincident with the zero mark on the speed circle (or the mark on the speed circle that is coincident with the zero of the correction circle), and this reading will be the desired error m yards resulting from the variation from standard in the density of the atmosphere due to the given barometric and ther- mometric readings. A multiplier carrying a negative sign (that is, one taken from a red point on the atmospheric condition curves) always gives a "short" (density greater than unity) ; and one carrying a positive sign (that is, one taken from a black point on the atmos^jheric condition curves) always means an " over " (density less than unity). 21 323 APPENDICES 23. Having shown how to manipulate the computer in detail, it will be seen that the process of use in the plotting room would be about as follows : FORM FOR USE IN CONNECTION WITH FARNSWORTH ERROR COMPUTER. Range, yards. Temperature of powder: Standard, 90° ; actual, — °. Atmospheric conditions: Barometer, — "; tlier., — ". Motion of gun: Speed, — knots; angle, — °. Apparent wind: Velocity, — knots; angle, — ° Sums. Preliminary errors. Motion of target: Speed, — knots; angle, — °. Final errors (signs to be changed to give " corrections ") . Errors in. Range. Yds. Short. Over. Deflection. Knots. Right. Left. •(a) By "angle" is meant that angle le.^s than 90° which the course of the firing ship, direction of the apparent wind, or course of the target ship makes with the line of fire. (b) The preliminary errors include all those that will presumably be known long enough in advance to afford reasonable time for their determination. (c) The temperature of the powder and the readings of the barometer and thermometer will be known before starting the approach. The first two lines of the above form may therefore be filled out when work begins, and will presumably remain constant throughout the action. (d) As soon as plotting begins and the proposed line of fire and range are determined with sufficient accuracy, the plotter determines the angles made by the course of the firing ship and by the apparent wind (the information relative to the latter being sent down by the spotter) with the proposed line of fire, and the errors for gun motion and wind are determined and entered in their proper columns. The algebraic additions necessary to give the preliminary errors are then made and entered. This leaves only target' motion to be accounted for, and as soon as the plotter has the necessary information he gives the " angle " and speed of the target ship, the errors caused thereby are taken from the computer and entered in their columns, and then two simple algebraic additions give the total errors required. The necessary corrections for application to the sights for the ranging shot can then be sent to the guns. 24. The computer is readily available for the solution of any right triangle, in addition to the purpose for which it was devised. In the case of angle from 84° to 9U°, the sines are practically equal to unity and the cosines are negligible, and APPENDICES 323 oppositely for angles from 0° to 6°. For this reason the graduations for these angles have been omitted from the circles. For examples in the solution of right triangles we have : (a) Given One Angle and the Hypothenuse to Find the Side Adjacent. — Given that the angle is 30° and the hypothenuse is 27. Put 27 on the correction circle in coincidence with zero on the speed circle. Then find 30° on the range circle and read across to the correction circle, where 23.5 will be found for the side adjacent. (27 X cos 30° =27 divided by sec 30° =23.4 by the traverse tables.) (b) Given One Angle and the Hypothenuse to Find the Side Opposite. — Given that the angle is 30° and the hypothenuse is 27. Put 27 on the correction circle in coincidence with zero on the speed circle. Then find 30° on the deflection circle and read across to the correction circle, where 13.5 will be found for the side opposite. (27 X sin 30° = 13.5 by the traverse tables.) (c) Given One Angle and the Side Opposite to Find the Hypothenuse. — Given that the angle is 30° and the side opposite is 15. Put 15 on the correction circle in coincidence with 30° on the deflection circle, and coinciding with the zero of the correction circle will be 30+ on the speed circle for the hypothenuse. (15 x cosec 30° = 15 divided by sin 30° = 30+ by the traverse tables.) (d) Given One Angle and the Side Adjacent to Find the Hypothenuse. — Given that the angle is 30° and the side adjacent is 15. Put 15 on the correction circle in coincidence with 30° on the range circle, and coinciding with the zero of the correc- tion circle will be 17.25 on the speed circle for the hypothenuse. (15 x sec 30° = 17.3 by the traverse tables.) (e) Given One Angle and the Side Adjacent to Find the Side Opposite. — Given the side adjacent as 15 and the angle as 30°, first find the hypothenuse as in (d), which is 17.25. Put 17.25 on the correction circle in coincidence with zero on the speed circle, and in coincidence with 30° on the deflection circle will be found 8.G5 on the correction circle for the side opposite. (17.25 x sin 30° =8.62 by the traverse tables.) (f) Given One Angle and the Side Opposite to Find the Side Adjacent. — Given the side opposite as 15 and the angle as 30°, first find the hypothenuse as in (c), which is 30 + . Bring 30+ on the correction circle into coincidence with zero on the speed circle, and in coincidence with 30° on the range circle will be 26+ on the correction circle for the side adjacent. (30+ X cos 30° = 30 + divided by sec 30° = 26+ by traverse tables.) (g) Given the Two Sides to Find the Angles and Hypothenuse. — The com- puter does not handle this ease as easily as the traverse tables ; but it is not one usually encountered in the class of work wliere the instrument would habitually be used. INSTETJCTIONS FOR USE. 25. To summarize, the following brief instructions should be used in connection with the instrument : (a) To Determine the Error in Range Resulting from a Variation from Stand- ard in the Temperature of the Powder. Error in Range. — (Use blue curve. Column 10 of range table.) (1) Bring the given range on the range scale into coincidence with the powder temperature curve, and clamp the radial arm and smaller disc together. (2) Determine the variation from standard (90° F.) in the temperature of the powder {20° ^t° = T°). 324 APPENDICES (3) Turn the smaller disc and radial arm together until the range scale line cuts the speed circle at the T° mark. Eead the desired error on the correction circle coincident with the zero of the speed circle. (4) A powder temperature higher than the standard always gives an increase in range, and the reverse. (b) To Determine the Error in Range Resulting from a Variation from Stand- ard in the Density of the Atmosphere. Error in Range. — (Use yellow curve, Column 12.) (1) Bring the given range on the range scale into coincidence with the density curve, and clamp the radial arm and smaller disc together. (2) Determine the value of the multiplier from the atmospheric condition curves for the given readings of barometer and thermometer. (3) Turn the smaller disc and radial arm together until the range scale line cuts the speed circle at the mark indicating the value of the multiplier determined from the atmospheric condition curves. Eead the desired error on the correction circle coincident with the zero of the speed circle. (4) A negative sign on the multiplier alwa3's means a " short," and a positive sign an " over." (c) To Determine Errors Due to an Apparent Wind of Known Velocity and at a Known Angle to the Line of Fire. Error in Range. — (Use red curve, Column 13.) (1) Eotate radial arm until wind range curve intersects range scale on runner at given range, and clamp radial arm to upper disc. (2) Eotate lower disc until range scale line on radial arm intersects speed circle at apparent wind velocity in knots. Clamp discs together; unclamp radial arm. (3) Eotate radial arm until range scale line intersects range circle at angle to line of fire at which wind is blowing. (4) Eead across by range scale line to correction circle, and note result; the desired range error in yards. (5) Determine sign of error by glance at plotting board. Error in Deflection. (Use green curve,. Column 16.) (1) Eotate radial arm until wind deflection curve intersects range scale on radial arm at given range, and clamp radial arm to inner disc. (2) Eotate lower disc until range scale line on radial arm intersects speed circle at apparent wind velocity in knots. Clamp discs together ; unclamp radial arm. (3) Eotate radial arm until range scale line intersects deflection circle at angle to line of fire at which wind is blowing. (4) Eead across by range scale line to correction circle, and note result; the desired error in knots. (5) Determine sign of error by glance at plotting board. (d) To Determine Errors Due to Motion of Gun (or Target) at Given Speed and Angle with Line of Fire. Error in Range. — (Use black curve, Column 15.) (1) Eotate radial arm until target and gun range curve intersects range scale on radial arm at given range, and clamp radial arm to upper disc. (2) Eotate lower disc until range scale line on radial arm intersects speed circle at speed of gun (or target) in knots. Clamp discs together and unclamp radial arm. (3) Eotate radial arm until range scale line intersects range circle at aligle to line of fire made by course of gun (or target). (4) Eead across by range scale line to correction circle, and note result; the desired range error in yards due to motion of gun (or target). /-t, <7f x'<7//j3/W£' <^^/-ye Xwa^/a' Civrfr a APPEXDICES 325 (5) Determine sign of error by glance at plotting board. Error in Deflection. (Use no curve.) (1) Eotate upper disc until zero of speed circle coincides with speed of gun (or target) in knots on correction circle. Clamp discs together. (2) Eotate radial arm until range scale line intersects deflection circle at angle to line of fire made by course of gun (or target). (3) Kead across by range scale line to correction circle, and note result; the desired error in knots due to motion of gun (or target) . (4) Determine sign of error by glance at plotting board. NoTKS. — 1. In all the above described operations, the position of the decimal point at each step must be fixed bj' the operator's general knowledge of the conditions in each case. 2. By " angle " is meant that angle less than 90° which the course of the firing ship, direction of apparent wind, or course of the target ship makes with the line of fire. 3. The corrections to be applied to the sights are numerically equal to the determined errors, but of opposite sign. 4. Variations from standard in powder temperature and density of atmosphere cause errors in range only; none in deflection. 26. In this paragraph is given the mathematical demonstration that the quantites given in the tables for the value of the density of the atmosphere and of the multipliers for Column 12 of the range tables are theoretically slightly in error. (a) The table of multipliers for Column 12 of the range tables to be found on pages 7 and 8 of Bureau of Ordnance Pamphlet No. 500, on the ]\Iethod of Computing Eauge Tables (and in Table IV of the Eange and Ballistic Tables, edition for use at the Xaval Academy) is based on the standard table of densities for given barometer and thermometer readings. This latter gives the ratio of the density of half-saturated air for a given temperature and barometric height to the density of half-saturated air at 15° C. (59° F.) and 750 millimeters (29.5275") barometric height. The values given in the density tables were computed from the formula : g^ 1.05498 ^ 16 * 29.4338 .92485 + .002036^ , in which H is the barometric height in inches, t is the temperature in degrees Fahren- heit, and Ft is the vapor pressure in saturated air at t° . (b) Throwing out all constant multipliers, this equation will take the form in which x, y and z are variables. Xow if we desire to detcrmme the values of 8 for different temperatures for any given barometric height, the barometric height becomes a constant also for the time being; that is, x in the above function becomes a constant, and the expression for the value of 8 becomes an equation of the first degree involving only two unknown variables. Therefore, all values of S for this particular barometric height must lie on the same straight line when the curve is plotted; in other words, the curve in question must be a straight line. The values given in th'' table do not exactly do this, and are therefore in error to the extent to which they deviate from this requirement. The errors are believed to be due to decimal differ- ences in computation, and not to be of material magnitude. (c) The transformations by which the values in the density table are trans- formed into values of the multipliers for Column 12 are simply arithmetical processes, and of such a nature that they do not invalidate the above law. The same criticism therefore applies to the values of the multipliers given in the table. APPENDICES 325 (5) Determine sign of error by glance at plotting board. Error in Deflection. (Use no curve.) (1) Eotate upper disc until zero of speed circle coincides with speed of gun (or target) in knots on correction circle. Clamp discs together. (2) Rotate radial arm until range scale line intersects deflection circle at angle to line of fire made by course of gun (or target). (3) Read across by range scale line to correction circle, and note result; the desired error in knots due to motion of gun (or target). (4) Determine sign of error by glance at plotting board. Notes. — 1. In all the atove described operations, the position of the decimal point at each step must be fixed by the operator's general knowledge of the conditions in each case. 2. By " angle " is meant that angle less than 90° which the course of the firing ship, direction of apparent wind, or course of the target ship makes with the line of fire. 3. The corrections to be applied to the sights are numerically equal to the determined errors, but of opposite sign. 4. Variations from standard in powder temperature and density of atmosphere cause errors in range only; none in deflection. 26. In this paragraph is given the mathematical demonstration that the quantites given in the tables for the value of the density of the atmosphere and of the multipliers for Column 12 of the range tables are theoretically slightly in error. (a) The table of multipliers for Column 12 of the range tables to be found on pages 7 and 8 of Bureau of Ordnance Pamphlet No. 500, on the ]\Iethod of Computing Range Tables (and in Table IV of the Range and Ballistic Tables, edition for use at the Naval Academy) is based on the standard table of densities for given barometer and thermometer readings. This latter gives the ratio of the density of half-saturated air for a given temperature and barometric height to the density of half-saturated air at 15° C. (59° F.) and 750 millimeters (29.5275") barometric height. The values given in the density tables were computed from the formula : g^ 1.05498 16 ' 29.4338 .924854-.002036i . in which H is the barometric height in inches, t is the temperature in degrees Fahren- heit, and Ft is the vapor pressure in saturated air at t° . (b) Throwing out all constant multipliers, this equation will take the form «=^(^!) in which x, y and z are varial)les. Now if we desire to determine the values of S for different temperatures for any given barometric height, the barometric height becomes a constant also for the time being; that is, x in the above function becomes a constant, and the expression for the value of 8 becomes an equation of the first degree involving only two unknown variables. Therefore, all values of 8 for this particular barometric height must lie on the same straight line when the curve is plotted; in other words, the curve in question must be a straight line. The values given in th" table do not exactly do this, and are therefore in error to the extent to which they deviate from this requirement. The errors are believed to be due to decimal differ- ences in computation, and not to be of material magnitude. (c) The transformations by which the values in the density table are trans- formed into values of the multipliers for Column 12 are simply arithmetical processes, and of such a nature that they do not invalidate the above law. The same criticism therefore applies to the values of the multipliers given in the table. APPENDIX C. ARBITRARY DEFLECTION SCALES FOR GUN SIGHTS. INTRODUCTORY. 1. In many cases, notably in turret sights, the system of marking the deflection scales of sights in "knots," as described in this text book, is no longer carried out; these scales being marked in arbitrary divisions instead, the manner of constructing and using which scales will now be explained. 2. The method of controlling deflection by means of " deflection boards " and " arbitrary scales '"' was devised for the purpose of relieving the sight setters of the responsibility of keeping the deflection pointer on a designated deflection curve. The principle upon which the method is based is in no way different from the standard method of controlling deflection by means of knot curves. It differs in the method of application, in that one curve sheet upon which the knot curves are drawn per- forms the functions of the curve drums formerly fitted upon each individual sight. Many of the sights still in service are adapted for the use of either method of deflec- tion control, and it will be seen by trying both methods that they give the same results, regardless of which one is used. 3. The method of bringing the point of impact on the target in deflection in no way differs from that of bringing the point of impact on the target in range, except that deflection correction controls the angle of the sight with respect to the axis of the gun in the horizontal plane, while range correction controls it in the vertical plane. If the point of impact be short of the target, or, in other words, too low, the sight is raised : if the point of impact is to the left in deflection, the rear end of the telescopic sight is moved to the right, and vice versa. In either case it is the angle between the axis of the telescope and the axis of the gun that is changed, for range in the vertical plane, and for deflection in the horizontal plane. 4. To arrive at a clear understanding of the principle of deflection, it should be comprehended that all deflection measurements can be reduced to angular measure- ments. If the horizontal angle between the axis of the gun and the line of sight be the same for all the guns of the same caliber firing, then the corresponding deflec- tion, whether measured in knots or in yards, will also be the same for all those guns. It is thus seen that the sights for all types can be so constructed that the unit of measurement for deflection is an angle. PRINCIPLE OF ARBITRARY SCALES. 5. In the method of controlling deflection by the use of " deflection boards " and " arbitrary scales," the unit of measurement, that is, the angle corresponding to one division of the scale, is the angle that is subtended by one-half of a chord of 0.2 of an inch at 100" radius ; that is, it is the angle whose tangent is .001. By using this unit of measurement, the divisions on the arbitrary scale {G, Plate III), are all equal to 0.1 of an inch on all deflection boards for all sights for all guns, and all deflection boards are therefore uniform in construction. The arbitrary scale fitted to each sight is graduated so that one division of the sight scale corresponds to this APPENDICES 327 standard angle, whatever the value of the sight radius, and the actual magnitude of each such division in fractions of an inch therefore depends upon the value of the sight radius, and is determined from it by proportion, as follows: X 0.1 , / I 100 ' 1000 where x (in fractions of an inch) is the magnitude of the arbitrary division, and I is the sight radius in inches. These arbitrary scales, when once graduated, become permanent, regardless of any change in initial velocity or other modifications affect- ing the trajectory. The necessary corrections to provide for a change in initial velocity, for instance, would be made on the curve sheet {J, Plate III), and expensive and troublesome modifications in the manufactured scales on the sights would there- fore be unnecessary. As the above-mentioned curve sheets are made on drawing paper, quickly and at small cost, it will be seen that changes in the ballistics of the guns could be made without great expense or delay in the supply of the necessary means for deflection control. 6. In the triangle under consideration, the " side opposite " to the angle adopted as the standard angular unit of deflection, that is, the angle whose tangent is .001, is sometimes known as a "mill," because the side opposite is always one one-thousandth part of the side adjacent. In this case it is therefore the angle that corresponds to a deflection of 1 yard at 1000 yards range, and to a deflection of 10 yards at 10,000 yards range, etc. GENERAL DESCRIPTION OF THE SIGHT DEFLECTION BOARD. 7. The " sight deflection board," as shown on Plate III, as furnished to ships, is simply a means of mechanically turning a determined deflection in knots into the units of the arbitrary scale, and at the same time applying the drift correc- tion for the given range. It consists of a wood or aluminum board. A, about 20" square. On each side is a rack, B, which is secured by wing nuts, C. Across the top, and also held by the wing nuts C, is a metal strip, D, which carries the sliding pointer, E. The scale of arbitrary divisions, G, slides up and down the board parallel to itself upon the racks, B, as guides. A pinion on each end of the shaft, F, runs upon the racks, B, and prevents canting of the scale, G. The sliding pointer, H, is carried upon the scale, G, for use in keeping track of the divisions of the scale used. The curve sheet, J , is cut to fit under the racks, B, where it is held from slipping, after being properly adjusted, by the wing nuts, C. In placing the sheet on the board, it must be so adjusted that the reference line, XX, will always be under the " 50 " mark of the scale G as the latter is run up and down from top to bottom of the board. (It will be noted on the plate that the line AM', which sliould intersect the 50 curve at zero yards range, is slightly to the right of that curve at the 1000-yard range mark at the top of the curve sheet, which is of course as it should be. The slight divergence of the 50 mark of the scale G from the line XX that is noticeable on the plate is undoubtedly due to parallax in taking the photograph, the camera apparently not having been set up directly in front of that point.) 8. The legend on the curve sheet shows for what sights, for what caliber of gun, and for what initial velocity it is to be used ; and also indicates, for the information of the spotters and sight setters, the value in knots at some given range to which the divisions on the arbitrary scale correspond. 338 APPEXDKT.S METHOD OF USE. 9. The deflection board is designed primarily for use in the plotting room, but it can be used at any other point that may be desired, such as the spotter's top or in the turrets. 10. When about to open fire, the knot curve to be used should be determined by computation (or by the use of the gun error computer) in the same manner as has been explained for the deflection sight scale marked in knots; but this would no longer be sent out to the guns as the setting of the sights in deflection. Instead, the pointer E is placed at the top to indicate the curve to be used (the 45-knot curve on Plate III). The scale G is then run down the board to correspond to the range to be used (14,000 yards on the plate). The pointer H is then run along the scale G until it is over the proper curve on the sheet (45 knots), and the reading under the same pointer on the scale G will then be the number of divisions of the arbitrary sight scale at which the sights should be set to give the desired deflection (40 divisions on the plate). As the curves on the sheet are the drift curves for the gun, the sight setting in arbitrary divisions of the scale thus found will of course include the drift correction. 11. As the range varies during the firing, the scale G is moved up and down to follow it, and the pointer H is moved to the right or left to keep it over the proper curve on the sheet (45 on the plate) . The pointer H will then always indicate on the scale G the proper sight setting in deflection in the markings of the arbitrary scale. 12. In case the spotter's corrections indicate the use of a new curve at any time, the pointer E is shifted to that curve, and the new readings for the arbitrary scale are read off from the scale G by the pointer H (which is now following the new curve) and sent to the sight setters. 13. By this process the sight setters are relieved of all responsibility in regard to the deflection setting other than that of setting the sight for the scale readings which they receive from time to time from the deflection operator, and it is no longer necessary for them to be continually following a drift curve on the sight drum as the range increases or decreases. CONTROL OF DEFLECTION WITHOUT THE USE OF KNOT CURVES. 14. If preferred, or if no deflection board be at hand, a table may be made up showing the value in knots of one or more divisions of the arbitrary dtflection scale at different ranges, from which the spotter can estimate first his initial sight setting in deflection, and afterwards any changes that he may desire to make. So far as the deflection in yards is concerned, all that is necessary is a knowledge of the fact that for all guns, at all ranges, with all arbitrary deflection scale sights, one division of the arbitrary scale causes or corrects a deflection in yards equal to one one-thousandth of the range; while for the transformation from "knots" to "arbitrary divisions" and the reverse, a table can be used similar to : Range in Yards. Deflection in — Knots. Divisions of Arbitrary Scale. 5000 loono 15000 20000 5 5 5 5 3.5 4.0 4+ 4.5 APPENDICES 329 It will be seen from the above table that, for the gun in question, the variation between knots and divisions is so slight for all probable battle ranges that no great error would result from assuming that 5 knots always equal 4 divisions at all such ranges. 15. The method of control described above involving the use of the board gives greater accuracy than the one using curve drums on the sights, as the deflection board permits the curve sheets to be made on a larger scale. It is not necessary, however, to continue the use of the deflection board after the initial data has been obtained for opening fire. The board may be used to determine the setting of the sights in deflec- tion by the arbitrary scale for firing the first shot; and after that the spotter can indicate the deflection changes in terms of the arbitrary scale, providing he knows approximately the value of the arbitrary divisions in knots or yards at the target, for the approximate range at which the firing is being conducted (in yards, this is one one-thousandth of the range in yards, as already seen) ; and, after shots are seen to be hitting at the proper point, they can then be held at that point by giving the spotter's corrections in terms of the arbitrary scale, as soon as the point of impact appears to creep to the right or to the left, and before it can creep ofi: the target. PROPOSED USE OF ARBITRARY DIVISIONS FOR RANGE SCALES. 16. The advisability of using the " arbitrary scale " method for ranges as well as for deflection (replacing "yards" in range) is under consideration, and should it be done, the same division (one mill) would be used. This, with a sight radius of 100 inches, would give clearly read divisions on the sight scale corresponding to angular differences of two minutes in arc in elevation. 17. The use of the arbitrary scale in range would avoid any changes in sight graduations due to possible changes in initial velocity, as the divisions of the arbitrary scale could be taken from a range chart drawn on paper. This chart could be so drawn as to permit of compensation for slight changes in initial velocity due to variations in the powder, in powder temperature, etc. Also, as all divisions of the sight scale would be of equal magnitude, this system would lend itself readily to the employment of some step by step mechanical means of setting the sights directly from the plotting room or elsewhere, without the interposition of any person as a sight setter. B C he to ids es. es, m, .on :tion board. PLATE III.— DEFLECTION BOARD. ATMOSPHERIC DENSITY TABLES, BEING REPRINTS OF TABLE III AND TABLE IV, FROM THE EANGE AND BALLISTIC TABLES, 1914, PRINTED TO ACCOMPANY THIS TEXT BOOK OF EXTERIOR BALLISTICS. NOTES. 1. By the nsc of tliese two tables, especially of Table IV, in conjunction with the range table for any particular gun, may be solved all practical problems relating to the use of the range table in controlling the fire of that particular gun, by the methods explained in Chapter 17 of this text book, on the practical use of the range tables. These two tables are all that is necessary, also, in conjunction with the range tables, for the solution of calibration problems, as given in Chapters 18 and 19. 3. If it be desired to solve general ballistic problems for any particular gun, however, it will be necessary to have at hand the other tables contained in the edition of Range and Ballistic Tables printed to accompany this text book. ATMOSPHERIC DENSITY TABLES, BEING EEPRINTS OF TABLE III AND TABLE IV, FROM THE EANGE AXD BALLISTIC TABLES, 1914, PRINTED TO ACCOMPANY THIS TEXT BOOK OF EXTERIOE BALLISTICS. NOTES. 1. By the iisc of tliese two tables, especially of Table IV, in conjunction with the range table for any particular gun, may be solved all practical problems relating to the use of the range table in controlling the fire of that particular gun, by the methods explained in Chapter 17 of this text book, on the practical use of the range tables. These two tables are all that is necessary, also, in conjunction with the range tables, for the solution of calibration problems, as given in Chapters 18 and 19. 2. If it be desired to solve general ballistic problems for any particular gun, however, it will be necessary to have at hand the other tables contained in the edition of Kange and Ballistic Tables printed to accompany this text book. INTRODUCTION TO TABLE III 1. Tliere are given in this table values of 8, the ratio of the density of half-saturated air for a given temperature and barometric height to the density of half-saturated air for 15° C. (59° F.), and 750 mm. (29.5275 inches) barometric height. These values are computed by the formula 8 = which H is barometric height in inches, t is temperature Fahrenheit, and Ft is the vapor pressure in saturated air at t°. H- r^t 29.4338 ^ .92485 + .0020361' TABLE III t[ 28 in. 29 in. 30 in. 31 in. tl 28 in. 29 in. 30 in. 31 in. tf 23 in. 29 in. 30 in. 31 in. U 28 in. 29 in. 30 in. 31 in. 1.073 1.112 1.150 1.188 25 1.017 1.053 1.088 1.125 50 .966 1.000 1.035 1.069 75 .917 .950 .982 1.016 1 1.071 1.110 1.148 1.186 26 1.015 1.05] 1.086 1.123 51 .964 .998 1.033 1.067 76 .915 .948 .980 1.014 2 1.069 1.108 1.146 1.184 27 1.013 1.049 1.084 1.121 52 .962 .996 1.031 1.065 77 .913 .946 .978 1.012 3 1.066 1.105 1.143 1.181 28 1.011 1.047 1.082 1.119 53 .960 .994 1.029 1.063 78 .912 .945 .977 1.010 4 1.064 1.103 1.140 1.178 29 1.009 1.045 1.080 1.117 54 .958 .992 1.027 1.061 79 .910 .943 .975 1.008 5 1.062 1.100 1.137 1.175 30 1.007 1.043 1.078 1.115 55 .956 .990 1.024 1.058 80 .908 .941 .973 1.006 6 1.060 1.098 1.135 1.173 31 1.005 1.041 1.076 1.113 56 .954 .988 1.022 1.056 81 .906 .939 .971 1.004 7 1.057 1.095 1.132 1.170 32 1.003 1.039 1.074 1.111 57 .952 .986 1.020 1.054 82 .904 .937 .969 1.002 8 1.055 1.093 1.130 1.168 33 1.000 1.036 1.071 1.108 68 .950 .984 1.018 1.052 83 .903 .935 .967 1.000 9 1.052 1.090 1.127 1.165 34 .998 1.034 1.069 1.105 59 .948 .982 1.016 1.050 84 .901 .933 .965 .998 10 1.050 1.088 1.125 1.163 35 .996 1.031 1.066 1.102 CO .946 .980 1.014 1.048 85 .899 .931 .903 .995 11 1.048 1.08G 1.123 1.161 36 .994 1.029 1.064 1.100 61 .944 .978 1.012 1.046 86 .897 .929 .961 .993 12 1.046 1.084 1.121 1.169 37 .992 1.027 1.062 1.098 62 .942 .970 1.010 1.044 87 .895 .927 .959 .991 13 1.043 1.081 1.118 1.156 38 .990 1.025 1.060 1.09G 63 .941 .974 1.008 1.042 88 .893 .925 .957 .989 14 1.041 1.079 1.116 1.153 39 .988 1.023 1.058 1.094 64 .939 .972 1.006 1.040 89 .891 .923 .956 .987 IB 1.039 1.077 1.113 1.150 40 .986 1.021 1.056 1.092 05 .937 .970 1.003 1.037 90 .889 .921 .953 .985 10 1.037 1.074 1.110 1.147 41 .984 1.019 1.054 1.090 06 .935 .908 1.001 1.035 91 .887 .919 .951 .983 17 1.035 1.072 1.108 1.145 42 .982 1.017 1.052 1.088 67 .933 .906 .999 1.033 92 .885 .917 .949 .981 18 1.032 1.0C9 1.105 1.142 43 .980 1.015 1.050 1.085 68 .931 .964 .997 1.031 93 .884 .910 .947 .979 19 1.030 1.067 1.103 1.140 44 .978 1.013 1.048 1.083 69 .929 .962 .995 1.029 94 .882 .914 .945 .977 20 1.028 1.065 1.101 1.138 45 .976 1.011 1.046 1.081 70 .927 .900 .993 1.027 95 .880 .912 .943 .975 21 1.026 1.063 1.099 1.136 46 .974 1.008 1.043 1.078 71 .925 .968 .991 1.025 96 .878 .910 .941 .973 22 1.024 1.061 1.097 1.134 47 .972 1.006 1.041 1.076 72 .923 .956 .989 1.023 97 .876 .908 .939 .971 23 1.021 1.068 1.094 1.131 48 .970 1.004 1.039 1.073 73 .921 .954 .987 1.021 98 .874 .906 .937 .969 24 1.019 1.056 1.091 1.128 49 .968 1.002 1.037 1.071 74 .919 .952 .985 1.019 99 .872 .904 .935 .967 2B 1.017 1.053 1.088 1.125 50 .966 1.000 1.035 1.009 75 .917 .950 .982 1.010 100 .870 .902 .933 .965 INTRODUCTION TO TABLE IV 1. Tliis table is to replace Table III for handy use in a certain specific case. Column 12 of the Range Tables gives the change in ranee resulting from a raria- tion of ± 10% in the density of the atmosphere from the standard. To use this data, by the use of Table 111, it is necessary to determine from Table 111 the percentage variation in density, and then apply this to the data in column 12. To use this table, however, take from it the multiplier corresponding to the given atmospheric conditions and from column 12 of the range tables the number of yards change in range caused by a variation in density of ± 10%, multiply both together, and the product, with the sign of the multiplier, will be the variation in range due to the variation from standard of the existing atmospheric con- ditions. TABLE IV. MULTIPLIERS FOR COLUMN 12 OF RANGE TABLES Arguments, Temperature and Barometric Pressure tl 28 in. 20 in. 30 in. 31 in. tl 23 in. 29 in. 30 in. 31 in. t( 28 in. 29 in. 30 in. 31 in. U 23 in. 29 in. 30 in. 31 in. —.73 —1.12 —1.50 —1.88 25 — .17 — .53 — .88 —1.25 60 .34 .00 —.35 —.69 75 .83 .50 .18 — .16 1 —.71 —1.10 -1.48 —1.80 26 —.15 — .61 -.80 —1.23 51 .36 .02 —.33 — .67 76 .85 .52 .20 — .14 2 —.69 —1.08 —1.46 —1.84 27 —.13 -.49 — .84 —1.21 62 .38 .04 —.31 — .65 77 .87 .54 .22 — .12 3 —.66 —1.05 —1.43 —1.81 28 —.11 —.47 — .82 —1.19 53 .40 .06 —.29 — .63 78 .88 .55 .23 —.10 4 —.64 —1.03 —1.40 —1.78 29 —.09 — .45 —.80 —1.17 64 .42 .08 — .27 -.01 79 .90 .57 .25 — .08 6 —.62 —1.00 —1.37 —1.76 30 — .07 — .43 —.78 —1.15 65 .44 .10 —.24 — .58 80 .92 .59 .27 — .06 6 —.60 — .98 —1.35 —1.73 31 — .05 — .41 — .76 —1.13 56 .46 .12 — .22 — .56 81 .94 .61 .29 —.04 7 —.57 — .95 —1.32 —1.70 32 —.03 —.39 — .74 —1.11 67 .48 .14 — .20 — .54 82 .96 .63 .31 —.02 8 — .55 — .93 —1.30 —1.68 33 .00 — .36 —.71 —1.08 58 .50 .16 — .IS — .52 83 .97 .65 .33 .00 9 —.52 — .90 —1.27 —1.65 34 .03 -.34 — .69 —1.05 59 .52 .18 -.16 — .50 84 .99 .67 .35 .02 10 —.60 — .88 —1.25 —1.63 35 .04 — .31 — .66 —1.02 60 .54 .20 — .14 — .48 85 1.01 .69 .37 .05 11 —.48 — .86 —1.23 —1.61 3G .06 — .29 — .64 —1.00 61 .66 .22 — .12 —.46 86 1.03 .71 .39 .07 12 —.46 — .84 —1.21 —1.59 37 .08 — .27 — .62 — .98 62 .68 .24 — .10 — .44 87 1.05 .73 .41 .09 13 — .43 — .81 —1.18 —1.56 38 .10 — .25 — .60 — .96 63 .59 .26 — .08 — .42 88 1.07 .75 .43 .11 14 — .41 — .79 -1.16 —1.53 39 .12 — .23 —.68 — .94 64 .61 .28 —.06 —.40 89 1.09 .77 .45 .13 15 —.39 — .77 —1.13 —1.50 40 .14 —.21 — .56 — .92 65 .63 .30 —.03 — .37 90 1.11 .79 .47 .15 16 —.37 — .74 —1.10 —1.47 41 .16 — .19 — .54 — .90 66 .65 .32 — .01 — .35 91 1.13 .81 .49 .17 17 —.35 — .72 —1.08 —1.45 42 .18 —.17 — .52 — .88 67 .67 .34 .01 — .33 92 1.15 .83 .51 .19 18 —.32 — .69 —1.05 —1.42 43 .20 — .15 —.50 — .85 68 .69 .36 .03 — .31 93 1.16 .84 .53 .21 19 — .30 - .67 —1.03 —1.40 44 .22 — .13 — .48 — .83 69 .71 .38 .05 — .29 94 1.18 .86 .55 .23 20 — .28 — .66 -1.01 —1.38 45 .24 — .11 — .46 — .81 70 .73 .40 .07 — .27 95 1.20 .88 .57 .25 21 — .26 — .63 — .99 —1.36 46 .26 — .08 — .43 — .78 71 .75 .42 .09 — .25 96 1.22 .90 .69 .27 22 — .24 — .61 — .97 —1.34 47 .28 —.06 —.41 — .76 72 .77 .44 .11 — .23 97 1.24 .92 .61 .29 23 — .21 — .58 — .94 —1.31 48 .30 — .04 — .39 — .73 73 .79 .46 .13 — .21 98 1.26 .94 .63 .31 24 — .19 — .66 — .91 —1.28 49 .32 — .02 — .37 — .71 74 .81 .48 ,15 — .19 99 1.28 .96 .65 .33 26 —.17 — .53 — .88 —1.25 50 .34 .00 — .35 — .69 75 .83 .60 .18 —.16 100 1.30 .98 .67 .35 LOAfTPtRIOD ] \2 HOME USE ^ ALL BOOKS AAAY BE RECALLED AFTER 7 DAYS ^Om NO. DD6, 60., ./r'''^^E'R^;^,^^;fO^^ ®s I YD [6103 raOBABILITY TABLE FOR USE WITH CHAPTERS 20 AND 21. a Probability of a deviation less than a in terms of the ratio 7 a 7 ' P. a 7 ' P. a 7' P. a 7 ' P. 0.1 .004 1.1 .620 2.1 .906 3.1 .987 0.2 .127 1.2 .662 2. 2 .921 3.2 .990 0.3 .189 1.3 .700 2.3 .934 3.3 .992 0.4 .250 1.4 .735 2.4 .945 3.4 .994 0.5 .310 1.5 .768 2.5 .954 3.5 .995 0.6 .368 1.6 .798 2.6 .962 3.6 .996 0.7 .424 1.7 .825 2.7 .969 3.7 .997 0.8 .477 1.8 .849 2.8 .974 3.8 .998 0.9 .527 1.9 .870 2.9 .979 3.9 .998 1.0 .575 2.0 .889 3.0 .983 4.0 .999 COLUMBIAN CLASP W0HCF.STra.WA55. flOLYOKC.MiSS. RQCIV1U£C(»«I, iFra«eFl£LJ3.l>lASS. nABTfOfiacOSN. WAL'KfttMJU. cincmNxri,oHio. san franciscq.cal. No. 25 5i^ INTERPOLATION FORMUL.^. For use with Table II, Ballistic Tables. (1) A =A,+ (2) V =V,+ (3) Z =Z,+ z-z, 100 AV 100 AzA + v-v. -A,)- [(A-A,) AV z—z A.,. 100 '"^^'J- V V (4) A"=A';-^^X v-v, AV + „]. A,.,Az,„ , ^-^^^ a,,,.+ {A'-A:) >\ AzA" ^ZA' AV When V — Ft = 0, (4) becomes (5) a"=a';-\-{A'-a',)^ ^ZA' In using the above formulie exercise great care to use each quantity with its proper sign. These formulae are correct for working from the next lower tabular value only; if work is to be from the next higher tabular value there must be a general change of signs in tlie formulae. Work from the next lower tabular value unless directed to the contrary. fORM NO. DD6, 60., .^T^^^^^i^^^^^^^^ DtKKbLEY, CA 94720 ®s YD I&I03 /• riL.hJL. €) PROBABILITY TABLE For use with Chjipters 20 and 21 and INTERPOLATION FORMULAE For use with Table II, Ballistic Tables, enclosed / \ 25 COLUMBIAN CLASP ^^ WOHCF^rra.lMii. nOLYOKC.MASS. ROClVlLLtCOW, OK ClNClHNXn^CHiO. SAN FRANC15CQ.C1L. 1^« • m t.i'i 5»^