T1ftPlT'f™»C~"'^ ",!c ""y .. ... fyxmll Hmwsiitg f BOUGHT WITH THE INCO PROM THE SAGE ENDOWMENT THE GIFT OF 1891 iitotg ME FUND Ji.jM.L'i.a.S:. Cornell Unlveralty Library UF820 .144 Handbook of problems in direct tire. olin 3 1924 030 764 447 Cornell University Library The original of tiiis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924030764447 HANDBOOK PROBLEMS IN DIRECT FIRE. Captain JAMES M. mOALLS, FIRST REGIMENT ARtTTLERY : IlJSTRUCTOR OF BALLISTICS, UNITED STATES ARTILLERY SCHOOL; Author of " Exterior Ballistics," "Ballistic Machines" etc. NEW YORK: JOHN WILEY & SONS, 53 East Tenth Street. 1890, Copyright, 1890, by James M. Ingalls. All Rights Reserved, Dbttmmond & Neu, New York. Ferris Bros., New York. PREFACE. This book was prepared while the Author was engaged in teaching balHstics to student officers at the Artillery School, Fort Monroe ; and most of the examples were selected from those which had been given out from time to time to the classes under his instruction, as exercises in the practical applications of the ballistic formulae which the more advanced students were required to deduce. It was suggested to the Author, by officers of high rank, that a collection of these and similar examples, in book form, would be of permanent value, not only to the Artillery, but also to the other branches of the service, both regular and militia. A very slight knowledge of mathematics is all that is re- quired for the solution of the most important of the examples. It has come to the Author's knowledge that the first twelve problems have befen taught successfully to non-commissioned officers whose whole stock of mathematics consisted of a little arithmetic and less algebra. In the later problems the symbol of integration has been introduced in a few instances — chiefly, however, for the sake of concise definitions. Wherever this symbol occurs it may be passed over without detriment to the practical applications. It is believed that this is the first book of the kind ever pub- lished in any /language ; and the Author trusts that this will excuse whatever faults of arrangement or of execution may be detected by the reader. The solutions of the first seventeen problems are based upon the method first given to the world by captain (now Lieutenant-Colonel) Siacci, of the Italian IV PREFACE. Artillery, in 1880, and which is now universally employed in Europe and America. In connection with Siacci's method the Author has here introduced the labor-saving, auxiliary equa- tions of Captain Scipione (also of the Italian Artillery), and which he has rendered available, for the first time, to English and American Artillerists by extensive tables, prepared ex- pressly for this work. The attentive reader will find scattered throughout the work methods and processes which he will seek for in vain elsewhere, and which it is hoped may be found use- ful to the practical Artillerist. At the suggestion of the publishers an appendix (Appendix I) has been added, giving a concise, but quite complete, de- duction of the formulas of Siacci's method, together with other matter which it Is hoped may be acceptable to the mathemati- cal reader. The Author has also added Appendix II on his own responsibility, giving the latest and best methods for the solution of problems in Mortar-firing. Of the tables given in the book, those computed by the Author are the following : Table of Altitude Factors, page 89 ; Table of the yalues of B, page 153 ; Tables 2, 3, 4, and 5 in Problem XXI; Tables I, II, III, and V at the end of the book. TABLE OF CONTENTS. Introduction. Definitions. Notation. General Formulae of direct fire. Formulae relating to the horizontal range. Auxiliary formulae. Ballistic coefficient. Coefficient of reduction. Density of the air. Ballistic tables. Auxiliary tables. Pages I to 15 Problem I. Given the muzzle velocity and data for the ballistic coefficient, to calculate the velocity at a given distance from the gun. Comparison of computed with measured velocities. Striking energy of a projectile. Formulse for striking energy in terms of metric units. Penetration of projectiles. Eleven examples. Pages 16 to 27 Problem II. Given the ballistic coefficient and the remaining velocity at a given distance from the gun, to determine the muzzle velocity. Three examples. Pages 28 to 30 Problem III. To compute the distance from the gun at which the muzzle velocity will be reduced a given amount. Five examples Pages 31 to 35 Problem IV. To compute the coefficient of reduction when the muzzle velocity and final velocity are both measured. One example Page 36 Problem V. To compute the time of flight when the muzzle velocity and distance passed over by the shot, are given. Three examples Pages 37 to 39 Problem VI. To compute the remaining velocity after a given time, taking into account the effect of the wind. One example Pages 40 to 41 iii IV TABLE OF CONTENTS. Problem VII. To compute the remaining velocity at a given distance from the gun, taking into account the effect of the wind. Two methods. Three examples. Pages 42 to 46 Problem VIII. To cpmpute the effect of a wind upon the range. Application to the 3.2-in. B. L. steel gun.* Remarks upon Problems VI, VII and VIII. Two examples. Pages 47 to 53 Problem IX. Given the ballistic coefficient, the muzzle velocity, and angle of departure, to compute the remaining elements. Two methods. Nine examples. Pages 54 to 66 Problem X. Given the range and angle of departure, to copipute the muzzle velocity. Two methods. Six examples Pages 67 to 73 Problem XI. Given the range and striking velocity, to compute the remaining elements of the trajectory. Three examples Pages 74 to 77 Problem XII. Given the muzzle velocity and range, to compute the remaining elements of the trajectory. Two methods. Three examples Pages 78 to 83 Problem XIII. Given the muzzle velocity and angle of departure, to compute the elements of the trajectory at the summit. Effect of altitude upon the flight of a projectile. Table of altitude factors. Penetration of armor. Five examples. Pages 84 to 95 Problem XIV. Given the muzzle velocity, to determine the angle of departure which will cause a projectile to hit an object situated above or below the level of the gun. Two methods. Horizontal range. Rigidity of the trajectory. Seven examples. Pages 96 to 108 Problem XV. The computation of coBrdinates for plotting a given trajectory. Two methods. Approximate expression for j/. Five examples. . Pages 109 to 118 TABLE OF CONTENTS. V Problem XVI. Given the elements of a trajectory and any ordinate, to compute the corre- sponding abscissa. Practical applications. Danger-space. Application of the principle of the rigidity of the trajectory. Approximate method for computing the danger-space. Sladen's method of computing danger-spaces. Danger-range. Point-blank firing. Effect of error in estimating distances. Fourteen examples. Pages iig to 142 Problem XVII. Given the range, the final velocity, and the maximum ordinate, to compute the initial velocity and the ballistic coefficient, for small angles of departure. Relation between weight and calibre of bullet. Inverse problem. Relation between velocity, weight of projectile, and powder charge. Three examples. Pages 143 to 151 Problem XVIII. To calculate the volumes and weights of oblong projectiles and their ballistic coefficients. Relative weights of oblong projectiles. Ballistic coefficients of different oblong projectiles. Similar oblong projectiles. Length of ogival head. Eighteen examples Pages 152 to 168 Problem XIX. Given the muzzle velocity, the angle of departure, and the range, to compute the ballistic coefficient and coefficient of reduction. Best method of computing the coefficient of reduction. Two examples Pages 169 to 176 Problem XX. To calculate the drift of an oblong projectile. General explanation of drift. Mayevski's formula for drift. Baills' formula for drift. Effect of wind upon drift. Didion's method of computing the deviating effect of wind. Maitland's formula for wind deviation. Formula for computing the area of transverse sec- tion of oblong projectile. Twist of rifling. Rotation of a projectile about its axis. Revolutions per second. Surface velocity of rotation. Angular velocity of projectile's rotation. Moment of inertia of an ogival head. Radius of gyra- tion of an ogival head. Moment of inertia of the cylindrical part of a projectile. Radius of gyration of body of projectile. Radius of gyration of a cored shot or shell. Weight of cored shot. Centre of gravity of an'ogival head. Total muzzle energy of an oblong projectile. Twelve examples Pages 177 to 201 Problem XXI. To determine the probability of fire and the precision of fire-arms. Prelim- inary considerations. Centre of impact. The sum of the squares of the vertical (or horizontal) deviations with reference to the centre of impact, is a minimum. Absolute deviations. Centre of impact on a horizontal target. Law of the de- VI TABLE OF CONTENTS. viation of projectiles. Mean quadratic deviations. Mean deviations. Relation between the mean quadratic and mean deviations. Expression for it in term* of the mean quadratic and mean deviations. General probability table. Probable deviation. Fifty-per-cent zones. Twenty-five-per-cent rectangle. Probable rectangle. Table for computing sides of rectangles having a given probability. Enveloping rectangle. Comparison of experiment with theory. Table for com- puting the width of a zone of given probability. Probability of hitting any plane figure. Curves of equal probability. Relations between the semi-axes of ellipses of equal probability and the deviations. Probability of a projectile falling within an ellipse of equal probability. Table for computing the semi-axes for a given probability. Area of probable ellipse. Equation of probable ellipse. Table for computing the probability of a. given ellipse. Probability of hitting a. given object. Supply of ammunition. Probability that at least one shot will hit the object. Criterion for rejecting abnormal shots. Probability of the arithmetical mean. Twenty-one examples Pages 202 to 246 Problem XXII. To compute a range table. General considerations. Range column. Angles of departure. Angles of elevation. Variations of the angles of departure or of elevation. Variations of the muzzle velocity. Time of flight. Drift. Angle of fall. Striking velocity and penetration of armor. Range table for 8-inch B. L. naval gun. Variation of the angle of departure due to a variation of the range. Variation of the range due to a variation of the muzzle velocity. Pages 247 to 255 Appendix I. DEDUCTION OF THE GENERAL FORMULA. OF piRECT FIRE. Resistance of the air to the motion of a projectile. Oblong projectiles. Ter- minal velocity. Spherical projectiles. Differential equations 'of motion. Cases which admit of integration in finite terms. Motion in vacuo. Trajectory in the air. Approximate equations of motion of direct fire. Expression for the ve- locity when the resistance varies as the square of the velocity. Effect of the wind upon the range and striking velocity of a prejectile fired with a small angle of elevation. Seven examples Pages ^57 to 284 Appendix II. FORMUL/E FOR MORTAR FIRING. Euler's method. Expressions for the coSrdinates x and y. Expression for the time. Otto's tables. Modification of Euler's method and Otto's tables. Siacci's method for curved and high-angle fire. Nine examples. Pages 285 to 299 Problems in Direct Fire. INTRODUCTION. It has been the object in the following pages to give prac- tical solutions of all the important problems of gunnery relat- ing to direct fire ; and to illustrate the solutions by numerous examples, fully worked out in the manner which a considerable experience has shown to be the most simple and concise. For this purpose logarithms are habitually employed in making the numerical computations, as it is believed that by their use a considerable saving of time and labor is effected, and the liability to error reduced to a minimum. In the absence of a table of logarithms, however, nearly all the examples may be worked by simple arithmetic. Five-place logarithms are sufficient for the correct solution of all gunnery problems ; and four-place logarithms can often be used to advantage. An excellent five-place table has been compiled by Wentworth and Hill, and published by Ginn & Co. of Boston. This is the table that has been used in this work. Definitions. — The following definitions of a few technical terms which will be constantly employed, are here given for convenient reference : The trajectory is the curve described by the centre of gravity of the projectile. It is divided into the ascending and de- scending branches, the point of division being called the sum- mit. The line of departure is the prolongation of the axis of the 2 PROBLEMS IN DIRECT FIRE. bore at the instant the projectile leaves the gun. It is there- fore tangent to the trajectory at the muzzle. The angle of departure is the angle which the line of depart- ure makes with the horizontal plane. The angle of elevation (or depressioii) is the angle which the axis of the bore, when the piece is laid, makes with- the hori- zontal plane. It is sometimes called the quadrant elevation because it is often determined by applying the quadrant to the face of the piece. Jump is the difference between the angle of elevation and angle of departure. It varies in value from an angle too small to be appreciable to one of a degree of arc or even more, ac- cording to the kind of carriage and platform employed. It also varies somewhat with the angle of elevation. It must be determined by experiment in each case. Muzzle velocity is the velocity of the projectile on leaving the piece. It is sometimes called initial velocity or velocity of projection. Remaining velocity is the velocity at any given point of the trajectory. Final velocity is the velocity the projectile has in the descending branch when at the level of the gun. The range is the horizontal distance from the muzzle of the gun to that point of the descending branch of the trajectory, called the point of fall, which is at the level of the gun. This term is also applied to the distance between the gun and the target, or between the gun and the point where the projectile strikes, whether above or below the level of the gun. The angle of fall is the angle which the tangent to the tra- jectory at the point of fall makes with the horizontal plane passing through the muzzle. Direct fire is with guns, with service charges, and angles of elevation not exceeding 15*^. Indirect (or curved) fire is with guns, howitzers, and mortars, -with reduced charges (and therefore low velocities), and angles of elevation not exceeding 15°. High-angle fire is when the angle of elevation exceeds 15°. INTRODUCTION. 3 Notation. — The following notation will be employed : g denotes acceleration of gravity, and will be taken at 32.16 f. s. w, weight of a projectile in pounds. d, diameter of a projectile in inches. tf„ density of air two-thirds saturated with moisture, ther- mometer 60° (F.) and barometer 30 inches. 5, density of air two-thirds saturated for any observed read- ings of the thermometer and barometer. c, coefficient of reduction, depending upon the kind of pro- jectile employed. C, ballistic coefficient = -^ -^„ . o cd V, muzzle velocity in feet per second. V, velocity at any point of the trajectory. v^, velocity at point of fall. z/„, velocity at summit of trajectory. 0, angle of departure. 6, angle which the tangent to the trajectory at any point makes with a horizontal plane ; positive in the ascending and negative in the descending branch. 00, angle of fall. This angle, which is really negative, will be regarded as positive. X, y, rectangular co-ordinates of any point of the trajectory, the origin being at the muzzle of the gun, and the axis of x horizontal. To designate the co-ordinates of a particular point subscripts will be used as ;r„, y^, co-ordinates of the summit, etc. t, time of describing any portion of the trajectory from the origin. X, horizontal range. T, time of flight from the origin to the point of fall. u, an auxiliary quantity, function of the velocity and inclina- tion, defined by the equation u = v •-. m„ will represent cos the value of u at the summit, and u^ at the point of fall. 5( V), S{u), etc., space functions. A{V), A{U), A{u), etc., altitude functions. PROBLEMS IN DIRECT FIRE. /(F), I(JJ), I{u), etc., inclination functions. T{y), T{U), T{u), etc., time functions. B{v), M{v), drift functions. £„, energy of projectile at any point of its course = )t-tons. £/, energy per inch of shot's circumference = foot-tons = — °- . nd W, velocity of wind in feet per second, Wf„ velocity of wind parallel to range in feet per second. W„, velocity of wind normal to range in feet^er second. p, resistance of the air to a projectile's motion, in pounds, t, thickness of armor in inches. General Formulae. — The following fundamental equations give the values of x, t, y, and 6 at any point of the trajectory, in terms of V, 0, u, and C. They are the basis of the solutions of most of the problems of direct fire. Their demonstration is given in Appendix i. x=C\S{u)-S{V)\, (I) '^ = col0l^(^)-^(^)|' (^> f=-^-^.14^^-(r)|,. (3) tan^=tan0-^^^|/(«)-/(F)|, .... (4) cos 6 U = V -; (t,\ cos • • V3/ Of these equations, (3) was first given by Major Siacci of the Italian Artillery ; and the method of solving trajectories which is based upon it goes by the name of " Siacci's Method." INTRODUCTION. 5 At the Summit of the trajectory the motion of the projectile is horizontal ; and therefore at this point B = o. Substituting this value of 6 in (4) and (5) and reducing, we have at the summit the following special equations which will be found of great use in the sequel : /K) = -^i^ + /(F). (6) cos (7) In these equations F(the muzzle velocity) and (the angle of departure) relate to the muzzle of the gun and may be con- sidered constant. All the other symbols refer to any point of the trajectory ; namely, x, y are the rectangular co-ordinates of any point, / is the time from the origin to the point {x,y), is the angle made by the tangent to the trajectory at the same point with a horizontal plane, and v is the corresponding velocity, u is an auxiliary quantity defined by (5). Equation {3) would be the equation of the trajectory were it possible to ehminate the variable u ; but as this is impossible, the trajectory is determined by a combination of (i) and (3). The manner of using these and the following formulas will be fully explained under the appropriate problems. Formulae relating to the Horizontal Range. — By giving suitable values to the variables we may deduce a set of five equations for any point of the trajectory. For example, if we make ^ = o and — ^ = co, we shall have the equations for the point of fall ; and the particular values of x, t, v, and u upon this supposition are designated by X, T, v^, u^. Equations (i), (2), (3), (4), and (5) then become, respectively, X=C\S{u^)-S{V)\, (8) T = PROBLEMS IN DIRECT FIRE. cos-ca Ua = Vu> :! (12) cos ^ ^ We may give to (i i) a form better adapted for computation by combining it with (lo), eliminating I{V); thus, C tan 00 = ( A{u^)-A{V) \ 2 cos" It is evident from (6) and (lo) that we have the relation and therefore (lo) and (13) may be written, with great economy of space, sin2 = C{/(«„)-/(F)}, (15) When and go are both small we have, approximately, 2 cos'

where x is the abscissa of any point whatever of the trajectory, a and b are however taken from the same tables, respectively, as A and B, though not re- ferring necessarily to the same things. INTRODUCTION. 9 Substituting in (22) and (23) for sin 2 its value from (18), they become and ^ = ;i;tan0(i-J-) .... (24) tan0 = tan 0^1 — ^j (25) If in (25) we make ^ = o, we deduce for the summit of the trajectory the identity m = A ■which has already been established. (See Eq. 14.) Ballistic Coefficient. — The ballistic coefficient (C) which appears in nearly all the preceding formulse depends upon the weight, diameter, and smoothness of the projectile, its steadi- ness in flight, and the density of the air it encounters. Its ex- pression is in which d is the diameter of the projectile in inches, w its weight in pounds, c a factor, called the " coefficient of reduc- tion," depending upon the kind of projectile used, its steadi- ness, etc., and-r^a factor depending upon the density of the air. Coefficient of Reduction. — For guns and projectiles simi- lar in every respect to those used in making the experiments upon which the tables are based (and which may be called the standard), the coefficient of reduction will be unity. If the qualities of the gun and projectile are such that the latter meets with a greater resistance than the standard of the same diam- eter, the effect is the same as though the area exposed to re- sistance (that is, the area of its greatest cross-section) were increased ; and therefore in this case c must be greater than unity. On the other hand, if the resistance to a projectile be less than the standard, c will be less than unity. The method lO PROBLEMS IN DIRECT FIRE. for determining the value oi'c in any case, by experiment, will be given farther on. For our new breech-loading guns and the Krupp guns we may assume c = 0.9 without much error. Density of the Air. — In the factor — ^, d^ is the standard density of the air to which the experiments upon which the tables are based were reduced ; and d is the observed density at the time of firing. The value of this factor for any observed temperature (Fahrenheit) and barometric pressure may be taken from Table III. In computing this table the air was sup- posed to be two-thirds saturated with moisture, which is not far from the case on our sea-coast ; and therefore the hygro- metric condition of the atmosphere need not be noticed, the only observations necessary being those of the thermometer and barometer. Example. At target practice the thermometer stood at 88°. 5 and barometer at 30.194 in. -What was the value of — ^? S From Table III we see at a glance that for a temperature of 88°.5 the value -r^for 30 inches is 1.056, and for 31 inches it is 1.022; while the difference between them is 0.034. Therefore -^ — 1.056 — 0.194 X 0.034 = 1.049. Ballistic Tables. — Table I gives the values of the functions Siv), A {v), I(v), and T(v) ; and, also, the drift functions B{v) and M{v), which will be described farther on ; and extends from v — 2800 f. s. to 2^ = 400 f. s. It is based upon a discussion of the values of Bashforth's coefficients K, which he determined from his experiments at Shoeburyness, between the years 1865 and 1880. The table was computed in 1884, and first appeared in the second Artillery School edition of the Author's work on Exterior Ballistics, January, 1885. For convenience of interpolation the first differences are INTRODUCTION. II given in adjacent columns ; and as the second differences rarely exceed eight units of the last order, it will hardly ever be necessary to consider them in using this table. Table II, the ballistic table for spherical projectiles, is based upon the experiments made by General Mayevski at St. Peters- burg, in 1868 ; and extends from v — 2000 to w = 450. It was computed in 1883, and is the only ballistic table for spherical projectiles, based upon Siacci's method, yet published. Formulae for Interpolation. — To find the values of S{v), A {v), etc., when v lies between two consecutive values of v as given in Tables I and II, and when second differences are taken into account, we proceed as follows : Let % and v^ be the two consecutive values of the argument between which v lies. Let v^ — v^^ h; and designate the first and second differences of the function under consideration, by ^, and A^. Then if we symbolize the function byy"(z') we shall have, since f{v) increases while v decreases, ^r \ ^f \ \ ' "<'-'" A """ — "" i- ^0 — '" \ ^» f{v) = /(z^„) + —^ A, ^- ^i -j^) -- , by means of which f{v) can be computed. Conversely, if f{v) is given and our object is to find v, we have V. — v In using this last formula, first compute — r — by omitting the second term of the second member (which is usually very small), and then supply this term, using the approximate value v — v of -^-T — already found. If the second differences are too small to be taken into ac- count, that is, less than eight units of the last order, the above formulae become, respectively, 12 PROBLEMS IN DIRECT FIRE. and ^ = ^0- J (/(^) -/(%)), which express the well-known rules of proportional parts. In Table I the values of k are as follows : From V = 2800 to z/ = 2200, h= $0; " V ^ 2200 " z^ = 1600, A = 10 ; " V — 1600 " V = 1320, A = 5 ; " V = 1320 " z; = 1 160, h= 2 ; " 1/ = 1 160 " 7^ = 4CX), ^ = I. And in Table II, From V = 2000 to z; = 1200, ^ = 10 ; " e/=i200"z/= 450,^= 5. Example i. Find 5(z/) from Table I, when v = 1847.6. We have v^ = 1850, f{v^ = S{v„) = 2916.9, ^ = 10, and J, = 38.2. 0/ N ^ , 1850— 1847.6 .-. S{v) = 2916.9 -| X 38.2 = 2926.0. Example 2. Find from Table II the value of A(y) when ■z/ = 1023.7. We have v, = 1025, /(z/„) = ^(z/„) = 159.15, ^, = 3.84, ^ = 5, and v„ — V = 1.3. .-. ^(1023.7)= 159-15 + ''^^g^"^^ = 160.15. Example 3. Suppose for an oblong projectile we have found S(v) = 12870.2. What is the value of z^? We find in Table I the first value of S{v) less than 12870.2 to be 12856.7, which corresponds to v„ = 719. We also find ^, = 23.6 and k = I. 12870.2 — 12856.7 ^ ^ 23.6 = 719 - 0.57 = 718.43- INTRODUCTION. 1 3 Example 4. What is ^(562.7) by Table II? We have t/„ = 565, ^(565) = 2620.0, ^ = S, ^1 = 104.3, and Jj = 4.8. . ■ . ^(562.7) = 2620.0 + Y X 104.3 - y X Y X 4-8 = 2620.0 + 48.0 — 1.2 = 2666.8. Auxiliary Tables. — We have computed three auxiliary- tables, for oblong projectiles only, which give the values of the auxiliary quantities A, B, and m, respectively. They are based upon Table I, and are to be used in connection with it. These tables have each two arguments, viz., -^ (called z), found in the first vertical column, and the velocity, V, in the upper horizon- tal column. They give the values of the functions to four decimals for equidistant values of s from 100 to 7000, and for V from 1200 f. s. to 2250 f. s. The constant difference between the values of z is lOO, and that between V is 50. In the col- umns A^ are given the differences between the consecutive values of the functions relative to the same value of V and cor- responding to an increase of 100 in the value of z ; and in the columns /J^ are given the corresponding differences for an increase of 50 in the value of V. In the interpolation formulae these differences are to be used as positive whole numbers. If z and V are given values of the arguments intermediate to those found in the tables and we wish for the corresponding values of the function [symbolized z-sjlz, V)], we have in which z„ and F„ are the next smaller tabulated values of the arguments to those given, and f{z„ , F„) is the corresponding tabular value of the function. If f{z, V) and Fare given to find z, we have from the above equation 14 PROBLEMS IN DIRECT FIRE. Similarly, \i f{z, V) and 2 are given to find V, we have Example i. What is the value of A when z = 5279.3 and F= 1623.4? We have s^ = 5200, F„ = 1600, /{z„ V„) = 0.1089, ^^ = 3°, and ^^=iS- .-./(.., F) = ^ = 0.1089 + ^X30-^X55; .-. ^ =0.1087. Example 2. Given 5 = 0.1430 and F= 1740, to find 0. We have FJ, = 1700; and running down the column we see that £•„ = 5200 and f{z^, F„) = 0.1455. Also, .^^ = 47 and ^. = 55- , 100 ( 40 , ) .-. z= 5200 + ^- 1 J5 X 55 + 1430 - I4SS ] = 5240. Example 3. Given m = 0.2400 and z = 5250, to find V. We have z, = 5200, V, = 1700, /{z„ V„) = 0.2236, ^, = 75, and ^^ = 105. .-. .F= 1700 + ^ { j^ X 75 + 2436 - 2400 [ = 173s f. s. It sometimes happens that the given value of z, or of V, is found in the table. In this case the interpolation is shortened since we have z — z„. or V— F„, as the case may be, equal to zero. Example 4. What is the value of A when z = 3947.3 and F'= 1400 f. s. ? We have ^f„ = 3900, ^, = 31, ^^-^„ = 47-3. A3,,V^ = 0.0916, and F„ = F= 1400 f. s. .•. ^ = 0.0916 + .0015 = 0.0931. introduction: i5 Example %. Given m — 0.1834 and z = 4300, to find V. We have z, = s = 4300, F"„ = 1650, /{z, , F„) = 0.1900, and .-. F„= 1650 + ^1 1900- 1834 [ = 1650 + 36 = i^^^ f* ^- Problem I. Given the muzzle velocity ( V), and data for the ballistic coefficient (C), to calculate the velocity {v) at a distance {pc) from the gun. Solution. First compute C by the formula ^~ S cd" taking the value of -~ from Table III, for the given tempera- ture and barometric pressure, and using the proper value of c for thfe gun and projectile under consideration. If no observa- tion of the state of the atmosphere has been made, and if, besides, the value of c is not known, the best that can be done will be to compute C by the equation ^- d'° For all smooth-bore guns and for the older rifled guns in our service c = i. For the new B. L. rifles c = 0.9 approxi- mately. Next, take from the proper table (Table I, for elongated, and Table II, for spherical projectiles), the function S{V) for the given muzzle velocity V. 'Then compute 5(«) by the equa- tion (derived from Eq. (i)), 5(«) = ^ + 5(F), in which X 16 PROBLEM I. 17 and take the corresponding value of u from the table. Then from (5) we have cos cos in which v, u, and B refer to the point of the trajectory whose abscissa is x. If the angle of departure is small, not exceeding 10°, B will also be small and the ratio of the cosines will be nearly unity. Under these conditions we may for practical purposes assume that v=u. This assumption, namely, that cos cos 6 = I, implies that the motion of the projectile is horizontal ; that is, that the trajectory is a horizontal right line ; and, therefore, that gravity, which acts vertically, neither increases nor retards the projectile's motion. Should the trajectory be so curved that the ratio of the cosines differs materially from unity, it will be necessary to know the values of 4> and B in order to compute V with the greatest accuracy, as will be exemplified in subse- quent problems. Example i. Calculate the final velocity of a 15-inch solid shot for a range of 1000 yards, the muzzle velocity being 1700 f. s. and atmosphere normal. We have given V= 1700, d= 14.87, w = 450, and X — 3000, to find v^. Note. — Hereafter we shall omit the subscript oa as being unnecessary, since the nature of the problem will determine what velocity is meant. 1 8 PROBLEMS IN DIRECT FIRE. We have the formulas r — "^ _X to compute, v. The work by logarithms may be concisely and conveniently arranged as follows : log w = 2.65321 2 log d = 2.34462 log C = 0.30859 log X = 3.47712 log 1474 = 3-16853 = log .s. (Table II) 5(F) = 798 « S{v) = 2272 -• . V — 1259 f. s. By eliminating C from the above equations, the expression for V may be written Employing this last formula, the computation by logarithms is as follows : 2 log d = 2.34462 log X= 3.47712 a. c. log w — 7.34679 log 1474 = 3.16853 S{V)= 798 S{v) = 2272 -•. ■z^ = 1259 f. s. PROBLEM I. 19 This last solution involves the writing of fewer figures than the first ; and is as short as it is possible to make it. Generally, however, in ballistic problems, the value of C must be deter- mined before the problem can be solved ; and then the first arrangement is preferable. We may also write the expression for v as follows : and perform the arithmetical operations indicated. Example 2. Calculate the final velocity of an 8-inch elon- gated projectile fired from the B. L. rifle, with a muzzle velocity the same as in Ex. i, for a range of 3500 yards. Here ^ = 8 ; t = 0.9 (and, therefore, cd'^ = 57.6) ; V=- 1700 ; St w = 290 ; X = 10500 ; and -5-' = i. We have, therefore, log cd'' = 1.76042 log X— 4.021 19 a. c. log w = 7.53760 log 2085.5 = 3-31921 (Table I) 5(F) = 3512.1 S{v) = 5597.6 .-. -V = 1264.5 f. s. These two examples show that with the same muzzle velocity the lighter elongated projectile has about the same final velocity for a range of 3500 yards as the heavier spherical pro- jectile has for 1000 yards. They illustrate very strikingly the superiority of rifled guns over smooth-bores, in carrying destruc- tive energy to the object to be destroyed. Example 3. With a charge of 8 pounds sphero-hexagonal powder the 4.5-inch siege-gun gives a muzzle velocity of 1400 f. s., weight of projectile 35 pounds. What is the final velocity for a range of 15 16 yards? thermometer 82°, barometer 29.75 Inches. Note. — For this gun i =1. 20 PROBLEMS IN DIRECT FIRE. ■ We have given F= 1400, ^=4.5, w = Si,X= 4548, and (J, log -r = 0.02202 log w = 1.54407 a.c. log d' = 8.69357 log C = 0.25966 log X= 3.65782 log 2501.3 = 3.39816 S{V) = 4878.6 S{v) = 7379.9 .• . V = 1028 f. s. Example 4. What would be the velocity in the above ex- ample at the middle point of the range? In this case we have x = 2274 ft. and the remaining data the same as above. log X = 3.35679 log C = 0.25966 log 1250.6 = 3.09713 S{V) = 4878.6 S{v) = 6129.2 .-. V = 1 178 f. s. Comparison of Computed with Measured Velocities. — This problem is useful for testing the accuracy of the ballistic tables by comparing the computed velocity of a projectile at a considerable "distance from the gun, with the velocity measured at the same point with a chronograph. The only velocities, measured at a distance from the gun, known at the Artillery School, are those taken at Meppen, and published from time to time by Krupp, in his valuable Experiences de tir ; and those executed by the Hotchkiss Ordnance Company at Gavre. In both these the metric system of units is employed ; and to reduce all the given data to Eng- lish units would involve considerable labor even with the help of the "Tables of Reduction" found at the, end of this book. PROBLEM I. 21 , We can, however, diminish the labor as follows : The cona- plete expression for v is, since ^ = ~^ "Ts » in which, if 'metric units are employed, w = 450 pounds. The energy of a projectile in foot-tons is given by the equation 4480^ For the same projectile the factor w 4480^ PROBLEM I. 23 is constant, and maybe computed once for all. In our example the value of this factor is 0.0031233 ; and, therefore, the expres- sion for the energy of a i S-in. solid shot in terms of the veloc- ity is .£0 = 0.0031233^/=. The work by logarithms would be as follows : log of multiplier = 7.49462 2 log V = 6.20006 log .£„ = 3.69468 .-. E„ — 4950.9 foot-tons. (b) 8-inch B. L. rifle; &= 1264.5 f- s. ; w = 290 pounds. For the 8-inch B. L. rifle the expression for the energy is E„ =0.00201287/'; whence log of multiplier = 7.30380 2 log V = 6.20384 log E = 3.50764 .-. E = 3218.4 foot-tons. Example 8. Required the energy per inch of shot's circum- ference, in the above example. The energy per inch of shot's circumference is generally held to be the proper measure, in comparing the relative effi- ciency of different guns, of the ability of a shot to penetrate armor. Since ;r^is the circumference of a shot in inches, we have, designating the required energy by Ei , E„ wv' E. nd a^Zondg ' in which, as before, for the same projectile, all the factors ex- cept v' may be consolidated. For the 1 5-inch spherical shot we have E^ = 0.0000668597/' ; 24 PROBLEMS IN DIRECT FIRE. and for the 8-inch elongated projectile weighing 290 pounds, £■, = 0.000080x3872/'. The answers are : For 15-inch spherical, E^ = 105.98 foot-tons; " 8-inch elongated, E^ = 128.06 " The following expressions for energy are applicable to all projectiles : E^ = 0.0000069407 wv' ; Ey = 0.0000022093 —7-. The logarithms of these multipliers are, respectively, For E„ , 4.84141 — 10. . " E, , 4-34426 - 10. Formulae for Striking Energy in Terms of Metric Units. — The formiilze given in the preceding article are adapted to English units only ; but it frequently happens that we wish to compare two guns with respect to energy, the data of one being in English and that of the other in French units. The follow- ing formulae give the energies in foot-tons, when the velocity is in metre-seconds, the weight of the projectile in kilogrammes, and the calibre in centimetres : E^ = 0.000 1 647 1 wv' ; £, = 0.00013317-^. The logarithms of the multipliers are 6.21672 — 10 and 6.12440 — 10, respectively. If the velocity should be given in foot-seconds, as it would be if computed by Table I, while the weight and calibre of the projectile were given in metric units, we should have the follow- ing expressions for the energies : E, = 0.000015302 wv' ; _ wv^ E^ = 0.000012371—3-. PROBLEM I. 25 The logarithms of the multipliers are 5.18474 — 10 and 5.09242 — 10, respectively. Example 9. What is the muzzle energy of a projectile fired from the Hotchkiss lo-centimetre rapid-firing gun ? For this gun we have the following data : V — 600 m. s. ; w = 1 5 kg. ; <^ = 10 cm. log of multiplier = 6.21672 log w = 1. 17609 2 log z/ = 5-55630 log E„ = 2.9491 1 .-. £„ = 889.43 foot-tons. The total energy is therefore 889.43 foot-tons. To deter- mine the energy per inch of shot's circumference we have : log multiplier = 6.12440 log wv' = 6.73239 (See above.) a. c. log d = 9.00000 log E^ = 1.85679 .•. E^ = 71.9 foot-tons. Example 10. Compute the energy of a projectile fired from the Krupp 40-cm. gun, at a distance of 3000 metres from the gun. For this example we have th.e following data : d = ip cm. ; w — 920 kg. ; F= 550 m. s. = 1804.5 f- s- ; X— 3000 m. ; and S = 1.206 kg. We must first compute the striking velocity at 3000 m. by the formula on page 21, as follows: const, log = 9.23591 log S = 0.08135 2 log d= 3.20412 log X=: 3.47712 a. c. log w = 7.03621 log 1083.2 = 3-03471 5(F) = 3092.3 5 (7/) = 4175-5 .•.«;= 1 547.1 f.s. 26 PROBLEMS IN DIRECT FIRE. For the total energy, we have const, log = 5.18474 log W := 2.96379 2 log V = 6.37904 log E, = 4-52757 •"• -^0 = 3369s foot-tons. For the punching energy, we have const, log = 5.09242 log wv^ = 9.34283 a. c. log^= 8.39794 log E, = 2.83319 .-, £, == 681 foot-tons. Penetration of Projectiles. — To calculate the penetration of wrought-iron we will use Maitland's " Formula of 1880," which according to Mackinlay's Text-book of Gunnery, edition of 1887, is the one " now generally employed." This formula is which gives the thickness (t) of wrought-iron plate penetrated, in inches, in terms of the striking velocity, weight, and diame- ter of the projectile. Example 11. How many inches of wrought-iron will the new 8-inch projectiles penetrate at 3000 yards from the gun — weight of projectile 290 pounds, muzzle velocity 1850 f. s., and c=- 0.9? We must first compute the striking velocity at 9000 feet from the gun, by the method already given, and then substi- tute this velocity in the above expression for t. The complete work is as follows : log ^=3.95424 log C = 0.70198 (Ex. 2) log 1787.6 = 3.25226 = log z 5(F) = 2916.9 5 (v) = 4704.5 .-. V = 1435.0 f. s. PROBLESt I. ^7 log w = 2.46240 log d = 0.90309 2 )i.5S93i 0.77965 logt; = 3-15685 a. c. log 608.3 = 7-21588 1.15238 = log 14.20 0.14^= 1. 12 T= 13-08 inches. Problem II. Given the ballistic coefficient (C) and thfi remainiug velocity (v), at a distance {X) from the gun, to determine the muzzle velocity (F). Note.— Hereafter Cwill be said to be "given" when the data upon which its value depends are supposed to be known. Solution. Compute 5( V) by the equations z = -7^ and S{y) = S{v) — s, and take the value of V from the proper table. Example i. The velocity of a 4.5-inch solid shot fired from the M. L. siege-gun, with 8 pounds of sphero-hexagonal powder, was found to be 1387 f. s. at 101.6 feet from the gun. What was the muzzle velocity ? Air normal, c = i. log X = 2.cx)689 log C = 0.23764 log 58.8 = 1.76925 = log 3 S(y) = 4944.3 5(F) = 4885.5 .-. F=i399f.s. Example 2. The proposed 1 2-inch B. L. rifle is to fire a pro- jectile weighing 800 pounds, which, it is expected, will pene- trate 16.25 inches of solid wrought-iron armor at a distance of 3500 yards from the gun. What must be its muzzle velocity? We have d= 12; w = 8oo-, X — 10500 ; c = 0.9 ; and log C = 0.79049. We niust first determine the striking velocity necessary 28 PROBLEM II. 29 to produce the required penetration. Solving Maitland's penetration formula with reference to v, we have ^ = 6o8.3|(r + o.i4^)W'^|, which becomes, by substituting for r, d, and w their values, V = 608.3 X 17-93 X A /-— . V 800 Computation of v : log 12 = 1.07918 - log 800 = 2.90309 2 )8.i76o9 9.08804 log 17.93 = 1.25358 log 608.3 = 2.78412 logz'= 3.12574 .-. z;= 1335.8 f.s. , The muzzle velocity is now computed as follows : log ^=4.02119 log C = 0.79049 log 1 701.0 = 3.23070 = log z S{v) — 5209.0 S(y) = 3508.0 .-. V= 1701 f.s. Example 3. Data same as in the preceding example except that the weight of the projectile is increased to lOOO pounds ; and, therefore, log C = 0.88740. The striking velocity now becomes V = 608 X 17-93 V0.012 = 1 194.8 f. s. 30 PROBLEMS IN DIRECT FIRE. We therefore have log X = 4.021 19 log C =■ Q.88740 log 1360.8 = 3-13379 = log ^ S(v) = 6021.9 S(r) = 4661.1 .-. F=i444f.s. It appears, then, from these last two examples that if the weight of the projectile be increased by lengthening, or other- wise, to 1000 pounds, the muzzle velocity may be diminished 257 f. s. and yet be as effective against armor at 3500 yards as with the higher velocity. It was shown by Hutton about a century ago, and has been verified by all subsequent investi- gators, that, the gun and charge of powder remaining the same while the weight of the projectile is made to vary, the muzzle velocities generated are very nearly inversely proportional to the square roots of the weights of the projectiles. .Therefore to determine in this case what would be the muzzle velocity of the 1000 lb. projectile with the same weight of charge as gave the 800 lb. projectile a velocity of 1701 f. s., we have the proportion 1/1000 : 1^800 :: 1701 : V; •, V=i7oi\/ = ^52if.s., ' y 1000 which is a considerably greater velocity than would be needed. The charge could therefore be reduced ; but whether the strain upon the gun would be less is a question in Interior Ballistics with which we are not here concerned. Problem III. Given the ballistic coefficient, the muzzle velocity, and the ter- fninal or striking velocity, to determine the distance from the gun. That is, we have C, V, and v given to compute x or X. Solution. Take from the proper table the values of S{_ V) and S{v) for the given values of V and v. Then x is computed by the equation x=C{S{v)-S{V)]. Example r. The muzzle velocity of a service projectile fired from the 8-inch B. L. rifle is 1850 f. s. At what distance from the gun must a target be placed in order that the striking velocity may be 1 500 f. s. ? (For the value of C, see Ex. 2, Prob. I.) We have S{v) = 4393.0 5(F) = 2916.9 log 1476.1 = 3.16912 log C = 0.70198 log X = 3.871 10 .•. X — 7432.0 ft. Note. It will be observed that X\%a. particular value of x. Example 2. At what range will an 8-inch elongated pro- jectile (w = 290 pounds) have the same energy as a iS-inch solid shot at a range of 1000 yards? The muzzle velocity of the latter is 1700 f. s., and that of the former 1850 f. s. We have found (Ex. 7, Prob. I) that the energy of a i S-inch solid shot at 1000 yards is 4950.7 foot-tons; and that the ex- pression for the energy of an 8-inch projectile is E„ = 0.0020I28Z'*. 31 32 PROBLEMS IN DIRECT FIRE. As the energies are to be the same, we have 0.00201282/" = 4950.7 ; ^ = \/a 49507 ,0020128* Employing logarithms we have log 4950.7 = 3-69467 log of divisor = 7.30380 2 ) 6.39087 log V — 3.19543 .*. V = 1568.3 f. s. Computation of X: S{v) = 4079.6 S{V) = 2916.9 log 1 162.7 = 3.06547 log C = 0.70198 log X = 3.76745 ••• ^ = 5854-0 ft. For all ranges, therefore, greater than 1950 yards, the strik- ing energy of the elongated projectile exceeds that of the much heavier spherical shot. And this superiority of the former goes on increasing as the range becomes greater. Example 3. The proposed i6-inch B. L. gun will fire a pro- jectile weighing 2300 pounds with a muzzle velocity of 2000 f. s. At what distance from the gun will its energy be 55500 foot-tons ? Suppose c = 0.9. To determine the striking velocity we have the equation 55500; 4480, 'd> ^5 5500 X448o ^U V w I ' PROBLEM in, 33 log 55500 = 474429 log 4480 = 3.65128 log^= 1.50732 a.c. log w — 6.63827 2)6.54116 log V = 3.27058 .-. V = 1864.57 i' s. Computation of x : 2300 X = 0.9 X 2 ^j 5(1864.57) -^(2000)} 5(1864.57) = 2861.7 5(2000) = 2368.2 log 493-5 = 2.69329 log 2300 = 3-36173 a. c. log 259 = 7.59176 a. c. log 0.9 ■= 0.04576 log X — 3.69254 .-. X = 4926.4 ft. Example 4. Given d ^= 20 inches, w — 4500 pounds, F = 2000 f. s., and E„ ="55500 foot-tons, to calculate X. It will be found that the striking velocity in this case is 1333.02 f. s., and the range 11898 yards, or more than seven times as great as the range in Ex. 3. Example 5. The Krupp 40 cm. gun, designed for the defence of Spezzia', fires a projectile 15.75 inches in diameter and weighing 2028 pounds, with a muzzle velocity of 1804.5 f. s. The new English iio-ton gun fires a projectile 16.25 inches in diameter, weighing 1800 pounds. What must be the muzzle velocity of the latter in order that its "racking energy at 4000 yards maybe the same as the former at the same distance? If we make c = 0.9 for both guns, we shall have for the Krupp gun, log C = 0.95841, and for the English gun, log C= 0.87932. 34 PROBLEMS IN DIRECT FIRE. First compute the striking velocity of the Krupp gun at 4000 yards = 12000 feet. logX= 4.07918 log C = 0.95841 log 1320.6 = 3.12077 S{V) = 3092.3 S{v) — 4412.9 .•. V = 1495.8 f. s. As the energy of the two projectiles is to be the same, we have, in order to determine the striking velocity of the Enghsh projectile, the following equation, in which the subscripts refer to the English projectile : ° 4480^ 448qg-' 1 w .•. V, = V\/ Computation of z', : log w = 3.30707 log w, = 3.25527 2)0.05180 0.02590 log V = z.xja.^j log v^ = 3.20077 •■• «'. = 1587.7 f. s. Computation of F, : log X= 4.07918 log C = 0.87932 log 1584.4 = 3.19986 = log a •S'(^.) = 3993.0 5(F;) = 2408.6 .-. F,= 1988.6 f. s. PROBLEM III. 35 The English projectile will, therefore, require a muzzle velocity 184 f. s. greater than the Krupp projectile in order to have the same racking energy at 4000 yards. The energy developed at this distance is given by the above formula and is 31389 foot-tons. Next, let us determine the muzzle velocity required by the English projectile in order that it may have the same armor- piercing energy, or energy per inch of circumference, as the Krupp projectile at 4000 yards. In this case we have, since the energies are, by hypothesis, equal, E, = ^ondg 448o;r^,^' Iwd. .: z» = z/A / — -J- Computation of v^ : log w = 3.30707 log d, = 1.21085 a. c. log w, — 6.74473 a. c. log d = 8.80272 2)0.06537 0.03268 log^= 3-17487 log v^ = 3.20755 .•. z'l = 1612.7 f. s. Computation of F, : S{v,) = 3883.1 2 = 1584.4 {z has already been computed.) ■ S{V,) = 2298.7 .'. V, = 2020 f. s. Problem IV. Given V, v, and x, to determine the coefficient of reduction (c). Solution. Take from the proper tables the values of 6'(F) and Si^). Then c is found from the equation _ djw S{:v) - SiV) '^'~Sd'' X Example i. At Meppen the velocity of a projectile 5.87 inches in diameter and weighing 73.855 pounds, was measured at two points of its trajectory 4662 feet apart. The velocity at the point near the gun was 1665.7 f- s-, and the velocity at the farther point was 1246.74 f. s. Observation of the atmosphere gave — = 0.973. Required the value of c for this projectile. S{v) = 5701.3 s{V) = 3655-6 log 2045.7 =3.31084 a. c. log J? = 6.33143 log w = 1.86838 a. c. log d?' = 8.46272 log -i = 9-98793 log r = 9.96130 .'.<: = 0.915 36 Problem V. To determine the time from the origin (t), when the ballistic coefficient {C), the horizontal distance passed over {pc), and the muzzle velocity ( V), are given. Solution. Equation (2) viz : - ^ \T{u)-T[y)\ COS may, when is small (for reasons already given), b^ written t=c\'nv)-T{v)\, and this, in connection with the equation (see Problem I) S{v) = z^s{y), solves the problem for small angles of departure. C will be computed as already explained. Example i. Compute the time of flight with the data of Ex. I, Prob. I. Here V = 1700, X = 3000 and log C = 0.30869. As the value of V has already been worked out we will not repeat the operation. We have, then, to complete our data, v = 1259 f. s. Therefore, T(v) = 1.445 TiV) = 0.433 log 1. 01 2 = 0.00518 log C = 0.30859 log r= 0.3 1 377 /. T' = 2.06 seconds. 37 38 PROBLEMS IN DIRECT FIRE. Example 2. Compute the time of flight with the data of Ex. 2, Prob. I. We have V= 1700, v = 1264.5 ^"d log C = 0.70198. T{v) = 3.0s 5 T{V) = 1.626 log 1.429 =0.15503 log.C = 0.70198 log r= 0.85701 .•. 7^= 7.19 seconds, 1 Example 3. In firing with the 8-inch converted rifle, the observed time of flight (corrected) was 6 seconds. If ^= 8 inches, w — 184 pounds, c= \ and F'= 1280 f. s., what was the range, supposing the atmosphere to be normal ? The equations to be used are and X=C\S{v)-S{y)\. Computation of v : log w = 2.26482 log d' = 1.80618 log C — 0.45864 log T= 0.77815 log 2.087 = 0.31951 T{V) = 2.g8s T{v) = 5.072 . ,-. v = 992.2 f. s. PROBLEM V. 39 Computation of X: S{v) = 7822.8 5(F) = 5509.7 log 23 1 3. 1 = 3.36419 log C = 0.45864 log X = 3.82283 .-. X = 6650.2 ft. So far no account has been taken of the wind ; that is, the air has been regarded as motionless. We will now consider the effect of a wind upon the range, time of flight and final velocity. Problem VI. Given the ballistic coefficient (C), the muzzle velocity (F), the observed time of flight ( T) and the direction and velocity of the ■wind, to compute the remaining velocity {v). Solution. We will assume that the effect of the wind upon the velocity of a projectile, is due to that component of the wind which is parallel to the range, or plane of fire. Let /3 be the angle which the direction of the wind makes with the plane of fire reckoned from the target round to i8o°.on either side of the plane of "fire. Then if ^Fis the velocity of the wind and Wf the component parallel to the plane of fire, we shall have Wt, = Wcos A When )S = o, we have (since cos o = i), W^= W. When /? = 90°, Wf = o\ and when ^ = 180°, W/,= - W. For val- ues of /S between o and 90° the component of the wind paral- lel to the plane of fire retards the motion of the projectile and is positive. When /J lies between 90° and 180° this component increases the projectile's motion and is negative. If yS = 90°, that is if the wind blows directly across the range, it will, in accordance with our assumption, have no effect upon the velocity in the plane of fire. Having determined Wp, compute v-\- Wphy the formula (see Appendix i). nv+W,) = ^+T{V+W,), from which v can at once be determined. The sign of Wp de- pending upon the angle ^, must not be overlooked in using the above equation. Should the wind come from the rear (in which case cos ^ would be negative), the above equation would become 40 PROBLEM VI. T{v-W,) = ^^TiV-W,). 41 For the use of Artillerists the velocity of the wind is re- quired in feet per second ; but as the anemometers furnished by the Government give the velocity in miles per hour, we must multiply this velocity by ^^ to reduce it to feet per sec- ond. That is Feet per second = ^ times miles per hour. Example i. The following data are taken from the record of firing with the 3.2-inch B. L. rifle (steel), at Sandy Hook, March 18, 1885 : r= 4 seconds, F= 1608 f. s., d= 3.2 inches, w = 13 pounds, W =^ 13.2 f. s., § — 32°, thermometer 20°.9and barometer 36.093 inches. As the result of a preliminary inves- tigation we will take c = 0.93. Required the velocity v at the end of the 4 seconds. We have IVf = 13.2 cos 32° = 11.2 f. s., V-\- Wp — 1608 -\- 11.2 = i6i9.2f.s., -^ = 0.920, and log C— 0.09895 o log T = 0.60206 log C = 0.09895 log 3.185 = 0.50311 T{V+ W,) = 1.833 T{v-\- Wp) = 5.018 .-. v-\-Wp = 996.3 ; V = 985.1 f. s. Without taking account of the wind we should have found V = 994.0 f. s. There is, therefore, in this case, a loss of veloc- ity due to the wind, of about 9 f. s. in 4 seconds. If C, V, V and Wp are given to compute Twe should make use of the formula T=C\T{v^ Wp)-T{V+ Wp)], or T=C\T{v- Wp)- T{V- Wp)\, according as the component of the wind acts against or with the projectile. Problem VII. Given the ballistic coefficient {C), the muzzle velocity ( F), the observed range {X) and the component of the wind parallel to the plane of fire {Wf), to compute the striking velocity (v) and time of flight {T). Solution. Compute the time of flight by Problem V, upon the supposition that there is no wind. The value of v will then be found by the equation (see Appendix i). ^C v+ W,) = ^^±^ + S{V+ W,), c or S{v - W,) = / "^^ + S{V- W,), C according as the component Wp acts against or with the pro- jectile. The assumption made in the solution of this problem, that the time of flight is practically uninfluenced by. the wind, is not strictly correct, though near enough for most practical pur- poses. The mean velocity of a projectile is increased or dimin- ished by the wind in nearly the same ratio as the range ; and, therefore, the time of flight is not practically affected by the wind. Example i. In firing with the 8-inch converted rifle the observed range was 2000 yards, with a wind blowing directly toward the gun of 30 f. s. What was the striking velocity, density of air normal ? Here log C =■ 0.45864 (Ex. 3, Prob. V) ; F = 1280 f. s. ; Wp = 30 f. s. and X = 6000 ft. To avoid confusion we will designate the value of v computed on the supposition that there is no wind by v^. 42 PROBLEM VII. 43 Computation of z/, by Problem I. log X= 3.778 IS log C = 0.45864 log 2086.9 = 3-31951 5(F) = 5509.7 S{tj^ = 7596.6 .•. v^ = 1009.7 Computation of T by Problem V. T{v,) = 4.846 T(V) = 2.98S log 1. 861 = 0.26975 log C = 0.45864 log T= 0.72839 .-. T= 5.35 seconds, log Wf,— 1. 477 1 2 log 160.5 = 2.20551 = log TWf X = 6000.0 = log 6150.5 = 3-78963 log C = 0.45864 log 2142.8 = S{V+ W,) = 5345-2 = 3-33098 S{v-{- Wf) = 748S.o /. v-\- Wp= 1018.7 Wf = 30.0 .-. V = 988.7 f. s, Example 2. In the above example suppose the wind to blow directly from the gun, the other data remaining the same. Compute the striking velocity. Here instead of adding 160.5 to ^as in Example i,it must be subtracted. 44 PROBLEMS IN DIRECT FIRE. X = 6000.0 TWt= 160.5 log 5839.5 = 3.76638 log C = 0.45864 log 2031.2 = 3.30774 s{v- w;) = s6S2.i S{v — Wf) = 7713-3 .: V— Wf= 1000.5 Wp — 30.0 /. V = 1030.5 Example 3. At Meppen, Nov. 26, 1880, an experimental shot was fired with the following data : d = 10.5 cm., zc; = 16 kg., S — 1.268 kg., V = 467.5 m. s. = 1533.82 f. s., W — 4.2 m. s. = 12.73 f- s-j ^ = i57°3o' and X = 1929 m. = 6328.9 ft. Required the final velocity v. To determine C in English units when the data are given, as above, in French units, we have, making c = 0.9 and d, = 1.206 kg., C"= 19.0587^. The operation is then as follows': log of multiplier = 1.28009 log w = 1. 2041 2 a. c. log S — 9.89688 a. c. log d' = 7.95762 log C = 0.33871 log W= 1. 13924 log cos /3 = 9.96562 log Wf = 1. 10486 .-. W^ = 12.73 f- s. V- Wp= 1521.09 f. s. PROBLEM VII. log ^=3.80133 log C = 0.33871 45 log 2901.5 = 3.46262 5(F) = 4236.1 'S'C^'O = 7137-6 /. «;, = 1050.6 f. s. T{v^ = 4.402 7];F) = 2.07s log 2.327 = 0.36680 log C = 0.33871 log = 0.70551 log W^= 1. 10486 log 64.6 = 1. 8 1037 A" =6328.9 log 6264.3 = 379687 log C = 0.33871 log 2871.9 = 3.45816 S{V- fF^) = 4294.8 S{v — Wf) = 7166.7 .'. V— Wf= 1047.7 .-. V = 1060.4 f. s. = 323.2 m. s. Measured velocity = 325.9 m. s. Second Method. — If we eliminate T from the equations and X± TlVy, there results S{v ± W,) T W,T{v ± W;) = ^+ S{V± W,) T W^T{V± W,), 4.6 PROBLEMS IN DIRECT FIRE. from which v ±. Wp can easily be found by trial, since all the terms of the second member are known quantities. By this method we find the steiking velocity independently of the time of flight. Taking the data of Ex. i we have, by substitution and reduction, the equation 5(7/ + 30) - 30 T^z/ + 30) = 7346.4- By making use of Table I, the value olv-\- 30, which satis- fies this equation, is easily found to be 1018.6, the same as be- fore. This justifies the assumption we have made, that the time of flight is not sensibly influenced by the wind ; and ren- ders this second method generally unnecessary. Problem VIII. Given the muzzle velocity ( V), the computed range {X'), and time of flight (T) to calculate the variation of the range {^X) due to a given value of Wp. Note. — Hereafter Cwill be supposed to be given unless otherwise stated. Solution. Compute v by the equation and then ^X by the equation AX= C{S{v + W,) - S{V+ W;)\ ^ {X-\- TW,), Of /1X= C\S{v -Wp)-S{V~ Wf)] - {X- TW,). The first formula is used when the direction of the compo- nent Wf is toward the gun ; in which case AX is negative. The second formula is used when the direction of Wp is toward the target ; in this case AX is positive. Example i. Compute a table of AXiox the 3.2-inch field-gun for a range of looo yards. We have F= 1608 f. s.; X= 3000 ft.; 7^=2.21 seconds; and log (7 = 0.10843. We will begin by making Wp=- \o i. s., whence V-\- Wp = 1618 f. i. ' log T = 0.34439 log C = o. 10843 log 1.722 — 0.23596 T{V^ W^ - 1.836 T{y + W;) = 3-558 .-. v^Wp= 1 166.5 f- s. 47 48 PROBLEMS m DIRECT FIRE. sv + w;) 6208.7 3860.0 log 2348.7 = 3.37083 log C = 0.10843 log 3014.8 = 3.47926 X -ir TWf = ioz2.\ AX^- 7.3 ft. In the same way may other values of AX be computed and arranged in a tabular form, as below. To this table are added the values of AX when Wp is negative, that is, when it increases the range. The remaining velocities are given in each case. x = 1000 yards T' = 2.21 seconds. ft. per second. feet. V ft. per second. ft. per second. feet. V ft. per second. + 10 - 7 I156 — 10 + 6 1 167 20 15 II51 20 13 1172 30 21 1146 30 20 I178 40 27 1 140 40 25 I183 50 34 "35 50 32 1189 60 44 1130 60 38 1194 We also add a similar table for X = 2000 yards and T = 5.13 seconds. X = 2000 yards 7" = 5.13 seconds. Wp ft. per second. AA- feet. ft. per second. Wp ft. per second. feet. V ft. per second. + 10 — 21 931 — 10 + 35 947 20 50 922 20 63 956 30 78 914 30 go 964 40 105 go6 40 119 972 50 133 897 50 147 g8o 60 165 889 60 174 gSg PROBLEM VIII. 49 It will be seen from the first of these tables that for a range of looo yards AX is approximately proportional to Wp ; but this approximation decreases as the range increases, and soon ceases to be of any value as a working principle, as is shown by the second table. Example 2. What effect would half a gale of wind (50 f. s.) blowing up or down the range, have upon the range of a four- inch projectile weighing 25 pounds and having a muzzle velocity of 1900 f. s. ? I. Suppose the wind blows up the range. Compute AX when X = lOOO yards and 2000 yards. Pi («) X = 3000 ft.; V = 1900 f. s.; c = 0.907 ; -^ = i ; log C = 0.23621 ; H^ = 50 f. s.; V-\- Wp— 1950 f. s. logX= 3-47712 log C = 0.23621 log 1 741. 4 = 3.24091 S(V) — 2729.2 ■S'K) = 4470.6 .-. V, = 1483.565 Tiv;) = 2.231 T(V) = 1. 191 log 1.040 = 0.01703 log C = 0.23621 log T= 0.25324 .•. T= 1.79 seconds. T jr = 1.040 T{V-^ Wp) = 1.096 T{v + Wp) = 2.136 .-. v+Wp^ 1513.67 50 PROBLEMS IN DIRECT FIRE. S{v + W;) = 4329.1 s{v+ w;) = 2546.4 log 1782.7 = 3.25108 log C = 0.2362 1 log 307 1. 1 = 3.48729 ^4- TW^ = 3089.6 ^^ = - 18.5 ft. = — 6.2 yards. (6) ^ = 6000 ft. log X= 3.77815 log C = 0.23621 log 3482.9 = 3.54194 S{V)= 2729.2 S{v^) = 6212.1 .•. w, = 1166.0 f. s. Tlv,) = 3.561 7^r)= 1. 191 log 2.370 = 0.37475 log C = 0.23621 log T = 0.61096 .•. r = 4.08 seconds. T r^ = 2.370 T{v-\- w;) = 1.096 r(z'+ Wp) = 3.466 .-. v^Wp= 1182.73 5(F+fF,) = 2546.4 log 3554.1 = 3.55073 log C = 0.23621 log 6122.7 = 3.78694 X-\- TWf = 6204.0 ^X= — 8.13 ft. = — 27.1 yards. PROBLEM VIII. 51 Commander H. J. May, R.N., gets — 6 yards and — 28 yards, respectively, as the deviations in these two examples. (See Proceedings Royal Artillery Institution, Vol. XIV, page 358, Table i.) 2. Suppose the wind blows down the range. («) X = 3000 ft. ; V—Wp= 1850 f. s. We have, as before, T -^ = 1.040 T{v- w;) = 1.291 7\v- -w,) = 2.331 .-. V-Wp: S{v- - w,). = log 4619.0 2916.9 1 702. 1 = logC== : 3.23099 ; 0.23621 X- log /IX = 2932.3 = 2910.4 346720 21.9 ft . =!= 7.3 yards, (^) X = 6000 ft.; V- -Wp^i :850 f. s. As before, T C ~ 2.370 T{V- -W,)^ I.29I 1452.6 Tlv - W;) = 3.661 .: V -lVf = i 149 f- s- S{v - Wp) = 6328.8 S{V- W;) = 2gi6.g log 341 1.9 = 3.53300 log C = 0.23621 log 5877.8 = 3.76921 ^-7^^^ = 5796.0 .-, JX= 81.8 ft. = 27.3 yards. 52 PROBLEMS IN DIRECT FIRE. Remarks upon Problems VI, VII and VIII.— The fofmulae of Problems VI, VII and VIII are deduced upon the hypothesis that the effect of a wind blowing parallel to the range is simply to increase or diminish the resistance the pro- jectile encounters. That is, if a projectile is moving nearly horizontally with a velocity v, the resistance of the air, if there is no wind, is considered proportiohal to w" ; but if the air has a velocity 'Wp parallel to the plane of fire, then the resistance is proportional to (z' + W^)", or {v — Wpf", according to the direction of Wp. (See Appendix i.) We have assumed in Problems VII and VIII, that the time of flight is not sensibly influenced by the Wind, since the effect upon the time of the variation in the range is nearly compensated by the corresponding variation in the mean velocity. To obtain some idea of how much the time of flight of a projectile propelled with a given velocity and angle of depart- ure is effected by a change in the density of the air, we have made the following calculations with the data of the last example : It will be shown by the next problem that the projectile of this example with a muzzle velocity of 1900 f. s. and the air at its normal density, would require an angle of departure of 2° 11' 33" for a range of 6000 feet, while we have already found the time of flight to be 4.08 seconds. Now (the muzzle velocity and angle of departure remaining the same), if we assume the density of the air to be two-thirds of its normal density, we must multiply C by f ; and performing the necessary operations we shall find upon this hypothesis that r= 4.21 seconds, and X= 6632 feet. That is, AX = + 632 ft., while AT \s, only + 0.13 seconds. Again, if we assume the air to be only one-half its normal density, we shall find JX — -j- 1025 feet and JJ=o.2i seconds. If we suppose the air to have no density, or in other words, that the projectile moves in vacuo, we shall have AX ^ 2582 feet, and ^7'=o.44 seconds. PROBLEM Viri. S3 On the other hand, if we assume the air to be twice its nor- mal density, we shall find AX=- — 1216 feet, and AT ^ —0.33 seconds. From these illustrations it is manifest that for flat trajec- tories, and values of AX not exceeding say 500 feet, no notice need be taken of A T. Problem IX. Given the ballistic coefficient (C), the muzzle velocity ( V) and angle of departure( = BC. B is taken from Table B with the arguments z (already found in determining the range) and V. Final velocity. For the final velocity we have (Eq. 1 2) cos (b V = u -; cos oa or, when does not exceed 5°, V =:-U. Example 1. The 4.5-inch siege-gun, with a charge of 8 pounds of sphero-hexagonal powder, gives to a solid shot weighing 35 pounds, a muzzle velocity of 1400 f. s. With an angle of departure of 2° 50', what would be the range, time of flight, angle of fall and final velocity, thermometer 7i".5 and barometer 29.59 inches? We have ^ = 4.5, w= 35,^= i, -^- = 1.036, log C= 0.25300, F= 1400, and = 2° 50'. .-. 20 = 5° 40'. log sin 20 = 8.99450 log C= 0.25300 log A = 8.74150 .-. A = 0.0551 With the arguments F= 1400 and A = 0.0551, we find from Table A, , 3 X 100 ,^ .s = 2600 + ^-^^^g — = i6ii.5. log z = 3.41689 log C = 0.25300 log X = 3.66989 .-. X = 4676 feet. 56 PROBLEMS IN DIRECT FIRE. z = 2611.5 s(y) = 4878.6 S(u) — 7490.1 .-. u = 1018.5 7\u) = 4.741 J(F) = 2.514 log 2.227 = 0.34772 log C = 0.25300 •log r= 0.60072 , .•.7'= 3.99 seconds. From Table B we find for F= 1400 and 2 = 2611.$, B = 0.0681. log ^ = 8.83315 log C = 0.25300 log sin 203 = 9.08615 .-. 200 = 7° 00' .'. aj = 3° 30' Finally, we have an account of the small values of (fi and go, V = u = 1018 f. s. Example 2. Compute the range, etc., for the 8-inch B. L. rifle for an angle of departure of 10°. We have d=2i, w = 290, c = 0.9, V= 1850, and <}> = 10°. log sin 20 = 9.53405 log C = 0.70198 log A = 8.83207 /. A = 0.06793 Therefore, from Table A, , 3 X 100 ^ = 4500 H ^^— = 4514 PROBLEM IX. 57 log z = 3.65456 log C = 0.70198 log X = 4-35654 ••• ■^= 22727 feet. 2 = 4514.0 5(F) = 2916.9 S{u) = 7430.9 .-. u = 1023.5 Tlu) = 4.685 7((F) = .1.291 log 3-394 = 0.53071 log C = 0.70198 log sec = 0.00665 log r= 1.23934 .-. T= 17.35 seconds. From Table Bwe find for V= 1850 and ^r = 4515,^ = 0.1010. log B = 9.00432 log C = 0.70198 log 0.5 = 9.69897 2 log sec = 0.01330 log tan ffi» = 9.41857 .■. Qj = 14° 41'' log u = 3.01009 log cos = 9.99335 log sec a> = 0.01442 log V = 3.01786 .-. z; = 1042 f. s. Example 3. " Find the range of the proposed 20-pounde B. L. gun of 3.4-inch calibre at 7" elevation, also the angle o descent ; muzzle velocity = 1650 f. s." (Proceedings of th Royal Artillery Institution, Vol. 15, page 364.) This example is worked out by Niven's method in th S8 PROBLEMS IN DIRECT FIRE. volume cited, by making c = 0.9 and jump = 6'. We there- fore have the following data : w = 20, = 9.27143 .•. 03 = io°35' By Niven's Method 10 37 Difference = 2' Example 4. Compute the range for the Krupp 24-cm. gun, with the following data : J = 24 cm., ze/ = 215 kg., V=: 529 m. s. = 1735.6 f. s., 0=8° 35' and 6 = 1.262 kg. PROBLEM IX. 59 Const log = 1.28009 (See Ex. 3, Prob. VII.) log w = 2.33244 a. c. log (3^" = 7.23958 a. c. log S = 9.89894 log C = 0.75105 log sin 2 (p — 9.47005 ' log ^ = 8.71900 .-. A = 0.0524 = 3400 + ^ I ^ X 30 + 524 - 536 } =3444 'iog^= 3.53706 log C= 0.75105 log X= 4.2881 1 .-. X= I94i4feet. Measured range = 19567 feet. Difference =153 feet. {Experiences de tir, No. 56.) Example 5. Compute the range for the Krupp 40-cm. gun, with the following data: a? = 40 cm., w = 920 kg., F= 550 m.s. = 1804.5 f- s., quadrant elevation 5° 21', jump = 14', = 5° 21' + 14' = 5° 35' and d — 1.206. Const, log = 1.28009 log w = 2.96379 a. c. log ^' = 6.79588 a. c. log S = 9.91865 log C — 0.95841 log sin 20 = 9.28705 log ^ = 8.32864 .-. A =0.0213 6o PROBLEMS IN DIRECT FIRE. .'. z = i8cx) H ] — X lo + 213 — 213 {• = 1806 14 I 50 ' ) log z = 3.25672 log C — 0.95841 log ^=4.21513 .'. X= 16411 feet. Measured range = 16391 feet. Difference = 20 feet. Example 6. Compute the range with the data of Ex, 5, ex- cept that 0= 15°. log sin 2 = 9.69897 log C— 0.95841 log A = 8.74056 .'. A ^ 0.0550 /. z = 3700 + ^1 4_1 X 29 + 550 - 537 [ = 3784 SO log z = 3.57795 log C = 0.95841 log X= 4.53636 .-. X= 34384 ft. = 6.512 miles. Example 7. Compute the range of the proposed 1 6-inch B. L. rifle with the following data : ^ = 16 inches, w = 2300 pounds, c = 0.9, V= 1850 f. s. and = 15°. log sin 2 = 9.69897 log C= 0.99925 log A = 8.69972 .-. A — 0.0501 .-. z = 3600 -I (501 — 489) = 3663 PROBLEM IX. 6 1 log z = 3.56384 log C = 0.9992s log X = 4.56309 .-. X — 36567 ft. = 6.926 miles. In the last two examples the projectiles would attain alti- tudes of more than 3000 feet, and consequently the actual ranges would be somewhat greater than those computed, on account of the less resistance the projectile meets with at high altitudes. This will be considered in a subsequent problem. SECOND METHOD. We have not considered it worth while to compute auxiliary- tables for spherical projectiles, on account of their less frequent use ; and, therefore, for this class of projectiles, in the absence of tables, we proceed as follows : We have from (10) the following relation, in which, of course, u refers to the point of fall, where j/ = o : A{u) -A{V) _ sin 2 S{u)-S{V)~ C ^'■^'^^ The second member of this equation consists entirely of known quantities; and in the first member^ {V) and S {V) are known. But as the relation between the S-functions and A-functions does not admit of a direct solution of this equation it is necessary to determine u by trial. We may deduce a near value of u, and one sufficiently accurate in many cases of curved fire, by the following method : We have from the origin to the summit, by (2), and from the origin to the point of fall. 62 PROBLEMS IN DIRECT FIRE. If we assume (what is approximately true) that we shall have, from the above equations, r(«) = 2rK)-r(F); which gives u by means of the T-functions, «„ being computed by Equation (6), viz., sin 2 =■ 0.00425 log T' = 1.04844 .•. T' = II. 18 seconds. 66 PROBLEMS IN DIRECT FIRE. I{u) = 0.379S7 • /(«,) = O.I 506 1 log 0.22896 = 9.35976 log C = 0.30849 log 0.5 = 9.69897 2 log sec = 0.00850 log tan GO = 9.37582 .•• oa = 13° 22 log u = 2.86100 log cos = 9-99575 log sec 00 = 0.0 1 193 log V = 2.86868 ,•• V = 739 f. s. Problem X. Given the range X and angle of departure (0), to compute the muzzle velocity ( F). SOLUTION. (FIRST METHOD.) Compute A and z by the formulae - sin 20 and X ' = -€' and then take from Table A the corresponding value of V, which is the velocity required. Example i. With a quadrant elevation of 7"27' the range with the 40-cm. Krupp gun was 6588 metres, or 21614.6 ft. If the jump was 14' and air normal, what was the muzzle velocity ? We have log C = 0.95841 and = 7° 41' log sin 20 = 942324 log C = 0.95841 log A = 8.46483 .-. A = 0.0292 log X= 4.33475 log C — 0.95841 log z = 3-37634 ••• -s = 2378.7 Therefore from Table A V= 1800 ^^\l^l X i6-\- 287 - 292 J = 1825 f. s. The measured velocities ranged from 1798 f. s. to 1828 f. s. Experiences de iir, No. 63. 67 68 PROBLEMS IN DIRECT FIRE. Example 2. The average range of five shots fired at Sandy Hook, Sept. 29, 1885, from the 12-inch experimental cast-iron B. L. rifle, with a quadrant elevation of 4°, was 1 1089.8 feet, weight of projectile 80 opounds, thermometer 72°.4, barometer 30.115 inches. What was the muzzle velocity? We have being given. The best way of accomplishing this will be shown by examples. Example 5. Wishing to ascertain the muzzle velocity of a shell fired from the lo-inch S. B. gun with a charge of twenty pounds of cannon powder ; and it being impracticable, from the position of the gun, to use a chronograph, the following data were selected from the record of target practice at Fort 7° PROBLEMS IN DIRECT FIRE. Monroe, July 7, 1887, for the purpose of determining the velocity by calculation : Range (mean of four shots), 6795 feet ; quadrant elevation 6° ; weight of shell 107 pounds ; thermome- ter 82°, and barometer 29.950 inches. We have X — 6795 ; = 6° + jump = 6° 10', say ; d = 9.87 ; w — 107 ; -^ = 1.045 ; ^ — i> and log C = 0.05986. The operation is as follows : log X= 3.83219 log C = 0.05986 log z = 3-77233 ••• "S = 5920 For a first trial assume V = 1 500. 5(r)=i4i3 s = 5920 S{u) = 7333 .•. u = 661.2 A(u) = 1291.1 A{V) = 19.6 log 1271.5 = 3.10431 log z = 3.77233 log 0.21477 = 9.33198 I(V) — 0.03072 log 0.18405 = 9.26494 log C — 0.05986 log sin 2

: I(u) = 0.39890 I{u,) = 0.187S7 log 0.21 103 =9.32434 log C" = O.I 1858 2 log sec = 0.00382 log 0.5 = 9.69897 log tan 00 = 9.14571 .*. ffi» = 7° 58' Computation of T: T{u) = 6.090 T{V) = 1.163 log 4.927 = 0.69258 log C = 0.1 1858 log sec = 0.00228 log r= 0.81344 .-. r= 6.51 seconds. PROBLEM XII. 83 Computation of v : log u = 2.85449 log cos = 9.99809 log sec CO = 0.00421 log V — 2.85679 .•. V — 719 f. s. For the shell, we have d = 9.87, w = 107, log C = 0.04075, JC = 6000 and V— 1493. The results are as follows : 0=5° 9'; CO = 8° 19'; T = 6.48 seconds ; V = 691.8 f. s. It will be seen from the above that the shot, though having a muzzle velocity 135 f. s. less than the shell, has a striking velocity, at 2000 yards from the gun, greater than the shell by 27 f. s. The time of flight and, therefore, the mean velocity are about the same for both projectiles. The shell has a less angle of departure and a greater angle of fall than the shot. Problem XIII. Gwen the ntiizzle velocity (F) and angle of departure (0), to calculate the elements of the trajectory at the summit. 'Solution. At the summit we have (see equations (i 8) and (25)) . sin 20 m„ = A = — pr^ . With the given value of V and that of m„ computed as above, take from Table m the value of ^„ ; whence -X, = Cz, . Next, with the arguments Fand z^ take 3„ from Table B» We then have, by an obvious modification of (24) (since, at the summit, m^ = A\ y^ = ~x, tan 0. m^ To compute the summit velocity, we have from (i 5) and the definition of A, I{u,) = A-{-IiV), and from (7), z/„ = M„ cos 0. To compute the time from the origin to the summit, we have ^» = c-^0^^w-^(^)^ Example i. Take the data of Ex. 2, Prob. IX, viz., F = 1850, = 10°, log C = 0.70198, and A=m^= 0.06793. From Table m we have z, = 2400 + -^(679 - 647) = 2484.2 ; 84 PROBLEM XIII. 85 and from Table B, b^ = 0.0360 + 0.842 X 0.0023 = 0.0379. log^o = 3-39519 log C = 0.70198 log x„ = 4.09717 .-. X, = 12508 feet. log tan = 9.24632 log d, = 8.57864 a. c. log m^ = 1. 1 6794 log Jo = 3-09007 .-. 7, = 1230 feet. A =0.06793 /(F) = 0.03727 /(k„) = 0.10520 .•. u, = 1299.2 log«o= 3-11368 log COS 4> = 9.99335 log^o = 3-10703 .-. V, = I279.S T{u,) = 2.903 /"(F) = I.29I log 1.612 = 0.20737 log C = 0.70198 log sec

^ for every 100 feet of altitude from /« = o to h = 9900 feet are given in the following table. To use it, look for the thousands in the first vertical column headed " h" and for the hundreds in the first horizontal column. At their inter- 88 PROBLEMS IN DIRECT FIRE. section will be found the decimal "^zxt of the -factor required, which must be annexed to unity, as in the second column. TABLE OF ALTITUDE FACTORS. h 100 300 300 400 600 600 700 800 900 o I . 0000 0036 0072 0108 0145 0181 02 18 0255 0292 0329 1000 1.0366 0403 0441 0479 0516 -0554 0592 0631 0669 0707 2O0O 1.0746 0785 0824 0863 0902 0941 0981 1020 1060 1 100 3000 1. 1 140 1 180 1220 1260 1301 0341 1382 1423 1464 1506 4000 I. 1547 1589 1630 1672 1714 1756 1799 1841 1884 1927 5000 I. 1970 2013 2057 2100 2144 2187 2231 2276 2320 2364 6000 I . 2409 2454 2499 2544 2589 2634 2679 2725 2771 2817 7000 1.2863 2909 2956 3003 3049 3096 3144 3191 3239 3286 8000 1-3334 338Z 3431 3479 3538 3576 3625 3675 3724 3773 9000 1.3823 3873 3923 3973 4023 4074 4125 4176 4227 4278 For direct fire the mean value of h, or mean height of the trajectory, is about two thirds the uncorrected maximum height, or height of summit. Therefore, to determine the value of h with which to enter the table, we have .the following rule : It will be near enough in practice to take the even hundred nearest the computed value of h. Example 3. Calculate the range and time of flight of Ex. 2, Prob. IX, making allowance for the height of the tra- jectory ; («) when = 10°, and (<5) when 0=15°. (a) = 10°. We have j„= 1230 feet (Ex. i). Two thirds of this, to the nearest hundred, is 800 feet, which is the mean height of the trajectory. .•. h — 800 ; and from the table we find the altitude factor to be 1.0292, by which we must multiply the value of C hereto- fore used. We therefore have log C= 0.71448. The further calculations are as follow: log sin 20 : logC: 9-53405 0.71448 log^ = 8.81957 A = 0.0660 PROBLEM XIII. 89. .•. s = 4400+ — 1 660 — 654^ = 4427.3. (Table A.) log z = 3.64614 log C= 0.71448 log X= 4.36062 .-. X= 22941 feet. z = 4427.3 s{y) = 2916:9 6:(F) = 7344,2 .-. M = I03I..2' r(«)= 4.598, T{y)^ 1. 291 log 3-307 = 0.5 1943 log C= 0,71448 log sec = 0.00665 - log r= 1.24056 .•. 7^= 17.40 seconds. The calculated range is, therefore, increased 224 feet when the diminished density, of the air, due to the height of the trajectory, is taken into account — a distance too great to be neglected in accurate firing. The difference in the, time of flight is very small, as was to be expected ; since the increased range is compensated for by the greater mean velocity of the projectile. (See Erob. VIIL) (iJ) = 15". In this case we havejj/„ =2504 feet, h = 1700 feet, altitude factor = 1.063 1 and log C=.o,728ss. log sin 2(f) = 9.69897 log C= 0.72855 log A = 8.97042 .-. A = 0.C9342 .-. s=SSoo + ~ (934 - 917) = 556S,-4- (Table A.) 90 PROBLEMS IN DIRECT FIRE. log ^ = 3.74550 log C= 0.72855 log X = 4.47405 .-. X= 29789 feet. Without using the altitude factor the computed range would have been 29128 feet ; a difference of 661 feet. The difference between the computed times of flight in the two cases is 23.89 — 23.65 = 0.24 seconds. Note. — When „ = 0.0357 + .155 X .0023 — .09 X .0019 = 0.0359. (Table B.) log s„ = 3.36464 log C = 0.95841 log.^o = 4-32305 log tan = 9.5 1 178 log <5. = 8.55509. a. c. log m„ = 1. 1 8916 logJ^„ = 3-57908 ■ .-. j/„ = 3794 feet. We have, therefore, k = 2500 feet, and the altitude factor = I.0941. Whence log C= 0.99747. log sin 20 = 9.76922 log C = 0.99747 log A = 8.77175 .-. A = 0.05912 PROBLEM XIII. 93 .-. 2=3900 + — |. 09 X 31 + 591 -579} =3970.4- (Table A.) log z = 3.59883 log C = 0.99747 logvf= 4.59630 .-. X= 39473 feet. = 1 203 1 metres. = 7.476 miles. The mean of eight shots iired with this gun at Meppen, April 29, 1886, with 18° elevation, was 39808 feet. The cal- culated range is, therefore, short of the actual mean range by only 335 feet ; and this difference can be accounted for by the jump of the gun. To determine the jump which will cause the calculated range to agree with the measured range in this example, we proceed as follows : We have X=: 39808 feet (mean range), V= 1804.5 f- s-; and log C = 0.99747, to calculate — o.o55flf. 78.3S2W/ ^^ The computation is as follows : log w = 2.96379 log d — 1.60206 2) 1.36173 0.68086 log V - 3.04523 a. c. log 257.065 = 7.58996 log 20.70 = 1.3 1605 0.05 5(^= 2.20 r = 18.50 inches. Problem XIV. Given the muzzle velocity (V), to determine the angle of de- parture (0) which will cause a projectile to hit an object situated above or below the level of the gun ; also the striking angle {ff), the striking velocity {v), and the time of flight (t). SOLUTION. (FIRST METHOD.) Let X and y be the co-ordinates of the given object (see page 3) and s its distance from the gun, whence s = v'jr" -\-y^ and x = V{s -\-y){s — y). Also, let 6 be the angular distance of the object above (or below) the level of the gun, and therefore y tan 6 = - . X If the object is above the level of the gun, y and e are posi- tive ; while if it is below the" level of the gun they are both negative. Compute z by the formula and with the arguments V and z take a from Table A. Then from (22) we have - = tan e = tan © i i ; { . X ( sm 20 ) Solving with reference to tan 0, we have tan = — ;=• I I — Vi — aC{aC-\- 2 tan e) \ , from which to compute 0. 96 PROBLEM XIV. Ci7 To compute 6 we take m from Table m with the arguments V and z ; and then (Eq. 23) i ^C \ tan = tan -j i — ; r f . ( sm 20 ; For the striking velocity we have S{u) = z + S{V), and cos = S.g77g6„ log tan e = 9.29727„ .-. = _, 11° if Computation of v : [It will be observed that u is the same in both examples.] log M = 3.17464 log cos = 9.99804 log sec 6 = 0.00838 logz/ = 3.18106 .'. v=isi7i.s. Computation of t : log C{ T{u) -T{V)\= 0.72750 log cos = 9.99804 log t = 0.72946 .'. t = 5.36 seconds. The striking velocity in Ex. i is 1474 f, s. ; and in Ex. 2 it is 1517 f. s. The difference is due to the action of gravity, which impedes the motion of the projectile in the one case, and assists it in the other. Rigidity of the Trajectory. — The two preceding ex- amples illustrate an important principle known as the Rigidity of the Trajectory,* which assumes that the relations existing between the elements of a trajectory and the chord represent- * Called by German writers das Schwenken der Bahnen, and by the French [hypothhe de la rigidiU de la trajectoire. 102 PROBLEMS IN DIRECT FIRE. ing the range, are sensibly the same whether the latter be horizontal or inclined to the horizon, within certain limits. This principle gives the following simple rule for deter- mining the angle of departure when the object aimed at is above, or below, the level of the gun : Calculate the angle of departure for a horizontal range equal to the distance of the object from, the gun, and add to it the angle of elevation (or depression) of the object ; which gives the angle of departure sought. Example 3. According to the range table the 8-inch M. L. rifle (converted) requires an angle of departure of 5° 43' for a range of 3000 yards. What would be the angle of departure supposing the gun to be 40 feet higher than the object aimed at? Here s and ;ir differ insensibly from each other and = gcxx) feet ; j)/ = — 40 feet. .'. tan e = — _1_ 9000 log 40 = i.6o2o6„ log 9000 = 3.95424 log tan 6 = 7.64782„ .•. e = — 15' ••• 0=5"43' + (- IS') = 5" 28' The above rule is applicable to all our sea-coast guns, which are but moderately elevated above the level of the sea ; and we have also shown that it is sufficiently accurate for high- powered guns even in the extreme case of the highest battery at Gibraltar. But with guns of less power, giving trajectories of considerable curvature, the angle of departure computed by the rule, for the Signal Station at Gibraltar, would be wrong by some minutes. This is illustrated by the following ex- ample. Example 4. " It was recently necessary to fire a 64-pounder converted gun with a charge of 8 pounds, giving a muzzle velocity of 1260 f. s., from the Signal Station at Gibraltar, 1270 PROBLEM XIV. 103 feet above the level of the sea. The object fired at was 2000 yards from the muzzle of the gun." Find the angle of departure. Here d = 6.3, w = 64.5, V = 1260, s = 6000, j/ = — 1270, c = 1 and log C = 0.21088. We will first compute

= 3° 55' 4" z = 3816.4 5(F) =3470-8 S{u) = 7287.2 .; u = V = 1036.4 f. s. Tlu) = 4.543 71; F) = 1.602 log 2.941 = 0.46849 log C = 0.19650 log T = 0.66499 .-. 7^= 4.62 seconds. '^ Proceedings Royal Artillery InstittUion, No. 14, Vol. XV. PROBLEM XV. "3 We next find the general expressions for y and B (by apply- ing numbers already found) to become . and y = 0.78807 (0.0620 — a)x, tan 6 = 0.78807 (p.0620 — m). The values of a and m must be taken from Tables A and m with the arguments F= 1710 and z = --. We also calculate the value of u for each value of x by the equation and then t by the formula t = S{T{u)-T{y)). The results are given in the following table : log <:: = 0.19650; F= 1710; 5(F) = 3470.8; 7\F) = i.6o2. X feet. z u=v a m y feet. « i 5400 5550 5700 5850 3434-7 3530.2 3625.6 3721.0 1074.8 1064.5 1054-7 1045.3 0.0537 •0557 -0578 -0599 0.1270 •1323 •1377 •1432 35^3 27.6 18.9 9^7 — 2° 56' 3 10 3 25 3 40 4.07 4.20 4-34 4.48 SECOND METHOD. When the values of A, a, and m are not obtainable from the tables, they may be computed as in the second method of Prob. XIV. The following is, however, preferable : Multiplying Eq. (3) by Eq. (i), and reducing by Eq. (6), we have 2 cos" \I{u:iZ-\-A{V)-A{u)\. 1 14 PROBLEMS IN DIRECT FIRE. In connection with this equation we use the following when. V and are given : _ ^ 2 = -^, and S{u)^z-{-S{V), sin 20 If is not given it will be necessary to compute it from V and X, as explained in the Second Method of Prob. XII. Example 3. Compute ordinates 100 yards apart for the 1000-yard trajectory of the Springfield rifle. Also the co-ordi- nates of the summit. Here d — 0.45, w = 500 grains = -^ pound, X = 3000, c= I, V— 1301 and log C= 9.54745. First compute I(u^ and — 2°g' i&' I{u^) = 0.54300 /(«,) = 0.24890 log 0.29410 = 9.46850 log C = 9.59656 / n'f log sin 200 = 9.06506 /, £» = 3° 20' 7' Substituting numbers already found in the equation I{u,)S{u) - A{u) = 2-^^^ + I{u,)S{V) - A{V) ; it becomes when f = 5.75, I{u,)S{u) - A{u) = 73.62 + 971.63 - 93.77 ; or 0.248905(a) — A{u) = 951.48, from which to find u. By a few trials we find that this equation is satisfied when u = 1519.64 and u = 791.03. The first of these values refers to the ascending branch, and is of no practical importance in this example. Using the second value we compute JJC by the equation /IX= C\S{u^) - S{u)] as follows : S{uJ) = 1 1499.4 S{u) = 1 1236.9 log 262.5 = 2.41913 log C = 9.59656 log ^X = 2.01 569 .-, ^X = 103.7 feet. PROBLEM XVI. 123 Next let J/ == — 5.75 feet. In this case we have I{u^S(u) — A{u) = - 73.62 + 971.63 - 93.77; or o.2489o5(m) — A{u) = 804.24, from which we easily find ti = 1696.7 and u = 767.94, as the two values of u which satisfy this equation. The first value of u belongs to a point in the ascending branch prolonged back- ward through the origin ; and the second value to a point in the descending branch prolonged through the 1000-yard point, or point of fall. Using this second value of u we find AX as follows : S{u)= 1 1739.3 S{u„) — 1 1499.4 log 239.9 = 2.38003 log C = 9-59656 log AX= 1.97659 .-. AX= 94.75 feet. The first value of /IX (103.7 f^et) is the breadth of the danger-zone on level ground, against infantry, when the gun is fired with its muzzle close to the ground, and aimed at the foot of the target. This zone therefore lies entirely within the 1000-yard range. The second value of AX (94.75 feet) is the breadth of the danger-zone when the gun is fired with its muz- zle at a height of 5.75 feet above the ground and at a point of the target at the same height. This zone therefore lies entirely without the 1000-yard range. The actual danger-zone lies partly within and partly without the range point ; and its breadth is a certain mean of the two, computed as above, depending upon the height of the muzzle of the gun. Application of the Principle of the Rigidity of the Tra- jectory. — The essential features of the principle of the rigidity of the trajectory may be concisely stated as follows : — (see page 1 01.) 124 PROBLEMS IN DIRECT FIRE. If, for a certain gun, 0', w' and «'„ refer to a given horizon- tal range (or chord) s, then the corresponding elements of a trajectory which shall pass through a point at the same dis- tance s, but which is above (or below) the gun by the angular distance e, may be determined by the relations = 0' + 6, B = -w' -\-e, Ug = U a' According to this principle a rifle should be sighted (within the prescribed limits) for' distance only ; that is, without refer- ence to the angular elevation (or depression) of the object above (or below) the level of the gun ; and then aimed directly at the object ; for, it is evident, that to the elevation, denoted by ' (to which the sights are set), e is added by simply point- ing the gun at the object. In laying heavy guns with the Zalinski sight, the vernier should be set to 0' (taken from the Table of Fire, for the given horizontal distance), and the gun then so manoeuvred that the axis of the telescope is directed on the object ; when if the jump of the piece has also been taken into account, the gun will have the proper elevation. Example 3. What would be the danger-space against Infantry {h = 5.75 feet), on level ground, in the preceding example, if the muzzle of the gun were 2 feet high (as in firing kneeling), and aimed at a point 4 feet from the ground and 1000 yards distant ? We have already found (page 122) 0' :^ 2° 9' 16" (Sighting angle) go' — 3° 20' ;" m'„ = 778.88 f. s. PROBLEM XVI. 125 We also have by a given condition, 2 tan e = ; 3000 whence e = 2' 18". Therefore, for the new trajectory we have <^ = 2°ii'34" ^=-3° 17' 49" Ug = 778.88 f. s. We have next to compute values of u in the descending branch for which y = 3.75 feet and j/ = — 2 feet. If u' and u" are these values, we shall have for the danger-space AX, AX = C{S{u") - S{u')\. The equations for determining u' and u" by trial are found to be o.2S2285(«') — A{u') = 939.07, and o.252285(«") - A{u") = 865.46; from which we find" ' u' = 782.46 and 770.94 S{u") = 1 1673.2 S{u') = 11421.5 log 251.7 = 2.40088 log C = 9.59656 log AX = 1 .99744 ••• ^-^ = 994 feet. 126 PROBLEMS IN DIRECT FIRE. Example 4. Compare the maximum danger-spaces against infantry covered by the Steyer carbine and Springfield rifle, respectively, for angles of elevation corresponding to different ranges up to looo yards, reckoning from the muzzle of the gun. The' results of the calculations, with the data upon which they are based, are given in the tables below. We have taken Data : V = 1608 f. s. = 9.5 5080. STEYER CARBINE. w = 246.9 grains; that is, the striking energy of the Springfield rifle bullet will be nearly double that of the hypothetical bullet for all ranges. Example 10. Compare the charges of powder required to give the two bullets of Ex. 8 the same muzzle velocity. We have from Sarrau's monomial formula, when the muzzle velocities are the same, '. = -©•©•■ But when the ballistic coefficients of the two bullets are the same, we have £»- — (Fj.?- . (^ Vf !^ y — !L d^ Wj / ' ' ' \dj \w^ I ze/j We therefore have the proportion That is, if two bullets have their calibres and weights so proportioned that their ballistic coefficients are the same, then the charges of powder necessary to give the bullets the same muzzle velocity are proportional to the weights of the bullets, and therefore proportional to their respective striking energies, as shown above. Example 11. If the calibre of the Lebel rifle be reduced from 0.314 in. to 0.291 in., what will be the length of the new bullet upon the supposition that the weights and ballistic coef- ficients of the two bullets are respectively the same ? (See page 148.) In this case we have the proportion •■•'. = f'. = S7'. = '-°»'- l62 PROBLEMS IN DIRECT FIRE. ' If the Lebel bullet is 3.75 calibres long, we shall have K = 375 - 0.37 = 3-38 ; .-. /, = 1.08 X 3-38 = 3-65 ; .•. Z, = 3.65 -\- 0.37 = 4.02 calibres. Example 12. Show that by increasing the length of a cored shot of the modern type from 3 calibres to i\ calibres, its striking velocity for any given range may be reduced about 10 per cent without diminishing either its striking or pene- trating energy. Let Wj and v^ be the weight and striking velocity, respec- tively, of a cored shot 3 calibres long, and w^ and v^ the same for a 3^-caIibre shot. Then, since the two projectiles are of the same diameter, and are assumed to have the same striking and penetrating energy, we have the following relation between their weights and striking velocities (see page 34) : But the weights of two oblong projectiles of the same diameter are directly proportional to their reduced lengths. Therefore -' = (^\^ = / A-o76 y ^ f2-24y . ^^ VJ [L,-o.76l \2.74) ' .•. V, = 0.9042^^, . Example 13. With the conditions of Ex. 12, suppose the muzzle velocity of the 3-calibre shot to be 2100 f. s., and the range such that the striking velocity is 1400 f. s. What would be the muzzle velocity of the 3^calibre shot for the same range ? By Problem II we have for the two shots the following •equations: PROBLEM XVIII. 163 whence, by division, But the ballistic coefficients of two oblong projectiles of the same diameter are proportional to their reduced lengths. Therefore 'a We have found in Ex. 12 that z/, = 0.90422/, = 0.9042 X 1400 = 1265,88 f. s., and Therefore we have A 2.24 ■} = — =!^ = 0.8175. k 2.74 S{v^ =4878.6 S{V^ = 2024.8 log 2853.8 = 34S542 log 0.8175 = 9.91250 log 2333.0 = 3.36792 S{v^ = 5589.8 S{K) = 3256.8 .-. K = 1763 f- s. We see from this that by increasing the length of the shot one half a calibre, the muzzle velocity may be reduced 337 f. s., and still have the same striking energy for the given range. The range required to reduce the velocity from 2100 f. s. to 1400 f. s. would, of course, depend upon the diameter and weight of the projectile — in other words, upon the value of C. For the new 8-inch navy gun, for example, the range would be 4129 yards ; while for the 6-inch navy gun it would be 2936 yards. Example 14. With the data of Ex. 13 deduce the relative charges of powder required for the two projectiles. l64 PROBLEMS IN DIRECT FIRE. To solve this example we will make use of Sarrau's mono- mial formula for slow-burning ^powder, viz., v — M- Ze;A For the same powder and gun, and assuming the density of loading to be the same for both projectiles, we have the follow- ing relation between the muzzle velocitieSj weights of projectiles and charges in the two cases : V, efo*^ which differs but very slightly from the corresponding formula for quick-burning powder given on page 148. From this for- mula we have But since we have '•-.. 0.9042 ' V ^ i The pressure upon the base of the heavier projectile is, therefore, slightly less than upon the lighter one ; and we may, in consequence, fairly assume that the same is true with refer- ence to the walls of the gun. The calculations of the last four examples have been made for long fighting ranges, viz., 4000 yards for 8-inch and 3000 yards for 6-inch guns. For these and still longer ranges the calculations show that a gun which fires a 3i-calibre shot with a muzzle velocity of about 1750 f. s. has the same efificiency for penetrating armor as a similar gun firing a 3-caIibre shot with a muzzle velocity of 2100 f. s. ; with a saving of 20 per cent of powder and with a less pressure upon the walls of the gun. Moreover, the trajectory of the heavier projectile is flatter 1 66 PROBLEMS IN DIRECT FIRE. for the same range than that of the lighter projectile ; and, therefore, more likely to hit the object aimed at. For ranges less than those given above the advantages of the heavier projectile over the lighter are less marked than for the longer ranges ; but they still exist. The pressure upon the walls of the gun may become a little greater for short ranges with the heavier than with the lighter projectile, but not enough greater to be of any consequence. Example i6. Our 12-inch B. L. rifle, with a charge of 265, pounds of powder, gives to a cored shot 3 calibres long, and having an ogive of 2 calibres, a muzzle velocity of 1800 f. s. What charge of the same kind of powder would be necessary to give to a similar projectile 5 calibres long a muzzle velocity of 191 5 f. s.? Also, what would be the maximum pressure in the gun ? In this example, as in the preceding, we assume that the powder chamber is enlarged as the charge increases in such a way that the density of loading remains constant. We have F, = 1800, V^ = 1915, Z, = 3, Z,, = 5, tt, = 265 and P^ = 34000, to find tt, and P^ . We first find /, = 3 — 0.76 = 2.24, and /, = s — 0.76 = 4.24. For the charge we have =- S{u)-S{V) • From these two equations we must find u by trial; and then C and c by the equations C- ^._ and '^'^T Cd" Example i. Firing at Meppen with a 20.93-cm. gun, the observed range for an angle of departure of 5° 38', and muzzle velocity of 1709.35 f. s., was 13441.8 feet. From this data de- termine the values of C and c. For this example we have d = 20.93 cm., w = 140 kg., V = 1709.3s f. s., 0=5° 38', X= 13441.8 feet, and S, -^ = 0.9781. An approximate value of C may be computed from the given data by omitting the factor c ; and then an approximate value of u by the equation s(u)^^^s{y). i6g 170 PROBLEMS IN DIRECT FIRE. From this preliminary calculation we find « = 1126; and as this is less than its true value' on account of having taken C too small, we will assume for a first approximation m = 1 1 50- The operations are as follows : S(u) — 6321.8 ^(n = 3473-5 5(a) -5(F) = 2848.3 A{u) = 328.27 A{V)= 70.74 log 257.53 = 2.41083 log 2848.3 = 3-45459 log 0.09042 = 8.95624 = log /(«„) /(F) = 0.04860 log 0.04182 = 8.62138 log Jf = 4.12846 a. c. log 2838.3 = 6.54541 log sin 20 = 9.29525 The real value of log sin 20 is 9.29087 ; .-. «", = — O.00438. Next assume u = 11 70, and it will be found that e^ = -{- 0.00202. We therefore have the proportion 438 + 202 : 20 :: 202 : 6.3; .*. u = 1 170— 6.3 = 1 163.7. To compute C and c we proceed as follows : S{u) — 6227.6 5(n = 3473-5 log 2754.1 = 3.43998 log X = 4.12846 log C = 0.68848 PROBLEM XIX. 171 W As the expression for c contains the factor -75 , in which w is expressed in kilogrammes and d in centimetres, we must, to avoid the necessity of reducing them to EngUsh units, multiply by the factor No. of pounds in one kilogramme (No. of inches in one centimetre)' ' The logarithm of this factor is 1. 15298 ; „-, ^ , ^ .-.. = [1.15298]-^;^. We have, therefore, log w = 2.14613 S log -^ = 9-99038 const, log = 1. 1 5298 a. c. log C= 9.31 152 a. c. log d" — 7.35846 log c = 9.95947 .•. c — 0.91 1 Example 2. Determine the values of C for different ranges,- for the 3.2-inch steel B. L. rifle. This gun was fired at Sandy Hook in' March, 1885, for the purpose of determining the ranges for differences of 2° in eleva- tion, beginning with 2° and ending with 20° elevation, the limit permitted by the carriage. The principal characteristics of this gun, and of the ammu- nition used in these experiments, are as follows : Calibre of gun, 3.2 inches Weight of gun, 791 pounds Length of bore, 26 calibres Twist, .... Uniform, one turn in 30 calibres Weight of shot, 13 pounds Radius of ogive, \\ calibres Powder charge, z\ pounds, Dupont's L. X. A. Density 1.706 Granulation 270 172 PROBLEMS IN DIRECT FIRE. Nineteen shots were fired for velocity, which gave, all re- ductions being made, a muzzle velocity of 1608 f. s. Twelve shots were also fired at a target 50 feet from the gun to deter- mine the angle of jump. The following table gives a summary of the firing, up to 10° elevation. The ranges and times of flight are each a mean of 10 shots. S The values of -^ are taken from Table III with the ob- o served barometric pressures and temperatures for arguments. The values of W^ are computed by the method given oa- page 40. Elevation. Jump. Angle of departure Mean observed range X„ (feet). Elevation of gun above strik- ing point Observed time of flight (seconds). i, i (feet). 2 4 6 8 ■, 10 t ti 21 00 22 15 22 45 23 15. 22 00 2 21 00 4 22 15 6 22 45 8 23 15 10 22 00 4755 7093 giog 10907 12451 14-3 16.6 12.0 I2.g 12.7 4.00 6.60 g.oo 11.45 13-80 0.920 0.932 0.942 0.942 0.942 + 11. 19 8.80 7.46 7.46 7.46 In this example we will endeavor to eliminate the influence of the wind upon the ranges : that is, we will determine, at least approximately, what the ranges would' have been had the atmosphere been calm during the firing. The direction of the wind-component parallel to the plane of fire {Wp) was in all cases from the target toward the gun, and therefore dimin- ished the ranges. As the equations of Problem VIII do not apply in this case, we will make use of the following empirical equation for computing AX:* * See BaKstique ExlMeiire, by Major Muzeau of the French Artillery. PROBLEM XIX. 173 in which F" sin 20 This equation gives a fair approximation for AX for mod- erate winds and ranges ; but for long ranges the results are somewhat too small, as the X in the second member, and in the expression for «, should be the range in an undisturbed atmosphere, whereas we necessarily use the observed range. The following is an example of the computation of AX by the above formula : We have V =■ 1608, = 2" 21', X= 4755 and 7^= 4. log F' = 6.41257 log sin 20 = 8.91349 a. c. log g = 8.49268 a. c. log X = 6.32285 log a = 0.14159 .-.«= 1.385s 2tf — I = 1. 7710 logX= 3.6771 5 log cos = 9.99963 a. c. log F= 6.79371 a.c. log {2.a — i) = 9.75178 log 2.31 =0.36386 T — 4.00 1.69 X 1 1. 19 — 19 feet = AX. The corrected range is therefore -^=4755 + 19 = 4774 feet. By the principle of the rigidity of the trajectory we may consider the observed ranges (corrected for wind as above) horizontal, provided we increase the angles of projection by the corresponding angles of depression of the point of fall below the level of the gun. That is, the new angle of projection will be determined by the equation 0' = + e. (See page 124.) 174 PROBLEMS IN DIRECT FIRE. The following table gives the angles of departure, upon the supposition that the ranges are horizontal ; and also the ob- served ranges corrected for wind : Elevation. Angle of depression (e). Ane^le of departure for horizontal range (0')- (feet). Corrected (range feet). 2 4 6 8 10 o 1 II o 10 i8 8 01 4 30 4 03 3 29 a 1 II 2 31 18 4 30 16 6 27 15 8 27 18 10 25 29 19 31 39 53 67 4774 7124 9148 10960 12518 The following is the computation of C for the angle of ele- vation of 2° : Assume for a first approximation, u = 990. S{u) = 7852.5 S{V) = 3903.7 3948.8 \{u) = 614.16 ^(n= 9377 log 520.39 = 2.71633" log 3 = 3-59647 logo.13178 = 9.11986 I{V) = 0.05867 log 0.073 1 1 = 8.86398. log X = 3.67888 a. c. log 2 = 6.40323 log sin 20 = 8.94639 True value = 8.94403 .-. e,= ~ 236 PROBLEM XIX. 175 Next we assume u = 1000, and find by a similar process ^, = +485; .-. 236 + 485 : 10 :: 236 : 3.3; .-. M = 990 + 3.3 = 993.3. And this value of u completely satisfies the above equations. The computation of C and c is as follows : S{u) = 7808.1 S{V) = 3903.7 log 3904.4 = 3.591SS logX= 3.67888 log C — 0.08733 log ^ = 0.10364 log f = 9-96379 a. c. log C = 9.91267 log c = 9.98010 <: = 0.955. Proceeding in the same way for the remaining angles of elevation, we have the following results : Computed Observed— Elevation. » log C c time of flight. computed lime of flight. 2 993-3 O.0S733 0.955 3.91 +0.09 4 865.6 0.08316 0.977 6.48 0.12 6 795-0 O.IOIO5 0.948 8.87 0.13 8 741.8 O.II433 0.919 II. 19 0.26 10 698.3 O.I2217 0.903 13.38 0.42 The mean of the first three values gives, for angles of ele- vation from 0° to 6°, which includes all ranges up to 3000 yards, c = 0.96. 176 PROBLEMS IN DIRECT FIRE. For angles of elevation exceeding 6°, or for ranges exceed- ing 3000 yards, c = 0.91. Best Method of Computing the Ballistic Coefficient. — The most accurate method of calculating the value of c is that given in Problem IV. But where the terminal velocities can- not be measured directly by a chronoscope, the above method is as accurate and convenient as any that can be devised. We might determine the values of C and c from the observed range and time of flight, taking account of the effect of the wind, by a combination of Problems VI and VII, which by eliminating^ C gives x±TWt _ s{u ± w;) - s{v± w;) T ~ T(u±JV^)-T{V± W,)' from which to determine u± Wphy trial. C would then be computed by the equation ^_ X±TW, s{u± w,)-s{v± w^y This method requires, however, that the time of flight should be known to within one tenth of a second in order to be even approximately correct ; and is, therefore, of no practical value. Problem XX. To calculate the drift of an oblong projectile. It is found by experiment that elongated projectiles having ogival heads, fired from rifled guns which, like those in our service, give a right-handed rotation, always deviate to the right in a calm atmosphere ; while those fired from guns which give a left-handed rotation, as with the French naval guns, deviate to the left. This deviation is called drift (French d&ivation). It is generally constant for the same gun and range, and can therefore be tabulated and allowed for in laying the gun. No entirely satisfactory explanation of this difficult subject can be given without the aid of the higher mathematics. The subject has been very fully treated by the following authors : . General Mayevski. Traits de Balistique Extdrieure. Paris, 1872, Le Comte de Sparre. Mouvement des Projectiles oblongs dans le cas du Tir de Plein Fouet. Paris, 1875. General Mayevski. On the Solution of Problems in Direct and Curved Fire, St. Petersburg, 1882. This work is written in the Russian language. A translation into Italian of the parts relating to drift may be found in the Revista di Artiglieria e Genio for 1884, vol. 3, page 81. Major Muzeau. Balistique Extdrieure. Lithographie de I'EcoIe d'Application de I'Artillerie et du G^nie, 1883. This work was first pubUshed in the R^vue d'Artillerie, vols. 12 and 13, Paris, 1879. Lieutenant J. Baills. Traits de Balistique Rationnelle, Paris, 1883. Major Astier. Mouvement des Projectiles oblongs. Prof. A. G. Greenhill. On the Derivation, or Drift, of Elon- J 77 178 PROBLEMS IN DIRECT FIRE. gated Projectiles. Proceedings Royal Artillery Institution, vol. 1 1 , page 1 24. In the following attempt to explain, without the aid of mathematics, the principal phenomena connected with the sub- ject of drift, we have derived great assistance from the fine work of Muzeau, cited above. We shall suppose the rotation of the projectile to be from left to right in the upper hemisphere as viewed from the rear of^ the gun. If it should have a left-handed rotation all the phenomena would be reversed. When an oblong projectile, properly centred in the gun, emerges from the bore, its axis sensibly coincides with the tan- gent to the trajectory described by its centre of gravity ; and the resistance of the air acting symmetrically in lines parallel to the direction of motion has a single resultant directed along the axis of the projectile, which is also the axis of rotation. There is, therefore, nothing at first to cayse the projectile to deviate from the plane of fire, or to change the direction of its axis. But, under the action of gravity, the tangent to the tra- jectory immediately begins to fall below the plane of its initial direction, while the rotation of the projectile, on the contrary, tends to keep its axis parallel to its original direction. The result of this slight separation of these two lines is that the resultant of the resistance of the air takes a direction oblique to both of them, making a small angle with the axis, which it cuts at a point called the centre of pressure, and which for ser- vice projectiles is always situated between the point of the pro- jectile and its centre of gravity. This resultant may be resolved into two components, one of which, and by far the larger of the two, acts in the direction of the axis of the projectile, and in opposition to its motion ; while the other acts normal to the axis through the centre of pressure, and tends to raise the point of the projectile and cause it to revolve around an axis perpendicular to the plane of fire, in such a way that if the projectile had no motion of rotation it would " tumble," as it is called. This is sometimes observed in practice-firing with our converted 8-inch rifles, when PROBLEM XX. 179 the projectiles " strip" or fail to take the groove. But if the projectile has a sufficient motion of rotation about its axis this effect is not produced. The upward pressure combined with the right-hand rotation causes the point of the projectile to move off slowly to the right, as may easily be verified with the gyroscope. In the next instant the same effects are repeated except that the resultant of the resistance of the air has changed its direction of action with reference to the axis, — or, rather, the axis has changed its direction with reference to the resultant, — which latter now supplies a component whose effect is to thrust the point of the projectile to the right, and which, com- bined with rotation, causes the point to fall, or droop, as it is called. The constant action of these forces has for effect to cause the axis of the projectile to describe a conical surface around the tangent to the trajectory from left to right, the apex of the cone being at the projectile's centre of gravity. This motion of the axis of the projectile around the tangent is called precession from analogy with the similar motion of the earth's axis around the axis of the ecliptic. The plane passing through the axis and the tangent, turning with the former around the tangent, the resultant resistance of the air which is always contained in this plane makes an in- creasing ang'le with the plane of fire, and furnishes a component whose effect is to move the projectile from this plane. The lateral displacement which the projectile thus suffers is called drift. The angular velocity with which the axis of the projectile turns around the tangent is very small, since it is in inverse ratio of the angular velocity of the projectile, which in direct fire is very great. It is not strictly correct to say that the axis of the projectile revolves around the tangent. The rotation of the projectile really takes place around an instantaneous axis which describes in the interior of the body a cone around the axis of the pro- jectile, and in space another cone around the tangent. The motion of the projectile is the same as if the first cone, consid- l8o PROBLEMS IN DIRECT FIRE. ered as attached to the projectile and carried along with it, rolled upon the surface of the second cone. The effect of this is to cause any point of the axis of the projectile to take up an epicycloidal motion around the tan- gent, and the axis to describe a sort of corrugated cone. This, motion of the axis is called nutation. In direct fire, however, the instantaneous axis sensibly coin- cides with the axis of the projectile ; and therefore the resist- ance of the air depends almost entirely upon the velocity of translation, as has been shown by experiment. From the above it follows that the effects of the rotation imparted to a projectile by the rifling are — 1st. To increase the stability of the projectile by overcom- ing the perturbating effects of the resistance of the air which tend to upset it. 2d. To keep the axis of the projectile near the tangent (a condition very favorable to long ranges) by impressing upon the first of the two lines a motion of rotation around the second. Both of these theoretical results are known to be true from experience and independently of any theory. Mayevski's Formula for the Drift of an Oblong Projec- tile. — Mayevski, following De Sparre's method, which is founded upon the hypothesis that the angle made by the axis of the projectile with the tangent of the trajectory at any instant is very small, and which holds true for direct fire, has deduced an expression for the drift, which, when modified for direct fire, and reduced to English units, is as follows : n h cos (j) ( S\u) — S{V) ^ ■' ) I0CX)0 In this equation - ^ ■ in which k is the radius of gyration of the projectile with refer- ence to its axis, and R its radius. Mayevski gives for the mean value of jii for cored shot of the modern type, ;* = o.53. The method of computing yu will be given further on. PROBLEM _ XX. l8l \ ^ is a quantity depending upon the length of the projectile, the shape of the head, the angle which the resultant resistance makes with the axis, and the. distance of the centre of pressure from the centre of gravity. Mayevski gives the following mean values : -T = 0.41 for projectiles 2.5 calibres long. = 0.37 " " 2.8 " " = 0.32 " " 3.4 " " n is the length of twist in calibres — that is, the distance the projectile advances, in calibres, while making one revolution. g = acceleration of gravity ; It = 3.1416. £{u), B{y), and M{V) are certain functions of the velocities, defined by the equations Their values are given in Table I. S(u) and S{V) are the space functions, already well known. Example i. Compute a table of drift for the cored shot of the 8-inch M. L. converted rifle. For this gun we have the following data : V = 1404 f. s. ; zc = 1 83 lbs. ; a^ = 8 in. ; c=z i ; n = 4^; fx = 0.53 ; -T = 0.41 ; ^= 32.16. Making these substitutions, and reducing, the formula for drift for this gun becomes ^ ( Biu) - B(V) ,,,,,, \ X D = O.I9S86 I ^5(Fy - ^(^) I c^?0 • A table of drift would, of course, form part of a range table, with the range as argument ; and would be computed after u and had been determined by Problem XII. We may there- fore consider these quantities known. As an example of the numerical work required, we will compute the drift for a range of 3000 yards or 9000 feet. For this rasge we have u = 968.8 and 0=5° 43'- 1 82 PROBLEMS IM DIRECT FIRE. From Table I we have S{u) = 8006.0 5(F) = 4858.5 z = 3147-5 B{ii) =72.480 £(F)= 17-381 log 55.099= 1.71144 logger = 3.49797 Iogo.01751 =8.24317 ilf(F) = 0.00848 log 0.00903 = 7-95569 const, log = 9.29195 log X= 3.95424 log sec' = 0.00650 logZ)= 1.20838 Z?= 16.16 feet. We have computed the following range table for this gun by methods already fully explained and illustrated. Note. — The computations are for the 8-inch converted rifles numbered above 28. For the guns numbered from \ to 28 the twrist of rifling is one turn in 60 calibres. RANGE TABLE FOR THE 8-INCH CONVERTED RIFLE. (Data taken from Ordnance Memoranda, No. 24.) Range « Striking D (yards). Velocity. yards. 500 / 44 r 47 1303 I. ID O.I 1000 I 33 I 43 1213 2.30 V 0.4 1500 2 27 2 50 II35 3-59 I.O 2000 3 27 4 08 1070 4.96 2.0 2500 4 32 5 38 1021 6.40 3-4 3000 5 43 7 14 982 7.92 4.4 3500 6 59 g 01 947 9-52 7-9 4000 8 21 TO 58 qi6 II. 19 II. I 4500 9 49 13 06 888 12.94 I5-I 5000 II 24 • 15 25 862 14.78 20.1 PROBLEM XX. 183 Example 2. Compute the drift of a shell fired from the 3.2- inch B. L. steel field-gun, for a range of 2500 yards. For this gun and range we have the following data : V= 1608 f. s. ; w = II lbs. ; d = 3.2 inches ; c = 0.96 ; log C = X 0.12137; «=3o; -T = o.37; and^=:32.i6. For shell we have M — 0.64. Applying these numbers, we have the following formula for computing the drift for this gun : For a range of 2500 yards we have u = 877.6 and = 4° 40'. B{u) = 134.860 BiV)= 10.732 log 124.128 = 2.09387 log 2 = 375369 log 0.02189 = 8.34018 M{V) — 0.00564 log 0.01625 = 8.21085 const, log = 9.22939 log X =r. 3.87506 log sec' = 0.00433 log 2?= 1.3 1964 .-. £> — 20.88 feet = 6.96 yards. Example 3. Compute the drift of a shell fired from the French 27-cm. gun, model of 1870-1873, for ranges from 1000 to 7000 metres. For this gun and projectile we have the following data : V= 505 m. s. = 1657 f. s. ; zc/ = 180 kg. ; d= 27 cm. ; c — 0.95 ; log C = 0.56780; « = 45; -T = 0.41 ; /i = 0.64 ; and^ = 9.81m. 1 84 PROBLEMS IN DIRECT FIRE. Making the necessary reductions, the formula for the drift for this gun becomes, in English units, ^ SB{u)-B{V) ^„„A X The following table gives the drift calculated by the above formula and reduced to metres ; to which is added, for com- parison, a column of drift taken from the official Table of Fire, which is presumably based upon observations : Rangre (Metres). Drift in Metres. Calculated. From the Table. lOOO 2000 3000 4000 5000 6000 7000 0.4 1.8 4.8 10. 1 18.5 31-2 49-9 0.4 1.8 4-7 9-9 18. 1 30.3 47-7 Baills' Formula for Drift. — A simple approximate formula for calculating the drift of oblong projectiles has been elabo- rated by Lieutenant Baills of the French navy, and is given in his Balistique Rationnelle, page 270, Eq. (23). This formula is, in English units, D = ^ cos 3 { I -f 0.0000184 ^(i + -j } f, It gives for the flatter trajectories of direct fire practically the same values for the drift as Mayevski's formula, with about the same labor. Example 4. Apply Baills' formula to the data of Ex. 2. In addition to the given data we must compute the time of flight, which we find to be 6.756 seconds. PROBLEM XX. I 85 We also find ^^ cos -=0.39842; I + 0.0000184 ^-f I -l — j =1.1231; 45.643536. .-. D = 0.39842 X 1.1231 X 45.643536 = 20.42 feet, which IS practically the same as that given by Mayevski's formula. Effect of Wind upon the Drift.— In Problems VI, VII and VIII we have given formulas for determining the effects of wind upon the remaining velocity and range of an oblong pro- jectile, which formulas are based upon the hypothesis that these effects are due to that component of the wind which is parallel to the plane of fire, and which, according to its direc- tion, increases or diminishes the resistance the projectile en- counters. The component of the wind which is normal to the range, and which we will designate by W„, plays a much more com- plicated role, being inextricably mixed up with the phenomena of the drift. The wind acts chiefly upon the head of the pro- jectile, as is apparent from the fact that the air by the rapid motion of translation of the projectile through it is greatly condensed around the head, which it closely embraces, and is thrown off in stream lines, which unite again in rear of the pro- jectile. Any sideway motion of the air will therefore cause an unequal pressure upon the head of the shot, the effect of which is to produce upon a rotating projectile a precession of the axis independent of that which causes the drift, and which greatly modifies it. Suppose, for example, that the projectile has a right-handed rotation and that the wind blows from the left. Its action upon the head of the projectile combined with the rotation will cause the point to fall or droop, bringing its axis more nearly into coincidence with the tangent than it otherwise would be, and thus diminishes the drift. The contrary takes 1 86 PROBLEMS IN DIRECT FIRE. place if the wind comes from the right. Let D be the drift in a calm atmosphere — or, simply, the drift; D^ the diminution of the drift due to a wind blowing from the left ; Z?/ the increase of the drift due to a wind blowing from the right ; and s the lateral displacement which the wind would give the projectile if it did not rotate ; then the expressions for the deviation from the plane of fire will be : For a wind from left and for a wind from the right in which D^ is different from Z>, . The difficulty of computing Z is still further increased by another cause. With the wind from the left, for example, the projectile, having a certain velocity of drift to the right, dimin- ishes more and more the effect of the wind, as the difference between the sideway velocity of the projectile and that of the wind becomes less. It can easily be shown that when the time of flight is considerable, the projectile at the point of fall has generally a greater sideway velocity than the wind, this veloc- ity being in extreme cases as much as 60 or 70 fee;f per second.* With the wind from the right the reverse obtains. Didion's Method of Computing: the Deviating Efifect of the Wind. — The following method of determining the wind deviation independently of the drift was devised by General Didion : Suppose a velocity equal and contrary to that of the wind to be impressed upon all the elements of the system — atmosphere, projectile and gun (origin of co-ordinates), — thus producing the conditions of a calm atmosphere. The angle which the new plane of fire makes with the primitive plane, as well as the variations ^ Fand ^0, produced by the motion impressed upon the system, are determined, and thence the value of s. The expression for this last is, by this process, Fcos y c^,c^. . . c„,w& shall have c^ = Va/ -|- b^, c^ = ^a^ + b^ . . . Therefore the sum of the absolute devia- tions is :2c = :2\/a'-^b\ which is essentially positive. We have already shown that ^a' and .2^' are minima, and therefore :Ec' = :2 a'' + 2b ^ is also a minimum. That is, the sum of the squares of the abso- lute deviations is a minimum. This is what we should expect from the symmetrical grouping of the shots about the centre of impact. Centre of Impact on a Horizontal Target. — Vertical tar- gets being necessarily of moderate size, are employed at the shorter ranges only. They should be used whenever practicable, because by so doing errors due to inequalities of the ground are eliminated. At long ranges we employ the ground (or the 2o6 PROBLEMS IN DIRECT FIRE. surface of the water if fired at sea) as a horizontal target. In this case we will take for the axis of y the trace upon the ground of the vertical target on which aim is taken ; and for the axis of x the parallel to the plane of fire, drawn through the left lower corner of the target. The centre of impact of the shots upon the ground will be determined as before by the co-ordinates ^. = ^ and Y. = ^. n n The deviations, which are always referred to the centre of impact, are classified on a horizontal target, as lateral and lon- gitudinal, the latter corresponding to the vertical deviations before considered. We may assume that the trajectory which passes through a point on the vertical target will also pass through the corresponding point of fall on the horizontal tar- get ; and also that the portion of the trajectory joining the two points is a straight line. If therefore x is the height of a par- ticular shot on a vertical target, and jr, the horizontal distance from the foot of the target to the point of'fall, we shall have X ^ x^ tan 00, in which oo is the angle of fall. By means of this formula we may transfer points from a horizontal to a vertical target, and vice versa. The lateral deviations will be practically the same for both targets. Law of the Deviation of Projectiles. — The deviations of projectiles are analogous to the errors committed in the direct measurement of a magnitude of any kind. In fact, if we fire a great number of shots against a target under precisely similar circumstances, it will be found that the points of impact are the nearest together in the immediate vicinity of the centre of fire, and that these points lie more and more scattered the further we recede from this centre. The likeHhood then of obtaining a particular deviation diminishes rapidly as the latter increases, a limit existing beyond which there will be no de- J'KOBLEM XXI. 207 viation. Errors of observation follow an entirely similar law ; and therefore we can apply to both the same general principles. We will confine ourselves at present to vertical deviations on a vertical target, since the formulas deduced will apply equally to horizontal or longitudinal deviations, and thus a great deal of repetition will be avoided. Let then, as before, «,,«,... a„ be the n vertical deviations of a large number of shots, and make ^/. n — I Then it may be shown by the calculus of probabilities that the probability P that the vertical deviation of an additional shot will not exceed a specified amount s is given by the integral P= ~ I e-''di=F{i), say, in which Ej2 Now suppose we fire a new series of shots under precisely the same conditions as those by means of which E^ was deter- mined. Then the probability 'of any shot of the series not having a greater vertical deviation than ±s will be given by the formula and therefore the probable number of shots which may be ex- pected to fall between the two horizontal parallel lines drawn, the one at a distance s above the centre of impact, and the other at the same distance below it, will be found by multiply- ing the number of shots by the fraction expressing the proba- bility. Mean Quadratic Deviations. — E^ is called the mean ver- tical quadratic deviation, and is computed from the vertical 2o8 PROBLEMS IN DIRECT FIRE. deviations by the equation given above. We may also com- pute E^ directly from the abscissas of the n points of impact, as follows : We have But, from (i), nx. - -^-; whence, substituting in the equation defining E^ , we have V n—i V 71 —1 . Similarly, if E^ is the mean horizontal quadratic deviation, we shall have «_I V M— .1 Mean Deviations. — We may also employ the mean vertical and horizontal deviations for finding the probable deviation P; and when the number of shots is very great, the labor will thus be considerably abridged. Let, then, e^ and Cy be the mean vertical and horizontal deviations, respectively ; that is, let :S.a , Sb ^, = — and e^ = — , n ' n in which the deviations are all taken with the positive sign. Then we have, for the probability of a deviation s, P=F{t), as before ; while t is given by the equation s PROBLEM XXI. 209 As the sums of the positive and negative deviations in each direction are equal in absolute value, the mean deviations can be more easily obtained by dividing the sum of the positive deviations by half the number of shots. Let x^ be the abscissa of any point of impact greater than X^ ; then the correspond- ing deviation will be expressed by x^ — X„, and their sum, sup- posing them m in number, will have the value :Sx,-mX,. The mean horizontal deviation will therefore be _ 'Sx, — mX^ Similarly, Relation between the Mean Quadratic and Mean Deviations. — When the number of shots is very great, the probabilities Pz=F{f) = F \EV~2) \eV7tJ P=F{t)=zF are practically the same ; and in the limit they would be iden- tical. But for a small number of shots the quadratic deviation is more to be relied upon than the mean. In the limit we should have the relation e_ _ y2_ 2IO PROBLEMS IN DIRECT FIRE. From this we obtain the following relations between E and e : = (f)' 1.253314^, and /2\* Q E = 0.797885^. Expression for tt in Terms of the Mean Quadratic and Mean Deviations. — We have from the above equation the following remarkable expression for the ratio of the circum- ference of a circle to its diameter, viz., . = 2(f). In these equations we have omitted the subscripts x and f, since the equations are general. General Probability Table. — The table opposite gives the values of P with i as argument ; and, also, the values of / with wP as argument. From this table we can take P when i is given, and, recip- rocally, t when P is known. If the probability be given we take the corresponding value of i from the table, and then compute the probable deviation s, either vertical or horizontal, by the equation s = ± tE V2, or s = ± te Vtt, according as we employ the quadratic or mean deviation. If a certain deviation i s, vertical or horizontal, be given, we determine its probability by computing t by the equation i = — — = or E V2 e Vtc and take the probability corresponding to t from Table i. This gives the probability that the deviation of a shot will not be greater than s in either direction from the centre of impact. PROBLEM XXI. 211 TABLE I. / P Diff. i P Diff. t P Diff. o.oo O.02 0.04 .0000 .0226 .0431 226 225 225 0.84 0.86 0.88 7651 7761 7867 Ill 106 102 1.68 1.70 1.72 0.9825 0.9838 0.9850 13 12 II •0.06 0.08 O.IO .0676 .0901 .1125 225 224 223 0.90 0.92 0.94 7969 8068 S163 99 95 91 .1-74 1.76 X.78 0.9861 0.9872 0.9882 II 10 9 0.12 0.14 0.16 .1348 .1569 .1790 221 221 219 0.96 0.98 1. 00 8254 8342 8427 88 85 81 1.80 1.82 1.84 0.9891 0.9899 0.9907 8 8 8 o.i8 0.20 0.22 .2009 .2227 ■2443 218 216 214 1.02 1.04 1.06 8508 8586 8661 78 75 72 1.86 1.88 1.90 0.9915 0.9922 0.9928 7 6 6 0.24 0.26 0.28 ■ 2657 .2869 ■ 3079 212 210 207 i.oS 1. 10 1. 12 8733 8802 8868 69 66 63 1.92 1.94 1.96 0.9934 0.9939 0.9944 5 5 S 0.30 U.32 0.34 .3286 .3491 .3694 205 203 199 1. 14 1. 16 1. 18 8931 8991 9048 60 57 55 1.98 2.00 2.25 0.9949 0.9953 0.9985 4 32 II 0.36 0.38 0.40 ■3893 .4090 .4284 .4475 .4662 .4845 197 194 191 187 183 182 1.20 1.22 1.24 1.26 1.28 1.30 9103 9155 9205 9252 9297 9340 52 50 47 45 43 41 2.50 2-75 00 0.9996 0.9999 I. 0000 3 I 0.42 0-44 0.46 0.0443 0.0888 ".1337 0.050 O.IOO 0.150 0.4B 50 0.52 .5027 • 5205 •5379 178 174 170 1.32 1.34 1.36 9381 9419 9456 38 37 34 0.1791 0.2253 0.2724 0.200 0.250 0.300 - 0.54 0.56 ■0.58 •5549 • 5716 • 5879 167 163 160 1.38 1.40 1.42 9490 9523 9554 33 31 29 0.3208 0.3708 0.4227 0.350 0.400 0.450 0.60 0.62 0.64 .6039 .6194 .6346 155 152 148 1.44 1.46 1.48 9583 9610 9636 27 26 25 0.4769 0.5342 0.5951 0.500 0.550 0.600 0.66 0.68 0.70 .6494 .6638 .6778 144 140 136 1.50 1-52 1.54 9661 9684 9706 23 22 20 0.6608 0.7329 0.8134 0.650 0.700 0.750 0.72 •0.74 0.76 .6914 .7047 .7175 133 128 125 1.56 1.58 1.60 9726 9745 9763 19 18 17 0.9062 I. 0179 1.1631 0.800 0.850 0.900 78 0.80 0.82 .7300 • 7421 •7538 121 117 113 1.62 1.64 1.66 9780 9796 g8ii 16 15 14 1.3859 I. 8214 2.3268 0.950 0.990 0.999 212 PROBLEMS IN DIRECT FIRE. Probable Deviation. — When the probability is one-half, that is, when Z' = J, we find from Table i that t = 0.4769, Therefore, calling the value of s in this case r, we have r=± o.476gE V2 = ± 0.6745^ ; or, in terms of the mean deviation, r = ± 0.4769^ Vtt = ± 0.8453^. r is called the probable deviation, that is, the deviation with respect to which the probabilities of obtaining greater or less deviations are equal. In other words, if we have fired a con- siderable number of shots at a target, under the same condi- tions, we may expect that half the deviations will be less than the probable deviation and the other half greater ; and, gen- erally, the probability of obtaining a deviation less (or greater) in absolute value, than the probable deviation is one-half. Fifty-per-cent Zones. — We may, therefore, expect to find one-half the points of impact on the target lying within a zone of indefinite length whose sides are horizontal right lines at distances from the centre of impact equal to -|- r and — r, and whose breadth is therefore 2r. This is called the fifty-per-cent horizontal zone, and its breadth will be denoted by Z^^. We therefore have Z, = 2 X 0.6745^^ = 1.349^,; or, in terms of the mean deviation, Z, = 2 X 0.8453^, = 1-691^:.. Similarly, we have for the breadth of the fifty-per-cent vertical zone, ^y= 1-3494'; or, in terms of the mean deviation, Zy = \.6giey. Twenty-five-per-cent Rectangle. — The intersection of these two zones determines a definite rectangle whose centre is PROBLEM XXI. 213 the centre of impact, and whose sides are parallel to the co- ordinate axes, and which will probably contain twenty-five per cent (fifty per cent of fifty per cent) of all the shots. This rectangle is called the twenty-five-per-cent rectangle. Probable Rectangle. — If we wish to determine a rectangle which shall probably contain fifty per cent of all the shots, we must determine the breadth of a horizontal and of a vertical zone, each having a probability equal to the square root of one- half; and, therefore, giving by their intersection a rectangle whose probability is y\Y.V\^^\. We have -/j = 0.7071 = P; and from Table i we find the corresponding value of ^ == 0.7438. Therefore, in this case, .f = ± 0.7438^' Vt. = 1.05 \(jE ; or, in terms of the mean deviation, J = ± 0.7438^4^ = 1.3183^. Multiplying by 2, and designating the side of a rectangle by S, we have the following expressions for the sides of the probable rectangle : 5, = 2.i04-£'^=: 2.637^^; Sy = 2.104^^ = 2.637^j,. Example i. From the record of firing with a M. L. rifled mortar, at Sandy Hook, April 20, 1886, we take the following data : No. of X y b round. (yards.) (yards.) a 178 4 — 93.11 -4.67 179 84 16 -9.11 -7-33 • 180 32 9 — 61. II -0.33 iSt 163 12 + 69.89 -3-33 182 209 + 115.89 -8.67 183 54 6 -39-11 — 2.67 184 56 10 -37-11 -1-33 185 144 12 + 50.89 -3-33 186 96 9 + 2.89 ho.33 Mean range 3357 yards. 214 PROBLEMS IN DIRECT FIRE. Adding the values of x and y, we find •2x = 838 and 2y = 7& ; ••• X = H^ = 93-", V.^^ = 8.67. We next find ^a' and 2l>' as follows : 2x' = 1 14334 and Sy' — 858 ; .-. 2a' = 1 14334 —^ = 36306.89 ; 26' = 838 - ^-^ = 182 ; y and finally, employing the quadratic deviations, / 36306.89 V ^.= i— ^ ) =67.37; The breadths of the 50-per-cent zones are, therefore, Z, = 1.349 X 67.37 = 90.88 yards ; Z, = 1.349 X 4-77 = 6.43 yards ; and these are the sides of the 25-per-cent rectangle. For the probable rectangle we have 5, = 2.104 X 67.37 = 141-75 yards ; Sy = 2.104 X 4.77 = 10.04 yards. The probable deviations are, of course, one-half the breadths of the 50-per-cent zones ; or, in round numbers, 45 yards and 3 yards, respectively. To ascertain the actual number of hits in these rectangles, for comparison, we make use of the last two columns of the above table, which give the values of a and 3, or the co-ordinates FKOBLEM XXI. 215 of the points of impact with reference to the centre of impact as origin. These are calculated by the formulas and a=^ X — X„ b^y- F,, We find the number of hits in the probable rectangle to be 3, anjd in the 50-per-cent rectangle 6. These numbers should be \ and \, respectively. Example 2. Thirty shots were fired at Meppen, December 20, 1880, at a target looo m. distant, with a 12-cm. siege-gun, weight of projectile 16.5 kg., weight of charge 4.5 kg. of pris- matic powder, giving a M. V. of 512 m. s. All the shots struck the target to the left of the line of fire, and all but three below the centre. The following table gives the co-ordinates of the points of impact with reference to a suitably chosen origin, in centimetres : No. X y No. - 1 1 ' \ No. - y J 35 75 II 35 55 21 60 20 2 50 40 12 65 65 22 20 3 80 55 13 65 70 23 25 4 40 25 14 65 45 24 25 95 5 150 75 15 55 25 25 70 65 6 145 25 16 70 80 26 75 7 no 17 35 55 i 27 60 70 8 75 10 18 75 25 ! 28 50 60 9 So 20 19 55 40 : 29 50 10 35 35 20 40 105 ! 30 50 35 Adding the values of x and y, we find "Sx = 1675 and 2y = 1440- 1671; 1440 .-. X,= ^ = 55-83 cm., and F, = ^- = 48 cm. 2l6 PROBLEMS IN DIRECT FIRE. We next find J'«" and 23° as follows : Sx^ = 131625, 2/ = 88850 ; .-. •2a^= 131625 - 30 X (55-83)'= 38104.17. :2b-' = 88850 - 30 X (48)° - 19730, and finally = (^T=3^-^^^ The breadths of the 50-per-cent zones are Z, = 1.349 X 36.248 = 48.9 ; Zy = 1.349 X 26.083 = 35-2; and these are the sides of the 25-per-cent rectangle. For the sides of the probable rectangle we have S^ = 2.104 X 36.248 = 76.3 ; Sj = 2.104 X 26.083 = 54-9. The actual number of points of impact in this case (deter- mined as in Ex. i), in the 25-per-cent rectangle, is 10 ; and in the probable rectangle, 16. These numbers are, by theory, 7^ and 15, respectively. Example 3. Fifty shots were fired with the gun of Ex. i, December 17, 1880, at an angle of elevation of 5°, giving a mean range of 2894.3 m. Taking for the axis of x a line drawn through the point of fall farthest to the left, and parallel to the plane of fire, and for the axis of 7 a line drawn perpendicular to the plane of fire through the point of fall giving the shortest range, we have the PROBLEM XXI. 217 following co-ordinates of the fifty points of fall upon the ground, in metres : No. r y No. ^ y No. ' y I 69 5-0 18 44 2.8 35 14 2.0 2 69 2-5 19 44 2.5 36 14 1.5 3 61 4.7 20 45 30 37 9 I.O 4 56 3-5 21 47 1-3 38 10 I.O 5 55 2.7 22 34 3-5 39 9 2.0 6 56 1-5 23 33 3.8 40 6 0.7 7 54 3-8 24 34 3-1 41 6 I.O 8 35 5.0 25 31 4.0 42 3 2.5 9 35 31 26 25 2.5 43 4 3.0 10 36 30 27 26 2.5 44 2 3-5 II 38 30 28 27 2.5 45 2 3.8 12 38 4-5 29 26 4.2 46 I 45 13 39 4-5 30 24 3-5 47 9 1.5 14 39 5-5 31 24 4.0 48 22 4.0 15 40 I.I 32 23 3-5 49 21 3-5 16 41 1-7 33 23 4.9 50 0.0 17 41 2.4 34 19 5-5 Remark. These sliots are not given in the order they were fired. From these co-ordinates we obtain '2x= 1463 and .5^= 1 50.1. The co-ordinates of the centre of impact with reference to the assumed origin are therefore 1463 X. = = 20.26 m. ; ° 50 ^ ' 150.1 3.002 m. For the mean deviations e^ and ey we have '2x, =■ 1 1 14, "Sy^ = 98.9, yn = 25, for deviations in range, and m = 24, for lateral deviations. Therefore 1 1 14 — 25 X 29.26 25 ■ 98.9 — 24 X 3.002 25 15.3 m.; = 1.07408 m. 21 8 PKOBLEMS IN DIRECT FIRE. We next compute '2x'' = 59605, and 2j^ = 537.44 ; whence 2«' = 59605 - ^^' =16797.62; :Eb-' = 537-44 - -~^ = 86.8398: and therefore /i6797.62\* £, 49 /86.8398\i ^ V 49 >' We are now prepared to compute the sides of the 2S-per- cent rectangle, the sides of the probable or any other proposed rectangle, and the probable deviations. We have, using the quadratic deviations, 1.349 X 18.515 = 25.0 m. )„. , r ^ 1 „ J- Sides of 2|-per-cent rectangle. 1.349 X 1-331= 1.8 m. 3 °^ ^ 2.104 X 18.515 = 39.0 m. ) c.., r u ui ,.1 „ )■ Sides of probable rectangle. 2.104 X 1. 331 = 2.8 m. ) '^ ^ The probable deviations are, respectively, one-half the sides of the 25-per-cent rectangle. Therefore the probable longitu- dinal deviation, or probable deviation in range, is 12.5 m. The probable lateral deviation is 0.9 m. The origin of co-ordinates is 22.5 m. to the left of the plane of fire. The mean lateral deviation from the plane of fire is therefore 22.5 — F„ = 22.5 — 3.002 = i9.4984n. to the left. The origin is also 2865 m. from the gun. The mean range is therefore 2865 + X„ = 2865 + 29.26 = 2894.26 metres. PROBLEM XXI. 219 Comparing these results with the experiments (as in Ex. i), we find 13 hits within the 2S-per-cent rectangle, whereas by theory there should be 12.5 ; also 22 hits within the probable rectangle instead of 25. One-half the shots have a longitudinal deviation less than the probable deviation, and the other half greater ; while of the lateral deviations, 24 are less and 26 greater than the probable deviations. If we employ the mean deviations instead of the mean quadratic deviations, we have 25.9 m, 1.8 m. i.6qi X 1 5-3 1. 69 1 X 1. 014 2.647 X 15-3 =40.3 m 2.637 X 1-074= 2.8 m r Sides of 25-per-cent rectangle. \ Sides of probable rectangle. When the number of shots is considerable as in this example, it makes but little difference in the results whether we employ the mean or quadratic deviations. Table for Computing Sides of Rectangles having a given Probability. — The following table is useful in solving examples similar to the above ; that is, when we wish to deter- mine the sides of a rectangle about the centre of impact, which will probably contain a given per cent of hits. The table is constructed precisely as has been already illus trated in the case of the probable rectangle. That is, we enter Table i, with the square root of the given probability/" as the argument, and take out the corresponding value of t. We then have -^ = 2 V2t — 2.8284/. TABLE 2. p 5 P 5 P .? E E E 0.05 0.568 0.40 1.802 0.75 2997 .10 0.815 •45 '•952 .80 3.237 ■15 1. 013 • 50 2.104 •85 3- 524 .20 1. 187 • 55 2.260 .90 3.898 .25 1-349 .60 2.425 •95 4-473 .30 '•503 •65 2.599 •99 5^6l3 • 35 1.653 .70 2.788 ■999 6^937 220 PROBLEMS IN DIRECT FIRE. In this table P is the given probabihty, or " probability per cent" as it is frequently called ; 5 is either side of the rectangle ; and E the corresponding quadratic deviation. If, in place of the quadratic deviation, we prefer to use the mean deviation, we must substitute 1.2533? ^or E. If the probable deviation is given, we employ the relation r E = ;= = 1.4826^. 0.4769 \2 Example if. Compute the sides of the rectangle whose centre is the centre of impact, which will probably contain 75 per cent of the shots of Ex. 3. We have in this case /*= 0.75, corresponding to which we find, from Table 2, 5 •g = 2.997. Therefore S, = 2.997^^ = 2.997 X 18.515 = 55.5 m., 5; = 2.997^, = 2.997 X 1.331 = 3-989 m., which are the sides required. The actual number of hits within this rectangle was 38, — agreeing with theory. Enveloping Rectangle. — For a probability of 90 per cent we find, from Table 2, 5 = 3.898^, and this rectangle includes nearly all the hits of Ex. 3. The rectangle which includes all the hits of a given number of shots is called the enveloping rectangle of the shots. We may say, as a general rule, that the sides of the enveloping rectangle do not exceed, respectively, 4 times the mean quadratic devia- • tions or 5 time's the mean deviations. That is, we may reason- ably expect that no deviation will exceed these limits. Any shot falling outside the enveloping rectangle must be regarded as abnormal. PROBLEM XXI. 221 Comparison of Experiment with Theory. — For purposes of comparison we have computed the following table with the data of Ex. 3, showing the agreement between theory and practice. The computations are similar to those. given in the solution of Ex. 4. The table explains itself. Given probability. Sides of Rectangle in Metres. No. of Hits. P s^ ^y Theory. Observed. 05 10.52 0.76 2.5 I 10 15.09 1.08 5 9 15 18.76 1-35 7-5 :i 20 21.98 1.58 10 II 25 24.98 1.80 12.5 13 30 27.83 2.00 15 16 35 30.60 2.20 17-5 19 40 33.36 2.40 20 21 45 36.14 2.60 22.5 21 50 38.96 2.80 25 22 55 41.84 3-01 27 5 27 60 44.90 3-23 30 28 65 48.12 3.46 32 5 28 .70 51.62 3-71 35 31 ■ 75 55-49 3-99 37.5 39 .80 59-93 4.31 40 43 .85 65-25 4-69 42.5 45 90 72.17 5.19 45 47 •95 82.82 5-95 47.5 49 This table gives a tolerably clear idea of the confidence that may be placed in the solutions of problems having reference to probability of fire. By comparing the last two columns it will be seen that the agreement between theory and practice, though near enough for all practical purposes, is not exact in a single instance. For example, in the probable rectangle where, by theory, there should be 25 hits, we find but 22. But it may be shown by the calculus of probabilities that the prob- ability of there being exactly 25 hits in the probable rectangle is quite small. To show this, we will make use of BernoulH's theorem of compound probabilities. If we designate by/- the probability of a success, and by ^^ = l — / that of a failure, the probability 222 PROBLEMS IN DIRECT FIRE. that in in + n trials there may be m successes and n failures is expressed by the equation 1.2.3.4 imA- n) (1,2.3.,. ^^){^ . 2 .3 . ..«)^ ^ In our example the number of trials is 50; and we wish to determine the probability that 25 of them shall be successes and 25 failures — that is, that out of the 50 shots 25 shall strike within the probable rectangle (whose probability is ^), and 25 without it. We, therefore, have m = n = 2t,;p — ^;q—l—^—^; and by substituting, the above expression becomes i5o5°Vioo;r /i\5° — 7-50 2550 X 5o;r\2/ ■ By cancelling factors common to the numerator and de- nominator of this fraction, we easily obtain P=--==o.iiH-. y257r We see, then, that the probability of exactly 25 out of the 50 shots falling within the probable rectangle is but ^^. That is, if we should fire 100 series, of 50 shots each, under PROBLEM XXI. 223 precisely the same conditions, we should have no reason to ex- pect that exactly 25 shots would fall within the probable rect- angle more than eleven times. In this case, what we learn from the calculus of probabil- ities is that the probability that the probable rectangle con- tains 25 hits is greater than that of its containing 20 or any other number fixed upon. Moreover, it follows from Ber- noulli's theorem that the probability that the probable rect- angle contains 25 -|- « hits is the same as that it contains 25 — « hits. We have a right, therefore, to infer that in several similar series of shots the number of projectiles falling within the probable rectangle will be very nearly one-half the shots fired. Example 5. With the data of Ex. 3, compute the prob- ability corresponding to a transverse zone extending 22.5 metres on both sides of the centre of impact. Also determine the breadth of the longitudinal zone which shall have the same probability as this transverse zone ; and, lastly, the probability of the rectangle formed by the intersection of these zones. We have here, 5^ = 2 X 22.5 = 45 ; E^— 18.515; and since 7. = 2 V2t, E we have, using the given numbers, 45 t = — ~^ = 0.8593. 21/2 X 18.515 ^^^ Entering Table i with this value of t, we find, by interpola- tion, IQ< P = 0.765 1 + ^Q X o.oi 10 = 0.77^7, which is the required probability. To find the breadth of the longitudinal zone having a probability equal to that of the transverse zone, we evidently 5v s,= ^E I-33I; and therefore X 1.33 I = 3-235 metres. 224 PROBLEMS IN DIRECT FIRE. have from the above equation, since t is the same for both zones, the relation and therefore We have found Ey = ""18.515 That is, the required longitudinal zone extends 1.64- metres on both sides of the centre of impact. The probability of the corresponding rectangle is 07757 X 0.7757 = 0.6017. This rectangle should, therefore, according to the theory, contain 0.6017 X 50 = 30 hits. The actual number was 27. Table for Computing the Width of a Zone of given Probability. — The following table facilitates the solution of examples similar to the above. The argument is either -=. or — , 11 e according as the quadratic or mean deviations are employed. 5 is the breadth of any zone either horizontal or vertical, transverse or longitudinal, whose centre coincides with the cen- tre of impact. Z', is the corresponding probability when quad- ratic deviations are employed ; and P^ the probability in terms of the mean deviations. The table was computed as follows : For each value of the argument, ~ the value of t was computed by the equation 5^ ~ 2 4/2^' and the corresponding value of P taken from Table i. PROBLEM XXI. 22S Similarly, for each value of - the value of t was computed by the equation 2 \n e and thenlhe value of P was taken from the table as before, TABLE 3. 5 5- - or - /", Diff. •Pj Diff. - or - />! Diff. ■Pj Diff. El E e 0.0 .0000 399 .0000 3:8 2.9 8529 135 •7527 159 O.I .0399 397 .0318 318 3-0 8664 124 .7686 152 0.2 .0796 396 .0636 317 3-1 87B8 116 .7838 144 0.3 .1192 393 ■0953 315 3-2 8904 106 .7982 139 U.4 .1585 389 .1268 312 3-3 9010 99 .8121 129 0.5 • 1974 384 .1580 311 3-4 9109 go .8250 123 0.6 .2358 378 .1891 308 3-5 9199 82 .8373 "7 0.7 .2736 372 .2199 30s 3.6 9281 76 ■ .8490 no 0.8 .3108 375 ■ 2504 300 3-7 9357 6g .8600 104 0.9 .3483 366 .2804 297 3-8 9426 62 .8704 99 I.O .3829 348 .3101 2gi 3-9 9488 57 .8803 91 I.I • 4177 338 •3392 287 4.0 9545 51 .8894 87 1.2 ■4515 328 .3679 280 4.1 9596 46 .8981 80 1-3 .4843 317 ■3959 276 4.2 9642 42 .9062 76 1-4 .5160 307 .4235 269 4-3 9684 38 .9137 71 1-5 • 5467 296 •4504 262 4.4 9722 33 .9208 65 1.6 .5763 284 .4766 257 4-5 9755 30 ■9273 62 1-7 .6047 272 .5023 250 4.6 9785 27 •9335 57 1.8 .6319 260 • 5273 242 4-7 9812 24 .9392 53 1.9 •6579 248 •5515 235 4.8 9836 21 .9445 49 2.0 .6^27 236 •5750 228 4-9 9857 19 •9494 45 2.1 .7063 224 .5978 221 5-0 9876 31 ■9539 80 2.2 .7287 211 .6199 212 5-2 9907 24 .9619 70 2-3 .7498 200 .6411 206 5-4 9931 18 .9688 57 2.4 .7698 189 .6617 197 5.6 9949 10 • 9745 48 2.5 .7887 176 .6814 189 5.8 9959 10 •9793 40 2.6 .8063 167 .7003 182 6.0 9969 31 •9833 167 2.7 .8230 158 .7185 175 00 I 0000 1. 0000 2.8 .838S 141 .7360 167 Example 6. Employing the data of Ex. 3, required the probable number of shots that we may expect to fall within a rectangle whose centre is the centre of impact, and whose sides are 5^ = 30 m., and Sy= 1.5 m. 226 PROBLEMS IN DIRECT FIRE. We have for the transverse zone, employing the quadratic deviation, 5 30 E 18.515 = 1.620. With this argument we find from Table 3, by interpola- tion, 20 P, = 0.5763 + — X 0.0284 = 0.5820, which is the probability for the transverse zone. Therefore the number of hits we may look for in this zone is 0.5820 X 50 = 29. The actual number found there is 27. For the longitudinal zone we have 5 1.5 -£ 1.331 Therefore, for this zone, P, = 0.4177 +^ X 0.0338 = 0.4268, which is the probability for the longitudinal zone. The probable number of shots in this zone is, therefore, 0.4268 X 50 = 21 ; which agrees with observation. The probability of the rectangle of intersection of these two zones is the product of the separate probabilities. .-. P = 0.5820 X 0.4628 = 0.2484, which is the probability required. The probable number of shots in this rectangle is, therefore, 0.2484 X 50 = 12 + , while the actual number is 14. Example 7. Required the dimensions of a vertical target large enough to receive all the shots of Ex. 3. Also determine the sides of the probable rectangle, and its position on the ver- tical target, supposing the mean angle of fall {00) to be 8° 11'. PROBLEM XXI. 227 We will suppose the target to' be placed with its left lower corner at the origin of co-ordinates. The shot having the longest range strikes the ground 69 m. beyond the target, which must therefore be at least 69 tan 8° 11' = 9.9 m. high to receive this shot. The greatest deviation from the axis of x is 6 m. The required dimensions are therefore 9.9 m. high and 6 m. broad. The mean lateral deviation and mean lateral quadratic de- viation remain the same for the vertical as for the horizontal target, together with all that has been deduced from them. While the longitudinal deviations must be multiplied by tan 00 to reduce them to vertical deviations. We therefore have for the sides of the probable rectangle S, = 39.0 tan 8° 1 1' = 6.6 m. ; Sy = 2.8 m. In the same way may the sides of any other rectangle on a vertical target be determined from those already found for the horizontal target. The co-ordinates of the centre of impact on the vertical tar- get are X^ = 29.26 tan 8° 11' = 4.2 m. ; X, = 3.0 m. Example 8. For the gun of Ex. 3 we have found the mean error in range to be 15.3 m., and in direction (lateral) to be 1.074 m., for a range of 2894 metres. At this range what is the probability of hitting with a single shot a horizontal target 41 m. by 2 m., the longer side being parallel to the plane of fire, and its centre coinciding with the centre of impact? We have e^= 15.3, ey= 1.074. Therefore for the trans- verse zone :?^^ = 2.680. e iS-3 228 PROBLEMS IN DIRECT FIRE. With this argument we find from Table 3, by interpola tion, 8 /!, = 0.7003 + — X .0182 = 0.7149. For the longitudinal zone we have S 2 1.074 = 1.862. Therefore for this zone 62 P, = 0.5273 + — X .0242 = 0.5423. The probability of the rectangle of jntersection is O.7149 X 0.5423 = 0.3877 ; which is the probability required. The probable number of shots in this rectangle is 0.3877 X 50 = 19; which agrees with observation. Probability of Hitting any Plane Figure. — In what pre- cedes we have given methods for computing the probability of hitting a rectangle whose sides are parallel to the co-ordinate axes, and whose centre coincides with the centre of impact. The same principles enable us to compute the probability of hitting, at least approximately, any plane figure. X A G E B F H Y Y L I C L > X PROBLEM XXI. 229 In the diagram, suppose to be the centre of impact of a number of shots upon a horizontal target, OX the direction in which longitudinal deviations, or deviations in range, are meas- ured, and OY, perpendicular to OX, the direction in which lateral deviations are measured. In the last example we have shown that, with the data there used, the probability of hitting the rectangle ABCD (not drawn to scale) is 0.3877, and that the number of shots actually faUing within this rectangle agrees with the probability. We may also assume, on account of the symmetrical group- ing of the shots about the centre of impact O, that the proba- bility of hitting the rectangle OEBI is one-fourth that of hit- ting the rectangle ABCD. The probability of hitting within the rectangle OFHI, or OEGL, is found in the same way as that of hitting within the rectangle OEBI. In the first case we should take 5^ =20.5 and Sy=^2 (see Ex. 8) ; and in the second case we should take 5j = 4i and S, = i. The probability of hitting within EBFH is found by sub- tracting the probability of the rectangle FHOI from that of OEBI. In a similar manner we should find the probabili- ties of hitting within the rectangles GBIL, EGKF and IHKL. Finally, the difference between the probabilities of the rect- angles GBIL and IHKL is the probability of hitting within the rectangle GBHK. In the same way we may divide up any plane figure into small rectangles, and the sum of their separate probabilities will be, approximately, the probability of hitting the figure. Example 9. With the data of Ex. 8, what would be the probability of hitting the deck of a ship represented by a rect- angle 80 m. by 16 m. — {a) when the ship is approaching the gun, bow on, and ip) when the ship is steaming perpendicular to the plane of fire ? We will suppose the centre of impact to be at the centre of the rectangle. 230 PROBLEMS IN DIRECT FIRE. 5 80 (a) We have for the transverse zone, - = — — = 5.229; and ^ i5"3 5 16 for the longitudinal zone, — = = iS- *" e 1.074 The respective probabilities from Table 3 are O.9639 and I (practically). The probability of hitting the deck is, there- fore, for each shot, 0.9639. 6" 16 = {U\ In this case we have for the transverse zone, - ^ ^> 'e 15.3 1.046; and for the longitudinal zone, — = = 74. The ^ ' ^ e 1.074 ' probability of hitting the deck in this case is, therefore, 0.3235, or one-third what it was in the former case. Example 10. If a zone of a certain breadth contains in per cent of a large number of shots fired, what is the breadth of another zone which will probably contain n per cent of the shots, supposing all the shots to be fired under precisely similar circumstances? The given "per cents" are, for a large number of shots, the respective probabilities of a shot falling within the given zones. With these probabilities we enter Table i and take out the corresponding values of /. Then, since the deviations (mean or quadratic) are the same for both zones, it follows that the breadths of the zones are proportional to the values of t. For example, for a 20-per-cent zone we find ? = 0.1791; and for an 8o-per cent zone, t = 0.9062. Therefore we have the proportion O.1791 : 0.9062 :: I : 5. That is, the 80-per-cent zone must be 5 times as wide as the 20-per-cent zone. Example 11. If the 50-per-cent zones (horizontal and ver- tical) are each 6 feet wide for a certain gun and range, what is the probability of hitting a target 6 feet square, if the centre of impact is in the middle of the lower edge? As the 50-per-cent vertical zone just includes the target, the probability for this zone will be 0.5. PROBLEM XXI. 231 We must next determine the probability of a horizontal zone doubje the height of the target, since the centre of impact is, by hypothesis, in the lower edge of the target. This is equivalent to determining the probability of a zone twice the breadth of the 50-per-cent zone. Then, since the breadths of zones for the same series of shots are proportional to the values of A we have for the required zone ' ? = 2 X 0.4769 = 0.9538. With this argument, we find from Table i the probability of the zone to be 0.8226 ; which is the probability of a zone twice the breadth of the 50-per-cent zone. As the target is in the upper half of this zone, we divide the above probability by 2, which gives for the probability of the half-zone 0.4113. The probability for hitting the target is therefore f = 0.5 X 0.41 13 =4 0.2057. If the centre of impact were in the centre of the target, the probabiHty of hitting would evidently be 7^=0.5 Xo.S =0.25. If the centre of impact were raised 2 feet on the target, the probability of the vertical zone would still be 0.5. For the horizontal zones, we must take one 4 feet broad to get the probability of hitting that part of the target below the centre of impact ; and one 8 feet broad to get the probabiHty of hit- ting the upper part of the target. The values of t for these two zones are, respectively, / = I- X 0.4769 = 0.3179, and / = f X 0.4769 = 0.6359. The corresponding zonal probabilities are, therefore, 0.3469 and 0.6315. One-half ' the sum of these probabilities is the probability of the horizontal zone which includes the target. Finally, multiplying this last probability by the probability 232 PROBLEMS IN DIRECT FIRE. of the vertical zone (0.5), we find the probability of hitting the target to be , P = 0.5 X 0.4892 = 0.2446. Example 12. There were fired at the Griison turret, at Bucharest, on December 31, 1885, and January i, 1886, 94 pro- jectiles from a Kruprp 2i-cm. rifled mortar, planted at a distance of 25 10 m. (2745 yards). The charge was 3 kg. of coarse-grained powder ; weight of projectile, 91 kg. The angles of elevation varied from 53° to 56° 30' ; and the angles of fail from 57° 30' to 61° 30'. Omitting shots numbered 61, 62 and 72, whose points of impact are not given in the work from which this data has been taken,* we find for the remain- ing 91 shots as fpllows : Mean deviation in range, 33.27 m. ^ e^ ; and mean deviation in direction, 9.905 m. = ^j,. The co-ordinates of the centre of impact with reference to the centre of the turret are, in range, x = -\- 0.277 m., and in direction, j/ = -j- 0.275 m. As the turret was 6 metres in diameter, it will be seen that the centre of impact of the 91 shots was within the circumference of the turret. This indicates very fine marksmanship ; but as the turret was not hit, it is interesting to know the probability •of hitting it. Instead of the turret we will take its circumscribing square ; and, further, will assume that the centre of impact coincides with the centre of the turret. We have S, 6 — = = 0.1803. e^ 33-27 The probability, therefore, of a shot striking within a trans- verse zone embracing the turret is, from Table 3, 0.0573. We also have — = = 0.6057. e, 9.905 * Revue Militaire Beige, vol. 2, i886, page 171. PROBLEM XXI. 233 Therefore the probability of a shot striking within a longi- tudinal zone including the target is, from Table 3, 0.1909. The required probability is, therefore, P — 0.0573 X 0.1909 = 0.0109. That is, in 100 shots we could not expect to hit the target more than once. The dimensions of the 25-per-cent rectangle and probable rectangle are 1.691 X 33-27 = 56.26 m. ) „. , . 1.691 X 9.905 = 16.75 m. } Sides of 25-per-cent rectangle. 2.637 X 33.27 = 87.74 m. I 2.637 X 9-905 = 26.12 m. j ^ ^ The enveloping rectangle, or the rectangle containing all the shots fired, was 200 m. (218.7 yards) long and 70 m. (76.55 yards) broad. Curves of Equal Probability. — A curve of equal probabil- ity is one for which the probability of projectiles striking its dif- ferent points is constant. In what has preceded it has been assumed that the perimeter of a rectangle enjoys this property ; but a little consideration will show that this assumption, though perhaps accurate enough for most practical purposes in gun-' nery, is not strictly correct. Curves of equal probability are ellipses (including the circle) ; and they enjoy the following characteristic properties, viz. : Of all the equal plane areas that can be considered on the target {vertical or horizontal), that bounded by an ellipse of equal proba- bility is that in which the probability of projectiles falling is the greatest. From this cause the areas bounded by the ellipses mentioned have been called areas of maximum probability. Viewing the question of probability of fire from a theoretical point of view, it appears natural to choose one of these ellipses as a criterion for forming a judgment of the precision of pieces; for example, that of 25 or 50 per cent of the shots. The area thus chosen can be considered to be made up of the different 234 PROBLEMS IN DIRECT FIRE. elliptic rings, which include the elements to which the same probability attaches ; to all the elements of which it is com- posed correspond greater probabilities than to those which are exterior to them, and the anomaly is not incurred which is met with in the 25- or 50-per-cent rectangles, where some shots are considered as acceptable whose probabilities are inferior to those of others which are rejected.* Relations between the Semi-axes of Ellipses of Equal Probability and the Deviations. — It may be shown that the principal axes of ellipses of equal probability (which we will take parallel to the co-ordinate axes) are proportional to the respective deviations (mean, mean quadratic, or probable) in the same directions. That is, if a and b are the semi-axes, we have the relations, when the number of shots is very great, a r^ E^ e". We also have b Ty Ey e^ a — kE, \^ = ke.Vn = 0.4769 and kr b = kE^V2 = keyV7t^ ^, ^ " 0.4769 k being a constant whose value may be determined for any given value of a or ^ by one of the above equations. Probability of a Projectile falling within an Ellipse of Equal Probability. — It may be shown that the probability P of a projectile falling within an ellipse of equal probability is given by the equation P= I -e-^, in which e is the Naperian base, and k a quantity defined by either one of the equations given in the preceding article. * Translation of an article on the Precision of Fire-arms, published in the Memorial d'Artilleria. By Captain P. A. Macmahon, R. A. The author desires to express his great indebtedness to this admirable paper. PROBLEM XXI. 235 From this equation the probability of projectiles falling within a given ellipse can be easily computed. Table for Computing the Semi-axes for a given Probability. — To determine the semi-axes of an ellipse cor- responding to a given probability, we have from the above expression for P, substituting for k^ its values in succession, 2a 2b = 2 4/ - 2 log, (i .- P) and 2a 2b 2V -7t log, (I - P), By means of these formulas the following table was com- puted : TABLE 4. 2« ih ria 2^ 2« lb 2rt 2^ p — or — or P ^-~ or ■ — — or — E E ^^ e.. E E e e X y X y X y X y ■05 0.641 0.803 0.55 2.527 3-168 .10 o.giS 1. 151 0.60 2.607 3-393 ■15 1. 140 1.429 0.65 2.898 3-632 .20 1.336 ,1.665 0.70 3.104 3-890 .25 r.517 i.goi 0.75 3-330 4-174 •30 1.689 2. 117 0.80 3.588 4-497 •35 1.856 2.327 0.85 3.896 4-883 .40 2.021 2.534 0.90 4.292 5-379 ■45 2.187 2.741 0.95 4.895 6.136 • 50 2.355 2-951 1.00 ■ 00 00 With the help of this table it is as easy to compute an ellipse corresponding to a given probability as to compute a rectangle. For example, for the probable ellipse we have ^=0.50; and therefore, from the table. 2« = 2.355^^= 2.95 If, 23 = 2.3555, = 2.951^^. "236 PROBLEMS IN DIRECT FIRE. For the 2t,-per-cent ellipse we have 2«= 1.5 !;£•,= 1.901^;,; 2b = 1.5 17^^ = i.QOifj,; and similarly for any other ellipse. Area of Probable Ellipse. — The area of the probable ellipse is nab = 4.3552^^^, = 6.841 i^^^j,; while the area of the probable rectangle is (see page 213) S^S, = 4.4268^,^^ = 6.95 38^;.^^,, which is greater than the former, as has been already stated. Equation of Probable Ellipse. — From the relations already given, it may easily be shown that the equation of an ellipse of equal probability is J/ = f^ V^^^' = ^ Vd' - x' = ^ Va' - x\ I-'x ^x "x In all the equations here given relating to ellipses of equal probability, the centre of the ellipse is supposed to coincide with the centre of impact, and the principal axes to he parallel to the axes X^ and Y„ . Example 1.3. Compute the probable ellipse and 25-per- cent ellipse with the data of Ex. 3. We have Ex = 18.515 m., and Ey = 1.331 m. Probable ellipse. /'=o.50 2« = 2.355 X 18.515 = 43.6 m. ; 2b = 2.355 X 1.331 = 3-1 m. 25-per-cent ellipse. f*= 0.25 2a = 1. 517 X 18.515 = 28.1 m.; 2b = 1. 517 X 1. 331 = 2.0 m. If we employ the mean deviations instead of the mean qiiadratic, we have e^ = 15.3 m. and e^ = 1.074. PROBLEM XXI. 237 Probable ellipse. P=: 0.50 2« = 2.951 X 15-3 =45-2 m. ; 2b — 2.951 X 1.074 = 3-2 m. 25.per-cent ellipse. P— 0.25 2« = 1. 901 X 15.3 = 29.1 m. ; 2(5 = 1.901 X 1.074 = 2.0 m. It will be seen that the maximum length and breadth (2a and 2<5) of the ellipse of equal probability for any given prob- ability are greater than the corresponding sides of the rectangle of the same probability. But as each point of the bounding ellipse has the same probability, it follows that in employing rectangles some hits are likely to be omitted which have a greater probability than others which are retained. Table for Computing the Probability of a given Ellipse. — We have, for the probability required, p= I -e-'^. Therefore, employing mean quadratic deviations, Multiplying by the modulus of the common system of logarithms, we have 1 ^, D\ 0434294s /2«V Similarly, 1 /x n\ 0.434 2945 f2b y Iog(i -/')=- 8— (^)- We also have, in terms of the mean deviations, 1 (i-P) = - "•4342945 f^V ^ _ 0.4342945 /2^y 238 PROBLEMS IN DIRECT FIRE. The following table was computed by these formulas. The , . 2a 2b , , , argument may be either -p- or -^ , accorcyng as the length or breadth of the ellipse is given ; and the required probability follows in the second column, under the heading /',. We may 2a 2b also take for the argument either — or — ; and then the ^x ^y probability sought will be found in the fourth column headed P,. TABLE 5. E -Pi DifE. A Diff. 2^ E P^ DifE. .P2 DifE. O.I .0012 38 .oooS 24 2.6 5704 270 .4161 241 0.2 .0050 62 .0032 39 2 7 5980 267 .4402 239 0.3 .0112 86 .0071 56 2 8 6247 258 .4641 238 0.4 .0198 no .0127 70 2 9 6505 248 .4879 235 0.5 .0308 132 .0197 85 3 6753 239 .5U4 231 0.6 .0440 154 .0282 100 3 I 6992 228 ■ 5345 228 0.7 .0594 175 .0382 115 3 2 7220 217 • 5573 223 0.8 .0769 194 .0497 127 3 3 7437 206 ■5796 218 0.9 .0963 212 .0624 141 3 4 7643 194 .6014 213 I.o .1175 229 .0765 153 3 5 7837 184 .6227 20S I.I .1404 243 .0918 165 3 6 802 T 173 ■ 6435 201 1.2 .1647 257 .1083 175 3 7 8194 161 .6636 195 1-3 .1904 269 .1258 1S6 3 8 8355 151 .6831 I8S 14 .2173 279 • 1-444 195 3 9 8506 141 .7019 182 1-5 .2452 287 .1639 204 4 8647 130 .7201 175 1.6 • 2739 293 .1843 211 4 I 8777 120 ■ 7376 167 1-7 .3032 298 .2054 219 4 2 8897 112 • 7543 ■161 1.8 • 3330 302 .2273 224 4 3 9009 102 .7704 154 1.9 .3632 303 .2497 229 4 4 9III 93 .7858 146 2 .3935 3<>3 .2726 234 4 5 9204 86 .8004 139 2.1 .4238 301 .2960 237 4 6 9290 78 • 8143 133 2.2 ■ 4539 299 • 3197 239 4 7 936S 71 .8276 125 2.3 .4838 294 ■3436 241 4 8 9439 64 .8401 119 2.4 • 5132 290 ■3677 242 4 9 9503 58 .8520 112 2.5 .5422 282 •3919 , 242 5-0 9561 .8632 Example 14. With the data of Ex. 12, what is the proba- bility of hitting the area bounded by an ellipse of equal probability whose length (20) is 50 metres ? Also calculate the breadth (2b) of the ellipse. In this example we have, employing mean deviations, 2a ■■ 50 m. ; e^ = 33.27 m. ; and e, = 9.905 m. PROBLEM XXI. 239 Therefore 2a so — = = 1.5029; and by interpolation from Table 5 we get P, = 0.1639 + ^ X 204 = 0.1645, which is the probability required. The value of 2b is found as follows : We have the relation 2b 2a 50 ^y ^x 33-27 But ey = 9.905. •■• 23 = — ^ X 9-905 = 14-886 metres. ' Example 15. What is the probability of hitting a vertical circle whose diameter is 16 inches, assuming the mean deviation in either direction to be 8 inches, and the centre of impact to coincide with the centre of the circle ? In this case we have 2« = 2/J = 16 inches, and ^^ = ^^ = 8 2a inches. We therefore have — = 2 ; and from Table =;, e -' f J = 0.2726. That is, a little more than one-fourth the shots would, in the long-run, strike the circle. The assumption made above that the mean deviations (vertical and horizontal) upon a vertical target are equal is approximately correct in small-arm practice, especially by ex- perts. In this case it will be evident by inspection of the target that the areas of equal density are approximately bounded by concentric circles whose centres are at the centre of impact. That is, the pencil of trajectories, approximately cylindrical and increasing in density toward its axis, is cut by 240 PROBLEMS IN DIRECT FIRE. the plane of the target, nearly normally. Not so, however, when the bullets strike the ground. The ground, regarded as a horizontal target, cuts the pencil of trajectories at a very acute angle, causing the circles of equal probability on the vertical tar|;et to elongate into ellipses whose longer axes are parallel to the plane of fire and whose shorter axes practically remain the same. If ^ is the radius of any circle of equal probability on a vertical target, and oo the mean angle of fall, or the angle with which the axis of the pencil of trajectories strikes the ground, then we shall have for the semi-major axis of the corresponding ellipse on the ground, approximately, a^ R cot 00. Example i6. What- is the probability of hitting an ellipse whose vertical axis is 20 inches and horizontal axis 16 inches, when the mean vertical deviation is lo inches and mean hori- zontal deviation 8 inches ? In this case we have 2a 2b The answer is therefore the same as in the last example. Example 17. A marksman, at the end of the practice season, finds that just one-half of all the shots he has fired at the 200- yard range have struck the bull's-eye. What is his mean deviation at that range ? As one half of a large number of shots fired at the target have hit the bull's-eye, we may fairly assume that the probabil- ity of his hitting the bull's-eye is one-half ; and that the centre of impact and centre of bull's-eye coincide. Let R be the radius of the bull's-eye. Then we have, since Tte P — i= I ~ e "'^ PROBLEM XXI. 241 in which the e in the exponent must not be confounded with the Naperian base. From this equation we get R \fn log, 2 We may find the numerical value of e when R is given by means of Table 4. For example, suppose i? to be 6 inches. Then, since P = 0.5, we have, from Table 4, 2R 12 — — — = 2.951. e e :. e = — — - = 4.06 inches. 2.951 That is, his mean deviation from the centre of the bull's- eye would be very nearly 4 inches. The same result would of course be obtained by working out the above formula. Probability of Hitting a given Object. Supply of Ammunition. — If we wish to open a breach, demolish a bomb- proof, destroy an armored work, etc., we generally know, from ex;periments and calculations previously made and tabulated for use, the number of shots from guns similar to those at our command which must strike the work in order to accomplish the desired result. The very important question then arises. How many shots must h^ fired to secure the necessary number of hits ? The answer to this question determines the amount of ammunition required and the time that must be allowed. Let / be the probability of hitting the given surface, or object, with one shot. For example, if the object were the probable rectangle, the value of/ would be J; if the 25-per-cent rectangle, /would be i; if the Griison turret as in Ex. 12, / would be about o.oi ; and so on. The value of p is deter- mined by experiment, in advance, by methods already given. Now, though we know that the probability of hitting, for example, the probable rectangle, or probable ellipse, is one- half, it by no means follows that one-half the shots will hit this 242 PKOBLEMS IN DIRECT FIRE. surface. The last two columns of the table on page 221 show that for the example there considered the actual number of hits was very approximately equal to pn ; where p is the probabil- ity for each rectangle, and n the whole number of shots fired. Therefore, if «„ is the number of shots which must hit the object to insure the desired results, we may determine roughly the number of shots that must be fired, by the equation P We can find the probabihty that the number of hits will not vary in either direction from n^ =^pn by more than a given number, y, in the following manner : Let k' 2«„(« — «„)' and t = hy; then the required probability is expressed by which may be taken from Table i, with t as the argument.* Example 18. Suppose 50 shots are fired from a gun under similar circumstances. What is the probability that the num- ber of hits in the probable rectangle will not differ from 25 by more than 2 ? We have « = 50,/ = |-, «„ = 25, and _y = 2. Therefore h^ = -^. 50 X 25 .•. h^\ and t = 2k-= 0.4. Therefore, from Table i, we find P = 0.43, the probability required. * Les Projectiles, by Major Jouffret, chapter 7. PROBLEM XXI. 243 Example 19. What is the value oi y in Ex. 18, when P = 0.9? We find from Table i that, for/* = 0.9, t r= 1.1631. There- fore 7 = 1=5.8. There is, therefore, a probability of 0.9, or practical cer- tainty, that out of 50 shots fired from the same gun, 19 at least will strike the probable rectangle. Probability that at least one Shot will hit the Object. — To determine the probability P that at least one shot out of n shots fired shall strike the object, we use the formula P=i-(i-/)", in which, as before,/ is the probability of hitting with a single shot. Solving this with reference to n, we have ^^log^(i^-P)^ log (I -p) which gives the number of shots that must be fired to insure a probability P that the object will be hit at least once. Example 20. What was the probability of hitting the Grijson turret at Bucharest (see Ex. 12) at least once in 100 shots? Here n = 100 and/= o.oi. Therefore /»= I -(0.99)-"° = 0.63, the probability required. There was, therefore, more than an €ven chance of hitting the turret with the shots fired. Example 21. How many shots would have to be fired at the Griison turret to secure a probability of -jSp- that it would be hit at least once ? We have log tVit log 100 - log 99 ^• As but 94 shots were fired at the turret it is not at all sur- prising that it was not hit. Criterion for Rejecting Abnormal Shots. — In nearly all extended series of shots there will be found some which differ 244 PROBLEMS IN DIRECT FIRE. SO much from the others and from the mean as to indicate that there is something abnormal about them — though we may not be able to say exactly what it is ; and we therefore reject them in determining the mean deviation or dimensions of the probable rectangle. In order not to leave this rejection to the arbitrary discre- tion of the computer, we give the following criterion, or rule, for rejecting abnormal shots. It was first given by Chauvenet, and has been taken from his Spherical and Practical Astronomy, Volume II, page 565. We have seen (page 207) that if ^4/2 e Vtt' then F{t) (the values of which are given in Table I) multiplied by n, the number of shots, gives the probable number of deviations less than s, in the direction of either co-ordinate axis ; and hence the quantity n — nF{i)= n [i — F{t)] expresses the number of deviations that may be expected to be greater than the limit s. But if this quantity is less than J, it will follow that a deviation of the magnitude s will have a greater probability against it than for it, and may therefore be rejected. The limit of rejection of a single doubtful shot ac- cording to this simple rule is, therefore, obtained from the equation ^ = n{i-F{t)\, or Having found F{t) from this equation, we take the cor- responding value of {t) from Table i, and then compute the limiting value of s by either of the equations s = Et Vli ^ i.4i42Et, s = et Vn = i.yy2<,et. PROBLEM XXI. 24S If it be found that any deviation {a or 3) is greater than s, the shot producing it should be rejected. Then with the remaining n—\ shots determine new values of E (or e) and s, and proceed as before — rejecting but one shot at a time. Example 22. Determine the limit of rejection of one of the following shots fired at Meppen, February 21, 1882, with a 21- cm. mortar. Weight of charge, 4 kg. ; weight of projectile, 91 kg. ; angle of elevation, 30° ; mean range, 3307.4 m. No. of shot. (Metres.) (Mattes.) 115 188 + 49-6 116 - 138.4 "7 157 - -18.6 118 171 - -32-6 119 176 ^ h37.6 Taking the point of fall of least range as the origin, the values of x are given in the second column, and the correspond- ing values of a in the third column. From these we find We also have whence '?^= 55-36 m. ^, ^ 2n— I t = 1.1631. Therefore the limit of rejection is s — 1.7725 X 1.1631 X 55-36= 114.1 m. The shot numbered 1 16 must therefore be rejected. This reduces the mean error in range to 9 metres, instead of 55.36 m. Probability of the Arithmetical Mean. — We have as- sumed (see page 203) that the values of the co-ordinates of the centre of impact, X^ and F„ , are, respectively, the arithmetical mean of the co-ordinates of the various points of impact ; which is strictly true only at the limit — that is, when the number of 246 PROBLEMS IN DIRECT FIRE. shots is supposed to be infinite. The arithmetical mean, how- ever, gives the most plausible (if not the most probable) values of these co-ordinates, and is taken as the basis of all applica- tions of the calculus of probabilities to the combination of direct measurements made upon a single quantity. It is shown by writers on the method of least squares that the probable error of the arithmetical mean is equal to the prob- able error of a single observation divided by the square root of the number of observations* That is, if r„ is the probable error of the arithmetical mean, we shall have r Vn or, substituting for r its values given on page 212 we have _ , 0-6745^ _ , 0-845 3^ Vn Vn As an illustration of the above, it will be found that, in Ex. I, the probable errors of the co-ordinates X^ and F„ are, respectively, ± IS- '5 yards and ± I-07 yards. We may, there- fore, write these co-ordinates as follows : -^0 =93-11 ± 15.15 yds.; Y^ = 8.67 ± 1 .07 yds. The co-ordinates of the centre of impact in Ex. 2 may be writcen, ^0 = 55-83 ± 4-46 cm. ; F„ = 48.00 ± 3.21 cm. Similarly those of Ex. 3 become X^ = 29.26 ± 1.77 m. ; F„= 3.00 ±0.13 m. •■ Elements of the Method of Least Squares, by Mansfield Merriman, Ph.D., page 147. Problem XXII. To compute a range table. A range table should be so constructed as to afford all the data necessary to enable the gun for which it was prepared to be properly and promptly layed in such a manner that its pro- jectiles may hit a given object whose distance from the gun is known ; and, also, to predict the probable effect of the shots upon the object. In its simplest form it consists of a series of computed trajectories pertaining to several different ranges taken as argument, disposed in regular order for ready refer- ence, so that any range may be readily found in the table, and with it all the elements of the corresponding trajectory. The constants upon which a range table is based relate chiefly to the projectile, and must be known in advance with all the precision possible. These are the calibre and weight of the projectile and the coefficient of reduction, from which are deduced the ballistic coefficient C for a standard density of the air; the muzzle velocity and the jump of the gun. With these data we can compute for a given range, by Prob. XII, the angle of departure, the angle of fall, the striking velocity, and the time of flight. These are the fundamental elements required in a range table ; but it should also give the variations of the angles of departure due to variations of the muzzle velocity and of the density of the air, the drift of rifled projectiles, the danger-space, etc., all of which will be considered in their proper places. Range Column. — The first column of a range table should contain the ranges for which the table is computed ; and which, as already Stated, constitute the argument of the table. The common difference of these ranges should be small enough to avoid unnecessary interpolations in taking out the angles of elevation. It would be better in most cases to extend the 247 248 PROBLEMS IN DIRECT FIRE. table sufficiently to avoid all interpolations ; that is, to make the common difference not greater than the probable error in the estimate of the range. For the longer ranges the differences could be taken greater than for the shorter ranges, which are presumably more accurately known than the former. Angles of Departure. — In the second column of the table are placed the angles of departure corresponding to the ranges on the same line in the first column. In computing these from the formula sin 24> =^ AC, proceed as follows : Write the logarithm of C upon a piece of paper, or, preferably, near the lower edge of a card, and place it successively above the logarithms of A found in the table of logarithms, writing down only the sums. This can easily be done and saves a great deal of labor. Next, find in the proper table the angles corresponding to these logarithmic sines, and write down the values of to the nearest minute, performing the division mentally. It will be necessary to correct the value of C from time to time for altitude, by the method of Prob. XIII. Angles of Elevation. — It must be remembered that the angles of departure in the second column are not the angles of elevation employed in laying the gun, but are generally greater by the angle of jump. This latter can only be determined by experiment, and when found must generally be subtracted from the values of to give the angles of elevation. The jump is positive for all guns so far as is known, with the exception of the Hotchkiss rapid-firing guns, which are said to have a nega- tive jump of about 5^ minutes. For these guns the jump must be added to the angle of departure. If the jump has been accurately determined by experiment, the column of " angles of elevation" may take the place of that of " angles of departure." But it is generally best to retain both columns. Variations of the Angles of Departure or of Elevation. — Following the column of angles of elevation should be PROBLEM XXII. 249 columns of the variations of these angles due to variations of the density of the air or weight of projectile, and of the muzzle velocity. We rriay deduce formulae for these variations as follows, th^ details of which are omitted on account of their great length : Take the variations of equations (10) and (8) upon the supposition that X and Fare constant, and reduce by means of equations (17), (18) and (20), and the values of the differentials of the S and .^4 -functions which are given in Appendix I. The result, all reductions having been made, is the following very simple expression for the variation of sin 20 due to a varia- tion of C: A sin 2^— —{B — A)AC\ or, taking the variation of the first member, A(})=- ^^^-—AC. ^ 2 cos 20 C If we make AC =^ ± — , and multiply the second member by 3438 in order to reduce A^ to minutes of arc, we have finally A A".(> T (sec- onds.) Drift (yards.) " Striking velocity. T (inches.) Maximum ordinate. (feet.) 500 1000 1500 I ig'.4 39-5 02 .4 o'.i 0.3 0.7 I'.O 2 .0 3 -o 0.7 1-5 2-3 0.1 0.3 0.7 0°2l' 42 1 09 iggg ■ 1904 1812 17-3 16.4 15-5 21.6 2000 2500 3000 I I 2 25 .8 52 .0 19 .4 1 -3 2 .1 3 4 -i 5 -3 6.7 3-2 4.0 5.0 1-3 2.1 3-2 1 38 2 II 2 4g 1725 1643 1564 14-7 14.0 13-3 99.6 3500 4000 4500 2 3 3 48 .1 19.4 52 -7 4 -3 6.0 7 -9 8 .2 g.6 II .1 6.0 7.U 8.1 4.4 6.3 8.5 3 31 4 19 5 II 1490 1419 1356 12.6 II. g II. 3 265.5 5000 5500 6000 4 5 5 28 .g 07 .0 47 -9- 10 .1 12 .g l6.3 12 .7 14.4 16 .4 9.2 10.4 II. 7 II. I 14-3 18. 1 6 OQ 7 16 8 29 1294 1238 I187 10.8 10.3 g.8 350.7 442.8 555-6 6500 7000 7500 6 7 8 31.8 19 .6 09 -3 19 .g 23 .g 28 .4 18 .4 20 .4 22 .6 13-0 14.4 15-8 22.4 27. g 34-2 9 48 11 13 12 46 I142 II03 1072 9-4 9.0 8.7 696.7 855-4 1044 8000 8500 gooo 9 9 10 01 .7 59 -0 57 -5 33 .4 38.5 43 -7 25 -I 27 .0 30 .1 17.2 18.7 20.3 41.2 49-3 58.5 14 23 16 05 17 49 1047 , 1027 lOII 8.5 8.3 8.2 1254 1497 1783 In computing the drift column the following data was em- ployed : w = 30 ; /^ = 0.64 ; t = 0.31 ; and g = 32.16. With these data and the given value of V, the drift formula reduces to ( (Bu) - 3.436 ) X D - 0.68759 \ ^ — - — ^-^^ - 0.002S3 cos' 4>. 254 PROBLEMS IN DIRECT FIRE. The formula for the variation of due to a variation of one- tenth of the value of C reduces, for this gun, to ' ^'0 = T ^^ {B - A), cos 20 ^ ' and for the variation of due to a variation of ± 50 f. s. in the muzzle velocity the formula becomes ^ ' cos 20 If Cand Fboth vary at the same time, the total variation of will be the sum of the partial variations, regard being had to the signs of these latter. Finally, the expression for the thickness of armor the shots will penetrate at the various ranges becomes, by Maitland's formula, in inches, r = 0.00918982' — 1. 12, or. if we prefer the French formula, we have, after reduction, r = (0.0036317/)"^'. The value of C was corrected for altitude for all ranges greater than 4000 yards, by the method fully explained in Prob. XIII. Variation of the Angle of Departure due to a Variation of the Range.— This variation is found at a glance from the range table. In the absence of a range table we may compute the variation by the following simple formula, which is easily deduced from the principle of the rigidity of the trajectory : AX A(f)— — 3438 tan w -^-. If the angle of fall is not known approximately, it may be computed by Prob. IX. It can generally be estimated with PROBLEM XX 11. 255 sufficient accuracy for use in the above formula, by comparison with the angle of departure. Example 2. For the 8-in. converted rifle we have, for a range of 2700 yards, 0=5° 17' when the air is normal. With a head-wind of 30 miles per hour the range would be shortened by 50 yards, according to the method of Prob. VIII. How much should the angle of departure be increased in order that the range may be 2700 yards? Here we have^= 2700, AX^ — 50, and (suppose) 00 = 6°. .-. A = ^"^^ ^ ^ tan 6° = 6% minutes of arc. ^ 2700 If we had assumed ao = 6° 30' we should have found Acp = 7'. Variation of the Range due to a Variation of the Muz- zle Velocity. — For this variation we have, for a variation of ± 50 f. s. in the muzzle velocity, X 2 cos 20 tan oa in which A^ is taken from Table A. Since, in direct fire, we have, very nearly, 2 cos 20 tan ca = sin 2gl) = BC, the above expression for AX may be written AX _ A X ~^ B' in which, it must be remembered, A^ is to be taken from Table A. Example 3. In Ex. i, Prob. IX, how much would the range be increased by increasing the muzzle velocity to 1450 f. s.? Here ^=4676 feet ; A^ — .0035 ; and B = .0681. •■•^^=^^;^' -40 tee.. APPENDIX I. DEDUCTION OF THE GENERAL FORMULA OF DIRECT FIRE. Resistance of the Air to the Motion of a Projectile. — A projectile leaves the gun in the direction of the axis of the bore, and with a given muzzle velocity ; and is thenceforward, during its flight, constantly subjected to the action of two forces, which alone determine, in connection with the initial conditions, the curve, or trajectory, which its centre of gravity must describe. These forces are the constant force of gravity, which acts vertically downward, and the variable resistance of the air, which acts in a direction opposite to that of the motion of the projectile at each instant. This last is a tangential, re- tarding force, whose dynamical equivalent is, designating the resistance by p, or, substituting the weight of the projectile for its mass, _ _g dv f'~ wdf It has been proven by many conclusive experiments that the resistance encountered by a projectile at any point of its trajectory is directly proportional to the area exposed to re- sistance, and also to some function of the velocity with which it is moving at the time under consideration. If we assume that the axis of the projectile coincides with the tangent to the trajectory throughout its flight, which is very nearly correct in direct fire, the area exposed to resistance will be the area of the surface of the ogival head. This area, as is easily shown 257 2S8 PROBLEMS IN DIRECT FIRE. by the integral calculus, varies as the square of the diameter of the projectile, that is, with d''. We may, therefore, write the expression for the resistance With regard to /[v) it is impossible with our present knowl- edge to determine its form. It has been shown, however, by Bashforth and other experimenters, that for velocities greater than 1330 f. s. and less than 790 f. s. the resistance varies very nearly as the square of the velocity ; but that, for velocities between these limits, the resistances vary more rapidly than the square of the velocity. Assuming in which A and n are to be determined by experiment, we have, for the expression for p. Ad'' and for the retardation. g dv Ad' A w dt w C Oblong Projectiles, — A discussion of Bashforth's experi- ments made by the author in 1883 gave the following as the most probable values of A and n for oblong projectiles having ogival heads struck with radii of i^ calibres : Velocities greater than 1330 f. s.: « = 2 ; log A = 6.1525284 — 10. 1330 f. s. > z' > 1 120 f. s. : « = 3 ; log ^ = 3.0364351 — 10. 1 1 20 f . s. > z/ > 990 f. s. : n = 6; log ^ = 3.8865079 — 20. 990 f . s. > z/ > 790 f . s. : « = 3 ; logA = 2.8754872 — APPENDIX I. 259 10. 790 f. s. > z" > 100 f. s. : n — 2; log ^ = 5.7703827 — 10. In applying these expressions for computing the resistance and retardation a projectile suffers, we must take d in inches, w in pounds and v in feet per second. The value of g is 32.16 f.s. Example i. Compute the resistance and retardation for the service projectile used with the 8-in. converted M. L. rifle, when it is moving with a velocity of 1350 f. s. Here d= 8 in.,w = 183 lbs., c= i, « =: 2andF= 1350 f.s. For the resistance we have log A =6.15253 2 log d — 1. 8061 8 2 log V = 6.26066 a. c. log g = 8.49268 log /3 = 2.71205 .•.p= 515.3 lbs. g For the retardation = — p we have w logp = 2.71205 log^= 1.56732 a.c. logw=: 7-73755 log Ret. = 1.95692 .-.Ret. =90.56 f. s. Example 2. Compute the resistance suffered by an 8-in. service projectile fired from the new navy rifle, when its ve- locity is 2000 f. s. Here a? = 8 in., w = 250 lbs., c = 0.9, « = 2 and v = 2000 f. s. In this example we must write the expression for p as follows : Acd' p — z/". 26o PROBLEMS IN DIRECT FIRE. Performing the calculations as in Ex. i, we find p = 1 131 lbs.; Retardation = 145-5 ^^^• That is, the resistance of 1131 lbs. would, if it remained con- stant for one second, diminish the velocity of the projectile by 145.5 f-s. Terminal Velocity. — Let Vt be the velocity of a projectile when the resistance is equal to its weight. We must have in this case V, Ad'' P= g whence ""' - \Ad'] • The velocity Vt is called terminal velocity, because it is easily seen to be the velocity toward which a body, faUing in a resist- ing medium like the air, continually approaches, and only reaches at infinity. Example 3. What is the terminal velocity of the projectile of Ex. 2, supposing it to fall point downward ? It will be found by trial that the terminal velocity lies between 1 120 and 990 f. s. ; and, therefore, n=6. Therefore log^= 1.50732 log w — 2.39794 a. c. log d"" — 8.19382 a. c. log /i = 16.1 1349 6)18.21257 log Vi = 3-03543 ••• ■Vt = 1085 f-s. Similarly, it may be shown that the terminal velocity of the projectile of Ex. i is 1030 f. s. Spherical Projectiles. — General Mayevski, from a discus- sion of his own experiments with spherical projectiles made at APPENDIX I. 261 St. Petersburg in 1868, deduced values for A and n which, reduced to English units, are as follows : Velocities greater than 1233 f. s. : Ad'' p = v' ; log ^ = 6.3088473 — 10. Velocities less than 1233 f. s. : Ad' I v'\ * p = ^'"^i + -j ; log ^ = 5.6029333 — 10 ; r = 610.25. Example 4. Required the diameter and weight of a solid, spherical, cast-iron shot whose terminal velocity in air shall be 1233 f. s. We have whence Ad'v' p=w=-^, w Av'' Let dj be the diameter of a similar shot whose weight (ze/,) is known. Then d^_d^ w w^ whence, by division, Ad^v' d: gw. The solid cast-iron shot for the 15-in. S. B. gun is 14.87 in. in diameter and weighs 450 lbs. Therefore, substituting in the above equation, we find d = yo inches. We also have d'w, w — —jj- — 59900 lbs. We see from this example that the terminal velocities of all service spherical projectiles are less than 1233 f. s. 262- PROBLEMS IN DIRECT FIRE. Solving the equation g for V, we obtain w .=vW I+-Hr7— i), 4«y * by means of which the terminal velocities of service solid shot may be computed. Differential Equations of Motion. — We will assume that the projectile, if spherical, has no motion of rotation ; and, in addition to this, in the case of oblong projectiles, that the axis of the projectile lies constantly in the tangent to the trajectory • also, that the air through which it moves is still and of uniform density. As none of these conditions is ever exactly fulfilled in practice, the equations deduced will only give what may be called the normal trajectory, or the trajectory in the plane of fire, and from which the actual trajectory will deviate more or less. It is evident, however, that this deviation from the plane of fire is relatively small; that is, small in comparison with the whole extent of the trajectory, owing to the very great density of the projectile as compared with that of the air. Let the muzzle of the gun be taken as the origin of rectangular co-ordinates, of which let the axis of X be hori- zontal and that of Y vertical. The retardation at any point of the trajectory whose co-ordinates are x and y, and at which the inclination of the tangent to the horizon is 6, in the direc- tion of the tangent, due to the resistance of the air, is, as we w have seen, - p;. and the corresponding retardation due to the action of gravity is ^sin ^. Therefore, the total retardation in the direction of motion is expressed by the equation dv g dt w " APPENDIX I. 263 The first term of the second member of this last equation is always negative, since the resistance of the air tends to reduce the velocity. The second term is negative in the ascending branch ; but in the descending branch sin d changes its sign and the term becomes positive. The reasons are evident. The velocities parallel to X and Y are, respectively, v cos and z' sin d\ and the accelerations parallel to the same axes are g g — p cos 6 and g-\-— p sin d. Therefore, d{v cos 8) g dt = — - p cos d, (i') w and d(v sin 6*) ^ . „ , ,. -5^-3 ^=-p-— -psn^ (2') di ^ w*^ ^ ' Performing the differentiations indicated in these equations, then multiplying the first by sin Q and the second by cos 6, and taking the difference of the products, gives "vdB „ , ,. -dF^-S^'^^O (30 Since the resistance of the air does not enter into this last equation, it must be an expression for the forces resolved nor- mally to the trajectory, that is, at right angles to the direction of the resistance ; as may otherwise be easily shown. Designate the horizontal velocity by v^ ; that is, let z/, = J/ cos 6. Introducing this into (i') and (3'), they become '^l^-^pzosO, (4') dt w' ^^ ' and '^=-^-'»: w 264 PROBLEMS IN DIRECT FIRE. whence, eliminating dt. dd wcos e ^ ' Eq. (4') gives — p cos ff w but, as we have already seen, page 258, w^~~C' ~ C cos" 6*' whence, by substitution, C _ . ..dv, ^i?=-^cos— »— • (7') The relations between the element of time and the elements of the trajectory at any point are given by the fundamental equations of mechanics, dx = v,dt. ; dy = v^ tan ddi ; and ds = v^ sec Odd {s being the length of ihe trajectorj' from the origin). Substituting for dt in these equations its value from (7'), they become ^^=-^cos-^^,; (80 C dv, dy = - ^sm e cos"-^ e ^^■, (9') C „ dv, '^'=~A '=°'"" ^ <^^ ('°') We may also make d the independent variable as follows : From (5') we have V, dd ^, . „ dt= ! — —„ = -dt?ine; .... (n') ^cos g ^ ' APPENDIX I. 265 ■whence, substituting as before, dx=z ^^tan^; (12') S v'' dy— ^ tan 6*^ tan ^ ; . . . . (13') ds= ^sec6'^tan6' (14') ^. The horizontal velocity may be eliminated from these last four equations in the following manner: We have, by hy- pothesis, p _ Av^ _ Av^ w ~ gC gC cos" B which, substituted in (6'), gives ' dd gC dv^ cos"+' d A w "+"' (ISO Both members of (15') can be integrated in finite terms when n is any whole number. Symbolizing the integral of the first member by (&), that is, making (^) ~ J cos"+' e ' and integrating between the limits and 6, to which cor- respond in the second member F, and «/, , we have Let i be the value of when F, is infinite ; or, what is the same thing, let 266 PROBLEMS IN DIRECT I^IRE. then we have Making, for simplicity, " - nA' and solving for ■v^, we have k ^. = • — ; (17') Substituting this value of v^ in (u'), (12'), (13'), and (14'), they become, respectively, k dta.n 9 dt=-- r; (18') ^■^=-7 ?; (19) k" tan Od tan ^7 = - T ; (20') /&" sec ed tan (9 , , ''^•f = -7 T (21') ^l(0-(^)l« Either of the three groups of equations which have been deduced above may be said to contain the whole theory of the motion of a projectile in the plane of fire when subjected to the resistance of a medium whose action can be expressed as a function of the velocity. But, unfortunately, the laws of resistance which admit of the complete integration of these differential equations are very few, and do not include any which are found to belong to our atmosphere — at least, not for the high velocities employed in gunnery. APPENDIX I. 267 Cases which Admit of Integration in Finite Terms. — The following are all the cases which can be solved in finite terms ; that is, iri terms of known functions : 1. As has already been shown, the velocity can be deter- mined in terms of 6 when n is any integer. 2. When the resistance is supposed to be zero, that is, in vacuo, all the equations of motion can be integrated, and 6 can be eliminated between the expressions for x and y, as will be shown in the next section. 3. When the resistance is considered constant for all veloci- ties we can deduce expressions for v, t, x, y, and .t in terms of e. 4. When M = I, all the elements except j maybe expressed in terms of 6, and also of x. The equation to the trajectory in this case is ^=(A + tan0);. + ilog.(i-|^). 5. The expression for ds can be integrated when the resist- ance varies as the square of the velocity, as will be shown further on. 6. Professor Greenhill has recently succeeded in deducing expressions for the elements of a trajectory in terms of elliptic functions when « = 3. Motion in Vacuo. — Making p = o, Eq. (4) becomes dv^ = o ; and, therefore, v^ , or the horizontal velocity, is in this case con- stant and equal to its initial value K, . Therefore, in vacuo, V cos 6 = Fcos (p. Integrating (11'), (12') and (13') between the limits and d gives, if ^^1 = F, , V t = -^(tan — tan 61) ; .... (22') 268 PROBLEMS IN DIRECT FIRE. X = —^ (tan — tan ^) ; .... (23') J/ = -^(tan> - tan' ^) (24') Eq. (14') becomes, when z', = F,, ^ COS 61 Making ——, = i tan 61 sec <9 + i loge (tan (9 + sec ^, cos 7 we have s = ^{{)-{e)) (250 Eliminating tan 6 from (23') and (24') by division and addi- tion, we have y~xtB.n4> — jyT^, (26') which is the equation of a parabola whose axis is vertical. A parabola is, therefore, the curve a projectile would describe in vacuo. Since a parabola is symmetrical with respect to its axis, the descending branch of the trajectory in vacuo is similar in every respect to the ascending branch, and the angle of fall is equal to the angle of projection, but with the opposite sign ; and, generally, for the same value of y in the two branches, tan d has the same value numerically, but with contrary signs ; being positive in the ascending branch, negative in the descending branch, and zero at the summit. For the whole range we evidently have x — X, and ^ = — 0. Making these substitutions in (23'), we have, for the* range, 2 F' F' sin 20!. X = — - tan = -. g g APPENDIX I. 269 Subtracting (23') from this last equation and reducing gives X— x = — (tan

) A {v^ sec 0)""^ ' APPENDIX I. 271 Integrating between the limits

dx\ - ^-dx = -^. Integrating, and making x and y both zero at the origin, where u ^ V, we have 2cos'0, ^. I{V) I r I(u)du Symbolizing the second member of this equation by A{u) {altitude-function), and observing the limits, we have ^^^ {y-xtB.n and . v^ = u^ cos 0, We have sin 20' V/ -i /: /(«„) = ^; -\- 7(2 100) = O.IOI26, which gives u^ = 1320.0 ; .-. v„ = 1320 cos 10° = 1300.0 f. s., an error of only 3.4 f. s. Example 2. Given F= 790 f . s. ; 0= 10° and C ■= i. Compute v^ by both the preceding methods. We must first compute the final velocity and angle of fall by the second method of Prob. X. We have, first, /(k„) — sin 20° + /(790) = 0.85988. Next, A(u^ — 1856.71 5(a„) — 1 1258.8 = 0.85988 ; 278 PROBLEMS IN DIRECT FIRE. from which we find, by trial, «„ = 577-23. Then, /(«..) - /K) . tan CO = ^ — r— ; — ; 2 cos .-. CO = 12° 15' 20". Lastly, cos „ , z/„ = « — — = S0I.7 f. s. cos CO ■' ' By the analytically exact method we have /(z^,) = 2 1(10°) + (12° IS' 20")} + /(790 cos 10°) = 1.32736 ; ••• ■2'> = 569-3 f- S-, and v^ = z', sec 12° 15' 20" = 582.5 f. s. We may also compare the velocities at the summit in this example, by both methods, with very little additional labor. We have found in the first case /(«„) = 0.85988, which gives u„ = 669.78. Therefore v^ — 669.78 cos 10° = 659.6 f. s. In the second case we have I{v^) = 2(10°) + /(790 cos 10°) = 0.89952 ; -•- V, = 659.1. Some writers on ballistics ignore the relation cos v^ — u ^, ° cos and assume that in all cases. A careful consideration of the above discussion would seem to show that this cannot be done safely except in cases where the ratio cos

- i°g ^1} A = C{S{v,)-S{V:)] (30') This equation gives the length of arc described by the pro- jectile from the origin, when the resistance varies as the square of the velocity, and is analytically exact. Applying numbers already known, we have .o»V = JrV- l-^(i32«.5) - •5(1857-35)1 = 13277-2 feet. APPENDIX I. 281 This value of s differs from that computed by quadratures by less than one foot. We may therefore assume the correct- ness of the above computed values of t, x, and y, so far as the formulce are concerned, and that is all we are interested in at present. '■ We will now compute t, x, and y for the arc (10° — 0°) by the approxiinate formulae of direct fire. We have for v, ^K) = tV% sin 20° + /(1 886) = 0.09633 ; ••• «o = 13444 ; .-. v^ = 1344.4 cos 10° = 1324 f. s. For X, we have X = JJ/ { 5(1 344.4) - 5(1 886) 1 = 13236 feet. To compute j/„ we employ the formula given in Prob. XV, page 113: ■^ = 2 cos' ^^^"'^^ + "^(^) ~ ^(''"^^ = '^97 feet. For t. we have 100 ^" = 1 8 cos 10° ^ ^('3444) - nim)\ = 8".4788. From the above calculations we may infer that, for high velocities, the general formulae of direct fire are practically cor- rect for angles of departure up to 10°, the errors amounting to less than one third of one per cent. To illustrate the accuracy of the formulae when the law of resistance is that of the cube of the velocity, we will continue this example from 6 = 0°, where the velocity is 1322.5 f. s., to ^ = —9°, and velocity = 1127.56 f. s. 282 PROBLEMS IN DIRECT FIRE. Without going into details, the results are as follows : By Quadratures. By the General Formulae. 0^-9° 0^-9° 7/-,° 5".9424 7187,6 ft. - 536-77 ft. 1127.56 f. s. S".9999 7191.1 ft. - 536.6 ft. 1128.9 f. s. For the arc extending from ^ = — 9° and velocity = 1 127.56 f. s., to 6' = — 13° and velocity = 1081.55 f. s., the resistance varies as the sixth power of the velocity. The values of the elements for this arc computed by quadratures and by the approximate formulae are as follows : By Quadratures. By the General Formulse. -9°^- 13° -9°^ -13° -9»J^-.3« ^-13° 2". 4390 2640.1 ft. — 512.09 ft. 1081.55 f. s. 2".44I2 2645.6 ft. -513-17 ft. 1083.8 f. 0. For the whole arc, extending from 0= 10° to ^= — 13°, we have by quadratures, taking the sum of the several separate results, the following values, to which are added the same ele- ments computed at one operation by the approximate formulae : By Quadratures. By the General Formulae. 10° t - 13° 10° ^ - 13" .o°J-i3° ^-13° 16". 84 23026 ft. 243 ft. 1082 f. S. 16". 87 23067 ft. 248 ft. 1083 f. s. The complete horizontal range, time of flight, etc., in this example, computed by the first method of Prob. IX, are as follows : X= 24057 feet; T= I7".8i; 00=^ 14° 35'; v„ — 1072 f. s. ; APPEJ^DIX I. 283 which may be regarded as very close approximations from an analytical point of view. Effect of the Wind upon the Range and Striking Veloc- ity of a Projectile fired with a Small Angle of Elevation.— In the following discussion it will be assumed that the trajec- tory is so slightly inclined to the horizon that the horizontal velocity of the projectile is practically the same as its real velocity, and also that the motion of the wind is horizontal and uniform during the flight of the projectile. The method is therefore only applicable to the flatter trajectories of direct fire, say for angles of projection not exceeding about 8°. We shall also assume that the effect of a wind blowing parallel to the range is simply to increase or diminish the resist- ance the projectile encounters. That is, if a projectile is mov- ing nearly horizontally with a velocity v, the resistance of the air, if there is no wind, is proportional to v^ ; but if the air has a horizontal motion Wp parallel to the plane of fire, then the resistance will be proportional to (z* -\- W^)" or {v — W^)", ac- cording to the direction of Wp. Upon this hypothesis the expression for the retardation (page 258) becomes dv A , „,. whence, considering the motion horizontal, we have C dv '^^^ ~A{v± ivjy-' and therefore, since dx = vdt, C vdv dx ^ — —. A{v± w^y The integration of the first equation between the limits V and V gives ^^ ^ ( {n - i)A{v ± W,)"-' ~ {^i^^)A{V± IV^)"-' ) ' 284 PROBLEMS IN DIRECT FIRE. or T=c\T{:u± w;) -T{v± w;)], as in Prob. VI. The expression for dx giveh on the preceding page may- be written "^■^ ~ A {v± Wff = _ ^ i ^'" ^t^'" \ ~~ Ai(v± W;)'-' "^ {v ± Wff ) C dv Integrating between the limits V and v, to which corre- spond X = and X ^= JC, and also ^ = o and ^ = T, we have ^=^1 (« _ 2)A{v ± Wff-' " (n-2)A{V± Wf)"-' \ "^ ^*^' or, finally, X=C{S{v± W;)-S{V± Wf)] T WfT, as in Prob. VII. APPENDIX 11. FORMULA FOR MORTAR FIRING. Euler's Method. — For practical muzzle velocities less than 800 f. s., the law of resistance is very nearly that of the square of the velocity.^ In this case « = 2, and (21') becomes _ C sec" Bde ; '^'--2A(i)-(ey but, by definition, when « = 2, {6) =/sec' edd = i tan ^ sec (9 + J log, (tan e + sec G) ; and, therefore, C d{ff) ds = 2A{i)-{0)- In this equation {i) is constant for a given trajectory, and is defined by either of the equations, (.) = C~ + {4>) = C^+ {d) = C^„ in which ^ ^^- (^^^ P^^^ ^^5-) Integrating the above expression for ds between the limits

6"), we shall have from (31') g ''ii)-W 288 PROBLEMS IN DIRECT FIRE. and g ^Kt)- (0) whence, subtracting the first equation from the second, mem- ber by member, thereby eliminating the factor [(/) — (0)], we have an equation which gives the length of any portion of a trajectory in terms of the inclinations of its extremities. If 6" differs but little from B' (say one degree), As will be very nearly a straight line having a mean inclination of \(d' + B"), and will be the hypothenuse of a right-angled tri- angle whose base is Ax and altitude Ay. We therefore have i» (i\ — (f)"'\ Ax = CM- log \l^_\ff\ cos W + 6"), k' li) — (6") Ay = CM- log '(^yir^ sin W + B"). Making and we have ^^ = l°g %^^ ^""^ ^^' + ^")' ^^ = i°g|pr{^si"*(«' + n Ax= CM- AS; g Ay= CM-AZ. g APPENDIX II. 289 For the horizontal range we evidently have X = -2 Ax = cm-:eah =CM-S, S g the summation extending from ^ = to ^ = co, or the angle of fall. To determine the maximum ordinate, the summation 2C is taken from 6* = to 6* = d, and is continued in the descend- ing branch to w (the angle of fall), determined by the condition :sAt: = o, since the sum of the negative increments of j/ in the descend- ing branch is equal numerically to the sum of the positive in- crements in the ascending branch. We therefore have Example 2. Compute the values of Ax and Ay with the data of Ex. I, for the arc comprised between 6' = 25° and d" = 24°. We have from Ex. i, (?) = 11.50253; and from Table IV, ((?') = 0.48269 and (6^") = 0.45953. We also have i{e' + 6") =: 24° 30' ; whence 11.50253—0.45953 - , AS = log — - — -^— — ^-^ X cos 24° 30' ^11.50253—0.48269 ^ •^ = 0.00082970 ; 11.50253 — 0.45953 . o f AZ = log — ^-^- ~^- X sm 24° 30' ^ ^ 11.50253 —0.48269 ^ "^ = 0.00037812. 290 PROBLEMS IN DIRECT FIRE. k' Multiplying CM — by these numbers, we have Ax — 86.54 feet ; Ay = 39.44 " also As = 95.10 " Expression for the Time. — We have for the time of describing any small portion of the trajectory, the expression Ax At= — , in which v^ is the mean horizontal velocity corresponding to Ax ; but from (32') we have VCk whence ^ AxV(i)-{e) _ VCk or, substituting for Ax its value already given, ,- Mk , At=VC AS V{i) - {ff). If we put A@ = AS V{i) - {0), the expression for At becomes At = VC A&. g We may compute A@ with great accuracy as follows ; Taking logarithms, we have log J© = log ^5 + i log [(0 - {9)\ APPENDIX II. 291 The two values of log [(«') — (^)] corresponding to the ex- tremities of As, are log \_{i) — ((?')] and log [(?') — {e")\ the first of which is too small, and the second too great ; whence, taking their arithmetical mean, log J0 = log ^^ + i log [(0 - (^0] + i log [(0 - {e")\ by means of which © can be computed. We then have g the summation extending the same as in determining the range. k The logarithm of M- is 1.62293. Tables. — General Otto of the Prussian Artillery published several years ago extensive tables of ^, C, and ©*, for values of beginning at 30° and continuing by intervals of S" up to 75". The argument for each of these tables is i, which must be first computed from the given data by the equation. (0 = Cy,- + (0), and then entering the proper table with this value of i, the values of ^, C, and are immediately found, and from which X, y^ , and T can be easily computed by the formulae already given. We give the following example illustrating this method. The tables referred to are those in Otto's work. Example 3. Compute the range, time of flight, angle of fall, and maximum ordinate of the trajectory described by the shot of Ex. I. We found (?) in Ex. i to be 11.50253; and therefore (Table i) 1 = 77° 29'. 18. Next, from Table 2, for = 6o°, and with the argument i — 77° 29'. 18, we find $ = 0.1402 and * " Taffeln fiir den Bombenwurf," Translated into French by Rieffel with ibe title " Tables Balistiques Genfirales pour le Tir £lev6." Paris, 1844. 293 PROBLEMS IN DIRECT FIRE. © = 0.4750. From Table 3, with the same arguments, we get oj = 63° 17' ; and from Table 4, Co = 0.0652. log 5 = 9.14675 log iI/-- = 4-39131 log C = 0.62697 log ^= 4.16503 .•. X= 14623 feet. log = 9.67669 k log M- = 1.62293 i log C = 0.31348 log T= 1.61310 .-. T= 4i".03. log Co = 8.81425 log CM— — 5.01828 log fo = 3-83253 ••• 7. = 6800 feet. Modification of Euler's Method and Otto's Tables. — Instead of taking t for the argument, we may shorten the cal- V culations and greatly abridge the tables by taking — rz for the V C argument. The new tables will be constructed as follows : Let then the first expression for (?) on page 285 becomes, by a sHght reduction, by means of which i can be computed for given values of ,\ u^avcosO. ) (38 ) The values of the functions (0) and {ff) are given in Table IV at the end of this work. The same table also gives the natural tangents to be used in connection with the functions. We may also deduce an approximate value for a by the following considerations : Since ds sec a = -^ , dx that is, the ratio of an element of the trajectory to its horizon- tal projection, it is evident that a suitable value for a would be s ds which is manifestly a mean of all the values of — between the dx given limits. As, however, this ratio cannot be found, General Didion, in his classic work Traits de Balistique, assumed for a APPENDIX IT. 303 the ratio of a parabolic arc (or trajectory in vacuo), whose extremities have the same inclinations as the arc of the trajec- tory in question, to its horizontal projection. To determine this ratio we have only to divide (25') by (23'), which gives X tan — tan d ' which is the same expression for a as before determined by other considerations. We have shown that the adopted value for a gives the true velocity of the projectile at any point of the trajectory ; and, also, the exact length of the curve described from the origin, when the law of resistance is that of the square of the velocity ; and we have thence assumed that it will also give sufficiently accurate values for the horizontal distance passed over, and for the time of flight, for the same law of resistance. To verify this assumption, and also to show the degree of approximation attained when the law of resistance is that of the cube or of the sixth power of the velocity, we must have recourse to quadratures for calculating the exact values of the definite integrals involved. .Example 10. — Let = 30°, B ■= 24°, and C = -ffj. Com- pute V, t, x,y, and s when F= 1886 f. s. by quadratures, and also by means of a. The following are the results : By Quadratures. By means of a. Difference. V t X y s 1400.4 f. S. 5". 888 8481.4 ft. 4381.9 ft. 9550.6 ft. 1400.4 f. s. 5"-894 8499.9 ft- 4392-7 ft- 9550.6 ft. ■ — o''.oo6 — 18.5 ft. — 10.8 ft. Of these elements v and s are, of course, the same by both methods of calculation ; while t, x, and y are too great when 304 PROBLEMS IN DIRECT FIRE. computed by means of a. The error is, however, less than one-fourth of one per cent. Let F= 1330 f. s., and B remaining as before. p a »'. By Quadratures. Using a. Difference. V mo f. s. 4761.6 ft. mo f. s. 4773-7 — 12. 1 ft. Let V = 1 120 f. s., cf) and 6 the same as before. p a v'. By Quadratures. Using a. Difference. V X 991.2 f. s. 3583.1 ft. 991.9 3558.4 ft. + 0.7 + 24-7 ft. It will be seen from the above that the values of x given by a are slightly too great when « = 3, and too small when « = 6 ; while the velocity is practically correct in both cases. We will now compute a complete trajectory and compare our results with actual experiment. In applying Didion's value of a to the computation of a complete trajectory at one operation, having given the muzzle velocity and angle of departure, we should compute a by the equation a = tan -j- tan 00' CO being the angle of fall, and considered positive. As the angle of fall is not known, we will suppose it equal to the angle of departure ; whence we have (0) tan Example 11. Compute the trajectory of the Jubilee^hots fired at Shoeburyness in April, 1888. APPENDIX II. 305 The data for this shot, as communicated to the author by Prof. A. G. GreenhiU, March 9, 1888, are as follows: «/=38olbs., ^=9.15 in., 0=40°, and F=236of. s. The calculations, which were made before the shots were fired, are as follows : From Table IV, we find (0) = 0.92914 log (0) — 9.96808 log tan =: 9.92381 log a = 0.04427 log F= 3-37291 log cos = 9.88425 log U— 3.30143 . .•. U = 2001.86 We will next compute the ballistic coefficient for the level of the sea, which we will designate by C, to be corrected later for altitude. The coefficient of reduction {c) we will take equal to 0.9 14, which is its theoretical value. log w = 2.57978 a. c. log d^ = 8.07716 a. c. log c — 0.03905 log C = 0.69599 The next step is to compute the height of summit in order to get a correction for the ballistic coefficient due to the de- crease of density of the air. From (31') we obtain, when 6* = o, /W= '-^ + nU) (37') Also, eliminating tan from (35') and (37'), and making z,^S{K)-.S{U), .306 PROBLEMS IN DIRECT FIRE. we have at the summit y, = ^\l{uy,^A{U)-A{u:)\. . . (380 log 2 = 0.30103 log tan = 9.92381 a. c. log a =9.95573 a. c. log C — 9.30401 log 0.30520 = 9.48458 7(f7) T= 0.02761 /(2^„) = 0.33281 .-. w„= 898.15 S{u,)= 9228.5 S{U) = 2361.7 log 6866.8 = 3.83675 log /(«„) = 9.52220 log 2285.32 = 3-35895 A{U) = 28.98 — A{u) = — 1001.17 log 1313-13 = 3-11831 2 log C= 1.39198 a. c. log 2 = 9.69897 logj/„ = 4.20926 .-. j/„ = 16191 ft. Following the rule given on page 88, we find 2 ^ = - X 16191 = 10794 ft. As this is beyond the limit of the table on page 88, it will be necessary to compute the altitude factor, or rather its' APPENDIX II. 307 logarithm, which can easily be done as follows : Designating it hy f, we have h 9 •■• log (log/) = log (log e) — log A. +log h. The value of log (log e) — log \ is 5.19374 — 10 ; and there- fore log (log/) = log k — 4.80626. In our example we have log -^ = 4.03318 const, log = 4.80626 log (log/) = 9.22692 •'■ log/— 0.16862 log C = 0.69599 log C = 0.86461 , 2 tan d) log =0.18057 log 0.20700 = 9.31596 I{U) = 0.02761 /(«,) = 0.23461 .-.?<„ = 988.1 We have from (35') when_y = o, A{u:)-A {U) _ 2 tan _ .s(^.„) - s{U) - -^^ + A^) - /(«.). We therefore have the equation A{u^ - 28.98 S(u^) — 2361.7 = 0.23461, from which to find m„ by trial, as explained in the second method, Prob. IX. We find by a few trials that u^ = 753-8. 308 PROBLEMS IN DIRECT FIRE. We are now prepared to compute the range as follows : 5(««,) = 12054.7 S(U)^ 2361.7 log 9693.0 = 3.98646 log C = 0.86461 a. c. log a — 9.95573 log X = 4.80680 .-. X = 64091 feet = 21364 yards. The ranges of the two shots fired April 15, 1888, were 21048 yards and 21358 yards, respectively. « Mean observed- range, 2 1 203 yards. Computed range, 21364 " Difference, 161 " The time of flight is computed as follows : T{u) = 10.009 T{U)= 1.002 log 9.007 = 0.95458 log C = 0.86461 log T= 1.81919 .•. 7"= 65.9 seconds, which is very nearly correct. TABLES. TABLE I. Jitillldio Table/or Ogiral-Headed Projectiles. u «(«) Diff A{u) Difi 700 Difl T{«) Diff E{u) Diff M{u) Diff 2800 000.0 1268 0.00 7 0.0000c 106 0.000 46 0.000 139 0.00107 6 2750 126.8 1292 0.07 21 0.00106 112 0.046 47 0.139 150 0.00113 6 2700 256.0 i3'5 0.28 36 0.00218 II? 0.093 49 0.289 161 0.00 1 19 7 2650 3S7-5 1341 0.64 54 0.60336 125 0.142 51 0.450 174 0.00 1 26 7 2600 521.6 '367 1.18 7' 0.00461 133 0.193 53 0.624 188 0.00133 8 2550 658.3 •393 1.89 93 0.00594 140 0.246 56 0.812 203 0.00141 9 2500 797.6 1422 2.82 "5 0-00734 149 0.302 57 [.015 220 0.00150 10 '2450 939-8 1452 3-97 140 0.00883 160 3-359 60 '-235 239 0.CO160 10 2400 1085.0 1481 5-37 166 0.01043 169 0.419 62 1.474 260 0.00170 ir 2350 1 233- 1 1514 7-03 '97 0.01212 180 0.481 65 '•734 283 0.00181 13 2300 1384-5 '547 9.00 331 0.01392 192 0.546 68 2.017 308 0.00193 13 2250 '539-2 1582 11.31 266 0.01584 205 0.614 72 2-325 337 0.00206 14 2200 1697.4 321 13-97 58 0.01789 43 0.686 '4 2.662 7' 0.00220 3 2lgo 1729.5 322 14-55 60 0.01832 44 0.700 15 2-733 72 0.00223 3 2l8o 1761.7 323 '5-'5 62 0.01876 44 0.715 15 2.805 74 0.00226 4 2170 1794.0 325 ^i-77 6i 0.01920 44 0.730 15 2.879 75 0.00230 3 2160 1 826.-5 327 16.40 65 0.01964 46 0-745 15 2-954 77 0.00233 3 2150 1859.2 328 17.05 67 0.02010 46 0.760 'S 3-031 78 0.00236 3 2140 1892.0 329 17.72 68 0.02056 46 0.775 16 3.109 79 0.00239 4 2130 1924.9 331 18.40 70 0.02102 47 0.791 15 3-18S 81 0.00243 3 2120 1958.0 333 19.10 73 0.02149 48 0.806 16 3-269 83 0.00246 4 2II0 '991-3 335 19-83 74 0.02197 49 0.822 16 3-352 84 0.00250 3 2100 2024.8 336 20.57 76 0.02246 49 0.838 16 3-436 86 0.00253 4 2090 2058.4 .^37 21.33 79 0.02295 50 0.854 16 3-522 87 0.00257 4 2080 2092.1 339 22.12 80 0.02345 5' 0.870 le 3.609 89 0.00261 4 2070 2126.0 341 22.92 82 0.02396 5' 0.886 17 3.698 91 0.00265 3 2060 2 160. 1 343 23-74 85 0.02447 52 0.903 17 3-789 93 0.00268 4 2050 2194.4 344 24.59 87 0.02499 53 0.920 17 3.8S2 94 0.00272 4 2040 ?228.8 346 25.46 89 0.02552 54 0.937 '7 3-976 96 0.00276 4 2030 2263.4 348 26-35 9" 0.02606 54 0.954 •7 4-072 99 0.00280 5 TABLE I. —Continued. u S(n) Di£f A{n) Difl I{u) Difl r(«) Diff B{u) Diff M{n) Ditf 2020 2OI0 2000 2298.2 2333-1 2368.2 349 351 353 27.26 28.20 29.16 94 96 98 0.02660 0.027 ' 5 0.02772 55 57 57 0.971 0.988 1.005 17 17 18 4-171 4-271 4-373 100 102 104 0.00285 0.00289 0.00293 4 4 5 1990 1980 .1970 2403.5 2439.0 2474.6 355 356 358 30- '4 31-15 32.19 lOI 104 107 0.02829 0.02886 0.02945 57 59 6c 1.023 1.041 1.059 18 18 4-477 4.584 4.692 107 108 III 0.00298 0.00302 0.00307 4 S 5 i960 1950 1940 2510.4 2546.4 2582.6 360 362 363 33-26 34-35 35-48 109 113 "5 0.03005 0.03066 0.03127 61 61 62 1.077 1.096 [.114 19 18 19 4.803 4.916 5-031 113 115 118 0.003 1 2 0.00316 0.00321 4 S 5 1930 1920 1910 2618.9 2655.5 2692.2 366 367 370 36-63 37-81 39.02 118 121 124 0.03 1 89 0.03^53 0.03318 64 65 65 1-133 ..152 1. 171 19 19 20 5.149 5.269 5-392 120 123 126 0.00326 0.00331 0.00337 5 6 5 I goo 1890 1880 2729.2 2766.3 2803.7 37> 374 375 40.26 41-53 42.83 127 130 133 0.03383 0.03450 0.03517 67 67 69 1. 191 1.210 1.230 19 20 20 5-518 5.646 5.776 128 130 134 0.00342 0.00348 0.00353 6 5 6 1870 i860 1850 2841.2 2878.9 2916.9 377 380 382 44,16 45-53 46-93 137 140 143 0.03586 003656 0.03727 70 7> 72 1.250 1.270 1.291 20 21 20 5.910 6.046 6.1 86 136 140 142 0.00359 0.00365 0.0037 1 6 6 6 1840 1830 1820 2955.1 2993-4 3032.0 383 386 388 48.36 49-83 51-34 147 •51 155 0.03799 0,03872 0.03946 73 74 76 1.311 1-332 1-353 21 21 22 6.328 6.474 6.623 146 149 152 0.00377 3.00383 0.00389 6 6 7 1810 1800 1790 3070.8 3109.8 3149-0 390 392 394 52.89 54-47 56.09 158 162 167 0.04022 0.04099 0.04177 77 78 80 I-37S 1.396 1.418 21 22 22 6.775 6.931 7.090 156 162 0.00396 0.00402 0.00409 6 7 7 17S0 T770 1760 318S.4 3228.0 3267.9 396 399 401 57.76 59-47 61.21 171 174 179 0.04257 0.04338 0.04420 81 82 84 1.440 1.463 1.485 23 22 23 7.252 7-419 7.589 167 170 174 0.00416 0.00423 0.00430 7 7 8 1750 1740 1730 3308.0 3348-3 3388.9 403 406 409 63.00 64.83 66.71 183 188 193 0.04504 0.04585 0.04676 85 87 88 1.50S 1-531 '-5S5 23 24 23 7-763 7.941 8.123 178 182 187 0.00438 0.00445 0.00453 7 8 8 1720 1710 J 700 3429.8 3470.8 3512.1 410 413 415 68.64 70.61 72-63 197 202 207 0.04764 0.04854 3.0494.5 90 91 93 1.578 1.602 i'.626 24 24 25 8.310 8.501 S.696 191 195 200 0.00461 0.00469 0.0047 S 8 9 8 TABLE I.— GoxTmuED. S{u) Difl A{u) Diff I{u) DiB r(«) Difl B{u) Diff M(u) Diff 1690 1680 1670 i66o 1650 1640 1630 1620 1610 1600 1595 1590 1585 1580 157s 1570 1565 1560 1555 1550 • 545 1540 1535 1530 1525 1520 1515 151C 1505 1500 1495 149a 14SS 3553-6 3595-4 3637-4 3679.7 3722 3765.0 3808.0 3851-3 3894.9 3938.7 3960.7 3982.8 4005.0 4027.3 4049.6 4072.0 4094.4 41 16.9 4139-5 4162.2 4185.0 4207.8 4230.7 4253-6 4276.7 4299.8 4323-0 4346.2 4369.6 4393-0235 418 42c 423 425 428 43c 433 436 438 22c 221 222 22; 223 224 24 225 226 227 228 228 229 229 33 231 232 232 234 234 4416.5 4440.1 4463.8 74.70 .76-83 79.01 81.24 83.52 85.86 88.27 90-73 93-25 95.84 97.16 98.49 99.84 IOI.2 102.60 04.00 05.4: o5.86 08.32 100.79 1 1 1.29 12.S0 114-33 US "7-45 1 19.04 120.65 122.28 123-93 125.60 236 127.29 237 129.01 237 130.7s 213 218 223 22S 234 241 246 252 259 3 133 135 137 139 140 142 144 4-6 147 150 •5 IS3 155 157 159 16 163 165 167 169 172 •74 17s 3.05038 0-051,33 0.05 229 3.05327 0.05427 0.05529 0.05632 0.05738 0.05845 0.05955 0.06010 0.06066 0.06 1 80 0.06238 0.06296 0.06355 0.06414 0.06474 0.06534 0.06595 0.06657 0.06719 0.06782 0.06846 0.06910 0.06975 95 96 98 100 102 103 106 107 0.07040 0.07106 0.07173 68 0.07241 0.0730c, 0.07378 1.65 1 1.676 1.701 1.726 1-752 1.778 1.804 1-831 1.858 1.885 1.899 •-913 1-927 1.941 '-955 1.969 •-983 1.998 2.012 2.027 2.042 2.057 2.07 2.086 2.101 2.117 2.132 2.147 2.162 2.178 2.194 2.210 2.226 27 14 14 14 14 14 14 14 15 14 15 15 IS 15 M '5 16 15 15 15 16 ]6 16 16 le 8.896 9.10 9-31 9.526 9.746 9-97 10.202 0.439 0.682 0.931 11.058 II. 186 1.316 11.448 11.581 1.716 11.853 11.991 12.131 12.274 12.418 12.563 12.71 1 12.861 13-013 13.166 13.322 13.480 13.640 13.802 205 210 215 220 225 231 237 243 249 127 12S 130 32 •33 •35 •37 •38 140 •43 •44 •45 148 150 152 •53 156 158 i6o 162 164 13.966 14.13: 14-30 166 169 171 0.00486 0.00495 0.00504 0.00513 0.00522 0.00532 0.00542 0.00552 0.00562 0.00573 0.00578 0.00584 3.00589 0.00595 0.00600 0.00606 0.006 1 2 0.00618 0.00624 0.00630 0.00636 0.00642 0.00648 0.00655 0.00661 0.0066S 0.00674 0.00681 0.00688 0.C0695 0.00702 0.00709 0,007 1 6 9 9 9 9 10' 10 10 10 ih 5 6 6 5 6 6 6 6 6 6 6 7 7 7 7 7 8 TABLE I.— Continued. « S(«) Die A{u) Difl Hu) Di£f r(i()Diff £{u) Diff -3/(«) Difl' 14S0 1475 1470 1465 1460 1455 1450 J445 1440 1435 »43c 1425 1420 1415 1410 1405 1 400 1395 1390 1385 1380 1375 1370 136; 1360 '355 '350 1345 1340 '^5 1330 1325 1320 4487.5 4SII-3 4S35-2 4559-2 4583.2 4607.4 4631.6 655.9 4680.3 4704.8 47294 4W4-1 +778.8 4803.6 4828.5 4853-5 4878.6 4003.8 4929.1 4954-5 4973-9 5005.5 5031.1 5056.8 5082.6 5 108.6 5134-6 5160.7 5186.9 ;2i3.2 5239-5 5265.8 5292.0 23S 239 24c 240 242 242 243 244 245 246 247 247 2.48 249 250 251 252 253 254 254 256 256 57 25 26c 26c 261 262 26 263 263 262 loC 32.50 34.28 36.09 37-92 39-77 41.65 43-54 45-47 47.42 49-39 51-39 53-42 55-47 57-55 59.60 61.80 63.96 66.15 68.37 70.62 72.90 75-21 77-55 79.92 82-3? 84.76 87-23 89-73 92.27 94.84 197-44 200.06 202. 6g 178 181 183 185 188 I 193 195 197 200 203 205 208 21 •214 216 219 222 225 22S 231 33 237 241 243 247 25c 254 257 260 262 263 107 0.07447 0.07517 0.07588 0.07660 0.07732 3.07805 007879 0.07954 0.08029 0.08105 0.081 82 0.08260 0.68338 0.084 1 8 0.08498 0.08579 0.0866 0.0S744 0.0S828 0.08913 0.08999 0.090S6 0.09173 0.09262 0-0935 0.09442 0-09533 0.09626 0.09719 0.098 1 3 0.09908 0.1 0004 O.IOIOl 2.242 2.258 2.274 2.290 2.307 2.323 2.340 357 2-374 2391 408 2.425 44 3 2.460 2.478 .496 .514 2-532 2.550 2.568 1.587 2.605 624 2.643 2.662 2.681 700 2.719 739 2.758 2.778 2.798 2.S18 4-47 4.645 4.821 4.999 5-179 5-362 5-548 5-736 5.926 6.119 6.316 6-5 6.716 6.920 7-1 7-339 7-552 7.768 7-9 8.311 8-437 8.666 8.899 9-135 9-374 9.6 9864 20. 1 1 5 20.369 20.627 0.SS9 21.150 21.417 •73 176 178 180 186 190 193 197 199 20 204 208 211 213 216 220 223 226 229 233 236 239 244 246 251 254 258 62 2!^r 267 iiol 0.00724 0.0073 000739 0.00746 0.00754 0.0076: 0.00770 0.00778 0.00786 0.00794 0.00802 0.0081c o.ooSig 0.008 2 8 0.00837 0.00846 0.0085 5 0.00864 0.00874 0.008S3 0.00893 0.0090 0.00912 0.00922 0.00933 0.00943 0.00954 0.00964 0.00975 0.00986 0.CO997 0.01008 0.01020 8 9 9 9 9 9 9 10 9 lO 9 10 10 II 10- II lO II II II II 12 5 TABLE I CO.N'TINUED. M S{u) Difl ^100 Dili /(») Diff 2'(u) Diff B{u) Diff 31 {u) Diff 1318 5302.6 106 203.76 loS 0.10140 39 2.826 8 21.527 109 0.01025 4 I3I6 5313-2 loe 204.84 108 0.10179 40 2.834 8 21.636 110 0.01029 5 1314 5323-8 107 205.92 109 0.10219 40 2.842 8 21.746 109 0.01034 4 I3I2 5334-5 10; 207.01 1 10 0.10259 40 2.850 8 21.855 no 0.01038 5 1310 5345-2 107 208. 1 1 III 0.10299 40 2.858 8 21.965 112 0.01043 S 1308 5355-9 108 209.22 III 0.10339 41 2.866 9 22.077 ••3 0.01048 5 1306 5366.7 108 210.33 112 0.10380 41 2.875 8 22.190 114 0.01053 4 1304 5377.5 108 211.45 "3 0.10421 4' 2.883 9 22.304 ••5 0.01057 5 1302 5388.3 IOC 212.58 114 0.10462 4' 2.892 8 22.419 116 0.01062 5 1300 5399.2 IOC 213.72 "5 0.10503 41 2.900 8 22.535 116 0.01067 5 1298 5410. 1 IOC 214.87 1 15 0.10544 42 2.908 9 22.651 118 0.01072 5 1296 5421.0 lie 216.02 •17 0.10586 42 2.917 8 22.769 1x8 0.01077 ^ 1294 5432.0 lie 217.19 ••7 0.10628 42 2.925 9 22.887 120 0.01083 5 1292 5443-0 I IC 218.36 118 0.10670 43 2-934 8 23.007 120 0.01088 5 1290 5454-0 III 219.54 119 0.107 1 3 43 2.942 8 23.127 121 0.01093 5 1288 5465.1 HI 220.73 I2C 0.10756 43 2.950 9 23.248 122 0.01098 S 1286 5476-2 111 221.93 I2C 0.10799 43 2-959 9 23.370 •24 0.0 1 103 6 1284 5487-3 112 223.13 122 0.10842 44 2.968 9 23-494 124 0.01109 5 1282 5498-5 112 224-35 122 0.10886 44 2.977 8 23.618 125 o.oi 1 14 5 1280 5509-7 113 225.57 123 0.10930 44 2.985 9 23-743 126 0.0III9 5 127S 5521.0 •13 226.80 124 0.1^974 45 2.994 9 23.869 127 0.01 124 6 1276 5532-3 •13 228.04 125 0.1 1019 45 3.003 9 23.996 129 0.0113d 5 1274 5543-6 113 229.29 125 0.11064 45 3.012 9 24.125 129 0.01135 6 1272 5554-9 114 230-54 127 0. 1 1 1 09 45 3.021 S 24.254 130 O.OI 141 S 12,70 5566.3 114 231.81 127 0.11154 46 3.030 c 24.384 '3^ O.OI 146 6 I26« 5577-7 114 233-08 129 0. 1 1 200 46 3-039 9 24-5 • 5 132 0.01152 6 1266 5589-1 J 15 234-37 129 0.11246 46 3.048 c 24.647 134 0.01158 5 1264 5600.6 115 235.66 •3' 0. 1 1 292 46 3.057 c 24.78 • •34 0.01163 6 1262 5612.1 116 236.97 •3^ 0.1 1338 47 3.066 S 24.915 .36 0.01169 6 1260 5623-7 116 238.2« 132 0. 1 1 385 47 3.075 S 25.051 •36 O.OI 175 6 1258 5635-3 i'7 239-6q 134 0.1I432 47 3-084 IC 25.187 138 0.01181 6 1256 5647.0 1 16 240.94 •34 3.1 1479 48 3-094 S 25-325 139 O.OI 187 5 1254 5658.6 117 242.28 • 36 3.11527 48 3-103 IC 25-464 140 0.01192 6 TABLE I. — CloxTiNUED. « S{u) ===== Difl A{ 0.12280 53 3.248 10 27.690 159 0.01287 & 1222 5850.7 123 265.18 •53 3.12333 53 3-25S 10 27.849 160 0.01293 7 12^0 586.3.0 124 266.71 153 3.12386 S3 3.268 10 28.009 161 0.C1300 7 1218 5875-4 124 268.24 155 0.12439 54 3.278 10 28.170 163 0.0 1 307 7 I216 5887.8 125 269.79 .56 0.12493 54 3.288 11 28.333 164 0.01314 7 I2I4 5900.3 125 271-35 '57 0.12547 55 3-299 10 28.497 166 0.01321 7 I2I2 5912.8 125 272.92 159 0. 1 2602 55 3-309 10 28.663 167 0.01328 7 1210 5925-3 126 274.51 160 0.12657 55 3-319 10 28.830 168 0.01335 7 1208 5937-9 126 276.11 161 0.12712 56 3-329 11 28.998 170 0.01342 7 1206 5950.S 127 277.72 162 0.12768 56 3-340 10 29.168 172 0.01349 8 1204 5963-2 127 27934 163 3.12824 57 3-350 II 29.340 173 0.01357 7 1202 3975-9 127 280.97 165 D.128S1 57 3-361 10 29-513 174 0.01364 7 1200 5988.6 128 282.62 166 3.12938 57 3-.371 u 29.687 176 0.01371 8 1198 6001.4 128 284.28 167 3.12995 S8 3-382 11 29863 177 0.01379 7 1196 6014.2 129 285.9.^ 168 0-13053 58 3-393 11 30.040 179 0.01386 8 I 194 6027.1 129 2S7.63 17c 0.13111 58 3-404 11 30.219 181 0.0139^ 7 ,1192 6040.C )13C 289.33 171 0.13165 59 3-415 11 30.400 182 0.01401 » 119c 605 3.C ) 13c 291.0/ 172 0.1322S 59 3.426 11 30.582 184 0.01409 8 II8S 6066.C )i3i 292.76 174 0.1328; 60 3-437 11 30.766 185 0.01417 8 TABLE I.— Continued. « S[v) Din Aiu) Difl /(,0 Dili T{u Diii Bin) Difl M{u) Diir 1186 5079.1 131 294.5c ■75 3-13347 60 3-44f II 30-95' 1 87 0.01425 7 1 1 84 5092.2 '31 296.25 '7/ 3.13407 60 3-45S 11 3'-'38 189 0.01432 8 1182 fit 05. 3 1 132 298.02 171 3.13467 61 3 •47c II 31-327 190 0.0144c 8 1 1 8c 5n8.5 132 299.80 '79 3.13528 61 3.481 II 3i-S'7 192 0.01448 8 117^' 6131.7 '33 301.59 181 3.13589 62 3-492 12 31-709 194 0.01456 8 1 176 6145.0 '33 30340 1S2 3. 1 365 1 62 3-504 II 3' -903 '95 0.01464 9 1174 6158.3 '34 305.22 184 0.13713 63 3-5'5 12 32.098 198 0.01473 8 1172 5171.7 '34 307.06 185 0.13776 63 3-527 II 32.296 1.99 0.0148 1 8 117c 6185.1 '35 308.91 186 0.13839 63 3-538 12 32.495 201 0.01489 9 116S 6198.6 '35 31077 18S 0.13902 64 3-550 n 32.696 203 0.01498 8 1 166 6212.1 '35 312.65 190 0. 1 3966 64 3-561 12 32.899 205 0.01506 "9 1164 6225.6 136 3'4-55 191 3.14030 65 3-573 II 33-'04 206 0.01515 8 1162 6239.2 ■36 316.46 193 0.14095 65 3-584 12 33-310 209 0.01523 9 1 160 6252.8 69 3'8.39 97 0.14160 32 3-596 6 33-519 105 0.01532 4 1159 6259.7 69 3'9-36 98 0.14192 33 3.602 6 33-624 106 001J36 5 1158 6266.6 68 320.34 98 0.14225 33 3.608 6 33730 106 0.01541 4 "57 62734 69 321.32 98 0.14258 33 3.614 6 33-836 1 06 0.01545 1156 62S0.3 69 322.30 98 0.14291 33 3.620 6 33-942 107 0.01550 •155 6287.2 69 323-28 99 0.14324 34 3.626 6 34.049 107 0.01554 1154 6294.] 69 324-27 99 0.14358 33 3-632 6 34-156 108 0.01559 i^53 6301,0 69 325.26 100 0.1439' 34 3-638 6 34-264 108 0.01563 1152 6307.9 69 326.26 100 0.14425 33 3-644 6 34.372 109 0.01568 1151 6314.8 7° 327.26 lOI 0.14458 34 3.650 6 34-481 109 0.01572 1150 6321.8 70 328.27 lOI 0.14492 34 3.656 6 34-590 no 0.01577 1 149 6328.8 69 329.28 lOI 0.14526 34 3.662 6 34.700 I II 0.01582 1 148 63357 70 330.29 102 0.14560 34 3.668 6 34.811 III 0.01586 1 147 6342.7 70 331-3' !02 0.14594 34 3-674 6 34.922 112 0.01591 1 146 63497 70 332.33 103 0.14628 34 3.680 6 35-034 112 0.01596 «!45 63567 70 333-36 'O3 0.14662 35 3.686 7 35-'46 112 0.01601 1144 63637 70 334-39 104 0.14697 34 3-693 6 35-258 "3 0.01605 "43 6370.7 71 335-43 104 0.1473 1 35 3-699 6 35-37' "3 0.01610 1 142 6377-8 70 33647 104 0.14766 35 3-705 6 35-484 "4 0.01615 1141 6384.8 7' 337-5' 105 0.14801 35 37" 6 35-598 "4 001619 TABLE 1.— CoNTmnED. S(u) Diff A{u) Diff /(«) Difl T(u)Diflf £{u) DiS M{u) Diff 140 139 138 137 136 135 134 133 132 131 130 129 128 127 126 125 12/1 123 122 121 120 119 118 117 116 'IS 114 113 112 II I 10 109 6391.9 6399.0 6406.1 6413.2 6420.3 6427.4 6434.6 6441 6448.9 6456, 6463.3 5470.4 6477.6 6484.8 6492.1 ''499-3 6506.6 3S'3-9 5521.2 6528.6 5536.0 6543 -4 6550.8 6558.3 6565.8 6573-3 6580.8 6588.4 6596.0 6603.7 661 1.4 6619.1 7» 71 71 7' 71 72 71 72 7 72 7 72 72 73 72 73 73 /3 74 74 74 74 75 75 75 75 76 76 77 77 77 ■ 78 08II6626.9I 78 338.56 339-61 340.67 341.73 342.79 343.86 344.94 346.02 347.10 348.19 349.28 350.38 351.47 352.57 353.68 354.79 355.90 357.03 358.16 359.30 360.45 361.60 362.76 363.92 365.09 366.28 367.47 368.67 369.88 37-1 .09 372.32 1373-55 Z7'^-79 105 106 106 106 107 108 108 loS 109 109 no 109 no HI II 111 >I3 "3 114 "5 115 116 116 117 119 119 120 121 121 123 '23 124 125 0.14836 0.1487 1 0.14906 0.1494 0.14977 0.15013 0.15049 0.15085 0.151 0.15157 0.15193 0.15229 0.1526 0.15302 0.15338 0.1 5 375 0.15412 0.15449 0.15487 0.155 0.15562 o. 1 5600 0.15638 0.15676 0.15715 0.15754 0.15793 0.1583: 1587: 0.15912 0.15952 0.15993 0.16033 35 35 36 35 36 36 36 36 36 36 36 36 37 36 37 37 37 38 37 38 38 38 38 39 39 39 39 4C 40 40 41 40 4 3.717 3.723 3.730 3.736 3742 3.748 3.75s 3.76 3.767 3-774 3.780 3.786 3.793 3.799 3.806 3.8 3.8 3.825 3.83' 3.838 3-844 3.851 3.858 3.864 871 3-878 3.885 3.892 3-898 3-905 3-912 3-919 3.926 35-712 35-827 35-943 36-059 36 176 36-293 30.41 1 36-530 36.649 36.769 36.889 37-0 37-131 37-253 37-375 37.498 37.622 37-746 37-871 37-996 38.122 38.250 38.378 38.508 38.638 38.770 38.902 39.036 39-170 39-306 39.442 39.5^0 39.718 "5 116 116 117 117 118 119 119 120 120 121 121 122 122 12 124 124 125 125 126 128 128 130 130 132 132 134 134 136 •36 '38 13S 140 8 0.01624 0.01629 0.01634 0.01638 0.01643 0.01648 0.01653 0.01658 0.01662 0.01667 0.01672 0.016; 7 0.01682 0.01687 0.0169 0.01697 0.01703 0.01708 0.01713 O.OI718 0.01723 0.01728 0.01734 0.01739 0.01745 0.01750 001756 0.0176 0.01767 Q.OI772 O.OI77S 0.01784 I0.OI790 TABLE I.— Continued. S{u) Diff A(u) Difl I{n) Dift T (u)Difl £(m) Dift M{ii) Diff U07 1 106 1105 1104 1 103 1 102 1 101 I too 1099 109S 109, 109C 109: 1094 1093 1692 1091 1090 1089 1088 1087 1086 1085 1084 1083 1082 1 08 1080 1079 1078 1077 1076 1075 6634.7 6642.5 6650.3 6658.2 6666.2 6674. 1 6682. 6690.2 6698.3 6706.4 6714.5 6722.7 673 1 .0 6739.2 6747-5 6755-9 6764.3 6772.7 6781.2 6789.7 6798.2 6866.8 6815.4 6824 6832.8 6841.5 6850.3 6859 6867.9 6876.8 68S5.8 6894.7 6903.7 84 84 84 85 85 85 86 86 87 87 87 88 88 88 89 90 89 90 91 376.04 377-30 378.57 379-85 381-. 14 382.44 383-75 385.06 386.38 387-71 389.06 390-4 391-78 393-15 394-53 395-93 397-34 398-75 126 127 128 129 130 13 131 132 133 J35 135 137 '37 138 14c 141 141 142 40s. I 401.60 403.05 404.50 405.97 407.45 408.94 410.44 411.95 413-47 415.00 416.54 8.10 419.66 421.24 7J43 '•45 •45 147 148 149 150 151 152 153 •54 •56 156 158 •59 0.16074 0.16115 0.16157 0.16198 0.1 6240 0.16282 0.16325 0.16367 0.16410 0.16453 0.16497 0.16541 0.16585 0.16629 0.16674 0.16719 0.16764 0.16810 0.16856 0.169O2 0.1 0.16995 0.17042 0. 1 7089 0.17137 0.1718s o-^7233 7282 7331 0.17380 0.17429 0.17479 0.17529 3-933 3-940 3-947 3-955 3.962 3-969 3-976 3-983 3-99 3-99' 4.006 4-013 4.021 4.029 4.036 4.044 4.051 4.059 4.067 4.075 4.083 4.091 4.098 4.106 4.114 4.122 ^••30 4.138 4.146 4-155 4.163 4.172 ^.180 7 39-858 39.998 40.140 40.283 40.427 40.5f2 40.719 40.866 41.015 41.165 41.316 41.468 41.621 41.776 41-932 42.089 42.247 42.407 42.568 42.730 42.894 43-059 43.225 43-393 43-562 43-732 43-904 44-077 44.252 44.428 44.605 44.784 44.964 140 •4 •43 •44 •45 •47 147 •49 •50 •51 •52 •53 •55 156 157 158 160 161 162 164 165 166 168 169 17c 172 •73 •75 176 177 79 180 182 0.01795 0.0 1 80 1 0.01807 0.01813 0.01818 0.01824 0.01830 0.01836 0.01842 0.01849 0.01855 0.0 1. 86 1 0.01867 0.01874 0.01 880 0.01886 0.01893 0.01899 0.01906 0.01913 0.01919 0.01926 0.01933 0.01940 0.01947 0.01953 0.01960 0.01967 0.01974 0.01982 0.01989 0.01996 0.02003 9 TABLE I.— Continued. u S{u) Diff A[u) Diff I{u) Difl r{u] Difl -B(«J Difl M(u) Difl- 1074 6gi2^ 9> 422.83 161 0.17580 51 4.185 £ 45.14^ )i84 0.02011] 7 1073 6921.9 92 424.44 162 0.17631 5' t-i97 9 4S-33C 1185 0.0201? 7 1072 6931-1 92 426.06 163 0.17682 5' 4.206 c. 45-515 186 0.02025 8 1071 6940.3 92 42769 164 0-17733 Si 4<2i4 9 45-70' 1 88 0.02033 7 1070 6949.5 93 ^29.33 165 0.17785 52 4.223 9 45-88g 190 0.02040 8 1069 6958.8 93 430.98 166 0.17837 S3 4.232 9 46.079 191 0.02048 8 1068 6968.1 94 +32-64 168 0.17890 S3 4.241 9 46.270 •93 0.02056 7 1067 6977-5 94 434-32 169 0.17943 S3 4.250 9 46.463 194 0.02063 8 1066 6986.9 94 436.01 171 0.17996 53 4-259 9 46:657 196 0.0207 ' S 1065 6996.3 95 437-72 172 0. 1 804.9 54 4.268 9 46.853 198 0.02079 8 1064 7005 8 96 439-44 173 O.I 8103 55 4.277 9 47.051 199 0.02087 8 1063 7015.4 96 441.17 175 O.I8I58 55 4.286 9 47.250 201 0.02095 7 1062 7025.0 96 442-92 176 0.18213 55 4-295 9 47-45 > 203 0.02102 8 I06I 7034.6 97 444.6s 177 0.18268 55 4-304 9 47-654 205 0.02110 8 1 06c 7044-3 97 446.45 178 0.18323 56 4-313 9 47.859 207 0.02 1 1 8 8 1059 7054.0 98 448.23 180 0.18379 56 4-322 10 48.066 208 0.02126 9 1058 7063.8 98 450.03 181 0.18435 56 4-332 9 48.274 210 0.02135 8 1057 7073.6 99 451.84 182 O.I 849 1 57 4-34' 9 48.484 212 0.02143 9 1056 70.83.5 99 453-66 184 0.18548 S7 4-350 10 48.696 213 0.02152 8 1055 ^■093 4 100 455-50 1 86 O.IS605 58 4-360 9 48.909 216 0.02160 9 1054 7103-4 100 457^36 187 0.18663 58 4-369 9 49-125 217 0.02169 8 1053 7113-4 100 459-23 189 0.18721 58 4.378 9 49-342 219 0.02177 9 1052 7123.4 lOl 461.12 190 0.18779 59 4-387 10 49.561 222 0.02 1 86 8 1051 7133-5 102 463.02 192 0.ISS38 59 4-397 9 49-783 223 0.02194 9 1050 7143-7 102 464.94 193 0.18897 59 4.406 10 50.006 22^ 0.02203 9 1049 7153-9 102 466.87 '94 18956 60 4.416 10 50.231 228 0.02212 9 1048 7164.1 103 4'jS.8 I iq6 0.19016 61 4:426 IC 50.459 229 0.02221 9 1047 7 '74-4 103 470.77 '97 0.19077 61 4-436 10 50.688 23' 0.02230 9 9 104.6 7184.7 104 472.74 '99 0.19138 61 4.446 9 50.919 233 0.02239 1045 7195.1 105 474-73 201 O.IOI99 .61 4-455 - TO 51.152 235 0.02248 10 1044 7205.6 105 476.74 203 0. 1 9260 62 4.465 10 51-387 237 0.0225 8 Q 1043 7216.1 io5| 478.77 204 0.19322 63 4-475 10 51.624 240 0.02267 0.02276 9 9 1042I 7226.6 m 4.80.8 1 206 0.19385 63 10 4.485 10 51.864 241 TABLE I. — Continued. Siu) Difl A{u) Difl I{u) Diff T{u) Diff B{u) Dif] M(u) Diff 1041 1040 1039 103S '037 1036 •035 ■034 1033 1032 1031 1030 1025 102S 1027 I02£ 1025 ro24 1023 1022 102 1 1019 1018 ID17 1016 1015 1014 1013 1012 101 1010 1009 7237.2 7247-9 7258/ 7269.3 7280.1 7291.0 7301-9 7312.9 7323-9 7335-0 7346.1 7357-3 7368.5 739M 7402.5 7414.0 7425-5 7437 7448-7 7460.4 7472.1 7483-9 7495-7 7507.6 7519.6 7531-6 7543-7 7555-8 7568.0 7580.3 7592.6 7605.0 482.87 484.95 487.04 489.15 491.28 208 209 21 213 214 493.42216 495-5S 497.76 499-95 502.17 504.40 506.65 508.91 1 1.20 513-50 515.82 518.17 520.54 522.92 525-32 527-75 530.20 53?.66 35-14 537-65 40.17 542.72 545-30 547-89 50.51 553-16 555.82 558.51 218 219 222 223 225 226 229 230 232 235 237 23S 240 243 245 246 248 251 52 55 258 59 262 265 266 269 272 0.19448 0.19511 0.1957s 0.19639 0.19703 0.19768 19834 19900 0.19966 0.20033 0.20100 0.201 0.20236 0.20305 0.20374 0.20443 0.20513 0.20584 0.20655 0.20726 0.20798 0.20871 0.20944 0.21017 0.21091 0.21 165 0.21240 0.21316 0.21392 0.21468 0.21545 0.21623 0.21701 07 68 68 69 69 69 70 71 71 71 72 73 73 73 74 74 75 76 76 76 77 78 78 79 11 4-495 4.505 4.516 4-526 4-537 4-547 4-558 4.569 +-579 4.590 4.600 4.611 4.622 4-633 1.645 4.656 4.667 4-678 4.689 4.701 4.712 4-723 4-735 4-747 4-759 4-77 4.782 4-794 4.806 4.818 4.830 842 4-S55 52.105 52.349 52-595 52.843 53-093 53-346 53.601 53-858 54.ns 54.380 4.644 54-911 55.180 55-452 55.726 56.003 56.283 56.565 56.849 7-136 7.426 57-719 8.015 8-313 8.614 8.918 59.224 59-534 59.846 5o. 1 62 50.480 00.802 51.127 244 246 248 250 253 255 257 260 262 264 267 269 272 274 277 280 282 284 287 90 93 296 298 30 1 304 306 310 312 316 318 322 325 328 6.02285 0.02294 0.02304 0.02314 0.02324 0.02334 0.02344 0.02353 0.02363 0.02373 0.02383 0.02393 0.02404 0.02414 0.02425 0.02435 0.02446 0.02457 0.02467 0.02478 0.02488 0.02499 0.02510 0.02522 0.02534 0.02545 0.02557 0.02569 0.02580 0.02592 0.02603 0.02615 0.02627 9 10 10 10 10 lb 9 10 10 10 10 11 10 II 10 II 1 1 10 II 10 II II 12 12 1 1 12 12 II 12 II 12 12 13 TABLE 1.— Continued. u S{u) Difl A{it) Difl J{u) Difi r(«)Diff £{u) Diff -«"(«) Diff 1008 1007 1006 1005 1004 1003 7617.4 7629.9 7642.5 7655.1 7667.8 7680.6 1002 lOOI 1000 999 998 997 996 995 994 993 992 991 990 989 988 987 986 98s 984 983 982 980 979 978 97/1 976I 7693-4 7706.3 7719-3 7732.4 7745-6 775S.8 7772.1 7785-4 7798-7 125 12f. 126 127 128 128 129 1301 131 132 132 133 133 133 134 7812.1 7825.5 7839.0 7852.5 7866.1 7879-7 7893-4 7907 7920.8 7934-5 7948.3 796 7975-9 7989.8 8003.7 8017.6 8031.5 8045.5 ■34 '35 135 136 136 137 >37 137 137 138 138 ■3& 139 139 r39 139 140 140 561.23 563.96 566.71 569.49 572.29 575-11 577.96 586.8 583-7^ 586.64 589.59 592.56 5-95-56 598-50) 601.65 604.74 607.8, 610.99 614.16 617-33 620.52 623.73 626.96 630.21 633-48 636.77 640.08 643-41 646.76 650.12 653-51' 656.92 660.35 273 ^75 278 280 28,2! 285 287 289 292 295 297 303 306 309 31 314 317 317 3 '9 32' 323 325 327 329 33' 333 335 336 339 341 343 345 0.21780 0.21859 0.21939 0.22019 0.22100 0.22182 0.22264 0.22347 0.22430 0.22514 0.22599 0.22684 0.22770 0.22857 0.22944 0.23031 0.23118 0.23206 0.23295 0.233S4 0.-3474 0.23564 0.23655 3.23746 0.2383; 3.23929 3.2402 1 0.24113 3.24206 3.24299 3,24392 3.24486 3.24580 IS 79 8c 8c 81 B2 82 83 83 85 85 86 87 87 87 87 88 89 89 90 go 91 9' 9' 92 92 92 93 93 93 94 94 95 4.867 4.880 4.892 4-905 4.918 4-930 4-943 4-955 4.968 4.981 4-995 5008 5.022 5-03S 5.048 5.062 5075 5.089 5.102 5. 116 5-'30 144 5.158 5.171 5.185 5.199 5-213 5.227 5.241 5-255 5.270 5.284 5.2.99 DI.455 61.786 62. 1 20 62.457 62.797 63.141 63.488 63-839 64- '93 64.551 64.912 65.276 65.644 66.016 66.392 66.771 67.154 67-540 67.931 323 68.717 69.115 69.515 69.918 70.3.24 70.732 71.144 71.558 71.975 72.395 72.818 73-245 73-674 33' 334 337 340 344 347 351 354 358 361 364 368 372 376 0.02640 0.02652 0.02665 0.02677 0.02689 0.02702 0.02714 0.02727 0.02739 0.02752 0.02766 0.02780 3790. 0.02793 0.02S07 ,02821 383 386 391 392 394 398 400 403 40S 408 412 414 4'7 420 423 427 429 432 0.02834 0.02848 0.0286 0.02875 0.02889 0.02904 0.02918 0.0293 0.02946 0.0296 0.02975 0.029S9 0.03004 0.03018 0.03033 o 03048 0.03064 0.03079 TABLE I.— Continued. S(«) Difi Ai^u) Diff I{u) Difi r(,0 Dm B{u) Difl 31 (u) Diff 97 b 974 97? 9;2 97' 970 969 968 967 966 9^ 96 + 962 961 960 959 958 95 956 955 954 953 95 951 950 C49 947 946 945 944 943 8059-5 8073-5 8087.6 8101.7 8115.8 8129.9 8144. S158.3 8172.5 8186.8 820 1. 1 8215.4 8229.8 >i2 14-2 8258.6 8273.0 8287.4 8301.9 8316.4 8331-0 8345-6 8360.2 8374-8 8389.5 8404- 8419.0 8433-8 S448.6 8463.4 8478.3 8493.2 8508-1 8523.1 663.S0 567.26 670.75 674.26 677.80 681.35 684.92 688. 692.12 695-75 699.41 703.09 346 349 351 354 355 357 359 36 363 366 368 370 706.70 372 710.51 714.26 718.03 721.81 725.62 729.46 733-32 737.20 741.10 745-03 748.98 752-' 756-96 760.98 765.02 769.09 773-18 '777.30 781.45 785.62 375 37; 378 3«' 384 386 38S 390 393 395 398 400 402 404 407 409 412 415 4'7 420 0.24675 c. 24770 0.24865 o. 2496 1 0.25057 0.251.54 0.25251 0.25348 0.25446 0.25544 0.25643 0.25742 0.25841 0.25941 0.26041 0.26142 0.26243 0.26344 0.2S446 o 26549 0.26652 0.26755 0.36858 O.C6962 0.27067 0.27172 0.27277 0.27383 0.27489 0.27596 95 95 96 96 97 97 97 98 98 99 99 99 100 IOC lOI lOI 101 102 103 103 103 103 104 105 105 105 106 100 107 107 0.27703 0.27811 0.27919 108 108 108 13 5-313 5-3^7 5-342 5-356 5-371 5-385 5.400 5-415 5.429 5-444 5-459 5-474 5.489 5-505 5.518 5-533 5-548 5 564 5-579 5-594 5.609 5.625 5.640 5-655 5.671 5.686 5-702 5.718 5-733 5-749 5-765 5.781 5-797 74.106 74-541 74-979 75.420 75.864 7'5.3ii 76.761 77.214 77.671 78.131 78.594 79,060 79-530 80.003 80.479 80.958 81.441 81.927 .417 82.910 83.407 83.907 84.410 84.917 85.428 85-9+2 •S6.460 86.981 S7.506 88.035 88.568 89.105 39.645 435 43*! 441 444 447 45c 453 457 46c 46? 466 470 473 476 479 483 486 49° 493 497 503 507 5" 514 51S 521 525 529 533 5.V 540 544I 0.03094 11.03109 0.03 1 24 0.03140 0.03155 3.03170 0.03186 0.03202 0.03218 0.03234 0.03250 0.03265 0.03281 0.03297 0.03313 0.03329 0.03346 5336: 0.03579 0.03396 0.03412 0.03429 0.03446 0.03463 0.03479 0.03496 0.03514 0.0353 0.03549 0.03567 0.03584 0.03602 0.0362c IS 15 16 >5 15 16 16 16 1-6 16 15 TABLE 1. — Continued. « S{u) Dia A{u) Difl J(«) Difl T{u) Diff B{u) Difl M{u) Diff 942 941 940 939 938 037 936 935 934 933 932 931 930 929 g28 927 926 925 924 923 922 921 920 919 918 917 916 915 914 913 912 911 910 8538.' 8553-1 8568.2 8583.3 8598.4 8613.6 8628.8 8644.0 8659.2 8674.5 8689.8 8705.2 8720.6 8736.0 8751.5 8767.0 8782.5 8798.0 13.6 8829.2 8844.9 8860.6 8876.3 8892.0 8907.8 8923.7 8939-5 8955.4 8971-3 8987.3 9003-3 9019.3 9035 -.4 789.82 794.04 798.29 802.56 806.85 811. 17 815.S 819.89 824.30 828.73 833.18 837.67 842.18 846.71 851.27 855.86 860.48 865.13 869.81 874.51 879.25 884.02 888.81 893-63 898.48 903.36 908.27 913.21 918.18 923-19 928.22 933-28 .J4S.37 422 425 427 429 432 435 437 441 443 445 449 45 1 453 456 459 462 J.65 468 470 474 477 479 482 485 488 491 494 497 50 503 506 509 513 0.28027 0.28136 0.28246 0.28356 0.28467 0.28578 0.28689 0.28801 0.28913 0.29026 0.29140 0.29254 0.29368 0.294S3 0.29598 0.29714 0.29830 0.29947 0.30064 0.30182 p. 30300 0.30419 0.30538 0.30658 0.30778 0.30899 0.31020 0.31142 0.31264 0.31387 0.31511 0.31635 0.31760 5.812 5.828 5.844 5.860 5-877 5-893 5.909 5.926 5.942 5.958 5-974 5.991 6.007 6.024. 6.041 6.057 6.074 6.091 6.108 6.125 6.141 6.158 6.175 6.192 6.210 6.227 6.245 6.262 6.279 6.297 6.314 6.332 6-349 30 189 90.736 91.288 91.844 92.403 92.967 93-535 94.107 94.683 95-263 95.847 96-435 97.027 97.623 224 98.829 99 439 100.05 [ 00.67 101.29 101.92 102.55 103.19 103.83 104.48 105.13 105.79 106.45 107.11 107.78 108.45 109.13 109.81 547 352 556 559 564 568 572 576 580 584 588 592 596 601 605 610 6J5 62 62 63 63 64 64 65 0.03638 0.03655 0.03673 0.03692 0.03710 0.03729 0.03748 o 03766 0.03785 0.03804 6.03823 0.03841 0.03860 0.03880 0.03899 0.03919 0.03938 0.0396 0.0398 0.0400 0.0402 0.0404 0.0406 0.0408 0.0410 0.0412 0.0414 0.0416 0.0418 0.0420 0.0422 0.0424 o.04?6 17 18 19 18 19 19 18 19 19 19 18 19 20 19 19 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 TABLE 1.— CONTINDED. u SW Diff A{u) Dift I{u) Diff T{n) Diff Bin) Dift M{u) Diff 909 9051.5 161 943-50 515 0.31885 126 6.367 18 1 10.49 69 0.0428 2 908 9067.6 162 948.65 5'9 0.32011 126 6.385 18 111.18 70 0.0430 3 907 9083.8 162 953.84 522 0.32137 127 6,403 18 1 1 1.88 70 0-0433 2 906 9 100.0 162 959.06 525 0.32264 128 6.421 18 112.58 7' 0-0435 2 905 91 16.2 163 964-31 529 0.32392 12^ 6.439 18 113.29 ,71 0.0437 2 904 9'32-5 163 969.60 §32 0.3252a 129 6.457 18 1 14.00 72 0.0439 2 903 9148.8 164 974.92 535 0.32649 129 6.475 18 114.72 73 0.0441 3 ■902 3165.2 164 980.27 538 0.32778 '30 6.493 18 115.45 73 0.0444 2 901 9181.6 164 985.65 54' 0.32908 130 6.511 18 116.18 74 0.0446 2 90c 9198.C 16s 991.06 545 0.33038 '3' 6.529 19 1 16.92 74 0.0448 3 899 9214:5 '^J 996.51 548 0.33169 '3' 6.548 18 117.66 74 0,045 ' 2 S98 9^3 1. c 165 1001.99 552 0.33300 132 6.566 '9 11840 75 0-0453 3 S97 9247.5 166 1007.51 555 0.33432 '33 6-5 85 18 119.15 76 0.0456 2 896 9264.1 166 1013.06 559 0-33565 133 6.603 '9 1 19.91 76 0.0458 2 895 P280.7 166 1018.65 562 0.33698 '34 6.622 18 120.67 77 0.0460 3 89^ 9297.3 167 1024,27 565 0.33832 '34 6.640 19 121.44 77 0.0463 2 893 9314.0 167 1029.92 569 0.33966 135 6.659 18 122.21 78 0.0465 3 ^92 9330.7 168 1035.61 573 O.34101 '36 6.677 19 1 22.99 79 0.0468 2 891 9347-5 168 1041.34 576 0.34237 136 6.696 18 '23.78 79 0.0470 2 Sgo 9364.3 168 1047.10 580 0.34373 137 6.714 19' '24-57 79 0.0472 2 889 9381-1 169 1052.90 583 0.34510 '37 ^•733 20 125.36 80 0.0474 . 3 888 9398.0 169 '058.73 587 0.34647 '38 6-753 19 126.16 81 0.0477 2 887 9414.9 170 1064.60 592 0.34785 '39 6.772 19 126.97 81 0.0479 3 886 9431.9 170 1070.52 595 0.34924 '39 6.791 20 127.78 82 00482 2 585 9448.9 170 1076.47 598 0.35063 140 6.811 '9 128.60 83 0.0484 2 ^84 9465.9 171 1082.45 602 0.35203 141 6.830 19 '29-43 83 0.0486 3 883 9483.0 '71 1088.47 606 0.35344 141 6.849 19 130.26 84 0.0489 2 882 9500.1 171 '094.53 609 0-35485 142 6.868 20 131.10 84 0.0491 3 38i 9517-2 172 :i 100.62 613 0-35627 '43 6.888 '9 '31-94 85 0.0494 2 58o 9534.4 172 1 106.75 617 0-35770 143 6.907 20 132.79 86 0.0496 3 ^79 95SI.6 173 1 1 12.92 621 0.35913 144 6.927 20 133-65 86 0.0499 2 S78 9568-9 173 II19.13 625 0.36057 '45 6.947 '9 '34-5' 87 0.0501 3 S77 9586.2 f75 1125.38 629 0.36202 '45 6-.966 20 '35-38 88 0.0504 2 TABLE I.— Continued. u S{u) Diff Aiu) Diff I(u) Diff 2'{u) Diff B(n) Diff J/.h) Dift 876 9603.5 ^74 1131.67 633 0.36347 146 6.986 20 136.26 88 0.0506 3 875 9620.9 174 1138.00 637 0.36493 146 7.006 20 137.14 89 0.0509 3 874 9638.3 '75 "44-37 64 r 0.36639 147 7.026 20 I3l8.03 90 0.0512 2 873 9655.8 175 1150.78 645 0.36786 148 7-046 '.9 138.93 90 0.0514 3 872 9673.3 175 1157-23 649 0.36934 149 7.065 20 139.83 91 0.0517 2 871 9690.8 176 1163.72 653 0.37083 149 7.085 20 140.74 91 0.0519 5 870 9708.4 176 1170.25 657 0.37232 150 7.105 21 141.65 92 0.0522 3 86g 9726.0 177 1176.82 662 0.37382 150 7.126 20 142.57 92 0.0525 3 868 9743-7 177 1183.44 665 0.37532 151 7,146 21 143.49 94 0.0528 2 867 9761.4 177 1 190.09 670 0.37683 152 7.167 20 144-43 94 0.0530 i 866 9779-1 .78 1196.79 675 0.37835 153 7.187 21 145-3; 95 0.0533 3 865 9796.9 178 1203,54 678 0.37988 153 7.208 21 146.32 96 0.0536 3 864 9814.7 179 1210.32 683 0.38I4I <54 7.229 20 147.28 96 0.0539 1 863 9832.6 179 1217.15 687 0.38295 155 7.249 21 148.24 98 0.0542 7. 862 9850.5 179 1 224.02 691 0.38450 156 7.270 20 149.22 98 0.0544 3 861 9868.4 180 1230.93 696 0.38606 156 7.290 21 150.20 99 0.0547 3 860 9886.4 180 1237.89 700 0.38762 157 7-3 'I 21 151.19 99 0.0550 3 859 9904.4 181 1 244.89 705 0.38919 158 7.332 22 152.18 100 0.0553 S 858 9922.5 181 1251.94 710 0.39077 158 7-354 21 153.18 lOI 0.0556 3 857 9940.6 181 1259.04 7>4 0.39235 159 7-375 21 154-19 102 0.0559 3 856 9958.7 182 1266.18 718 0.39394 160 7.396 22 155.21 102 0.0562 2 855 9976.9 ■83 1273.36 723 0.39554 161 7.418 21 156.23 103 0.0564 3 854 9995-2 183 1280.59 728 0.39715 162 7-439 21 157.26 104 0.0567 3 853 10013.5 183 1287.87 732 0.39877 162 7.460 21 158.30 105 0.0570 :i 852 1003 1. 8 184 [295.19 737 0.40039 ■63 7.481 22 159.35 106 0.0573 3 85. 10050.2 184 1302.56 742 0.40202 164 7-503 21 160.41 106 0.0576 5 850 10068.6 185 1309-98 746 0.40366 164 7.524 22 161.47 107. 0.0579 3 849 10087. 1 185 i3'7-44 752 0.40530 165 7.546 22 162.54 108 0.0582 3 848 10105.6 ,85 1324.96 756 0.40695 166 7.568 22 163.62 109 0.0585 3 847 10124.! 1 86 1332.52 761 0.40861 167 7.590 22 164.71 110 o.g588 3 846 10142.7 186 1340.13 766 0.4102? 168 7.612 23 165.81 no 0.0591 3 845 10161.3 187 1347.79 771 0.41 196 168 7.635 22 166.91 11 1 O.OS94 4 844 10 1 80.0 188 1355-50 776 0.41 364 1( 169 1 7.657 22 16802 112 0.0598 3 TABLE 1.— CoA"nNDED. u ^^('0 Din A") Dm 1(11) Diff Tiu Djif «(") D\h M(u) Din 84^ 842 84 1 ioi'98.8 10217.5 10236.3 187 18S 189 1363. 2G 1371.07 1378-93 7S1 786 791 ■4'533 ■4 '703 41874 170 171 172 7-679 7.701 7-723 22 22 22 169.14 170.27 171.41 "3 114 114 .0601 .0604 .0607 3 3 3 840 839 83S 10255.2 10274. 1 10293.0 189 189 190 1386.81 I394.S0 1402. 82 796 802 807 42046 .42218 4-'392 172 174 '74 7-745 7.768 7.790 23 22 23 '72-55 173-70 174.86 "5 116 "7 .0610 ,0613 .q5i6 3 3 4 837 836 835 10312.0 10331.0 10350.1 190 191 191 1410.89 I4I9.0I 1427.18 S12 817 823 42566 12741 .43917 175 176 176 7-813 7.836 7.85S 23 22 23 176.03 177.21 178.41 118 120 120 .0620 .0623 .Q626 3 3 3 S34 833 S52 10369.2 10388.4 101.07.6 192 192 '93 1435-41 1443.69 1452.02 82S 833 S39 ■43093 ■43^71 -43 i t9 178 I7S iSo 7.8S1 7.90.1 7.923 23 24 23 179.61 180.82 1 82.04 121 122 123 0629 .0632 .0636 3 4 3 83- 830 829 10426.9 10446.2 10465.6 193 194 194 1460.41 i,|68.S5 '477-35 844 850 855 -13629 .43S09 •4399° 180 181 IS2 7-951 7-974 7-997 23 23 24 183.27 1 84.5 1 185.76 124 125 126 ■0639 .0642 .0645 3 3 4 828 827 826 10.4.85.0 10504.4 10523.9 '9-1 195 195 1485.90 1494.51 1503.18 861 S67 872 .44172 ■44354 -4^538 1S2 1S4 184 8.021 8.044 S.06S 23 24 23 187.02 18S.28 189.55 126 127 129 .0649 .0652 .0656 3 4 3 825 824 823 10543.4 10563.0 10582.7 196 197 197 1 5 1 1 .90 1520.69 152952 879 883 890 .44722 .44908 ■45094 1S6 186 188 S.091 S.115 8.139 24 24 24 190.8.1 192.14 193-44 130 130 J 32 .0659 .0663 .0666 4 3 4 822 821 820 10602.4 10622.1 1 064 1. 9 i97 I'gS igS 1538.42 1547-3S 1556.39 896 901 908 .452S2 -45470 •45659 183 189 190 8.163 S.187 8.211 24 24 24 f94-76 196.08 197.42 132 134 '35 .0670 .0673 .0677 3 4 '4- S19 S18 817 1 066 1. 7 1 068 1. 6 10701.6 199 200 200 1565-47 1574.61 1583.S0 914 919 925 ■45849 46040 .46231 191 191 193 S.235 8.259 8.284 24 25 24 198.77 200.12 201.48 135 136 '38 .06S1 .0684 .0688 3 4 4 816 815 814 1 072 1. 6 1 074 1. 6 1 076 1. 7 200 201 201 1593-05 1602.37 1611.75 932 938 945 .46424 .46618 .46812 194 '94 196 8.308 8-333 S-357 25 24 25 202.86 204.25 205.65 '39 140 142 .0692 .0695 .0699 3 4 4 8'3 812 10781.8 10802.0 10822.3 202 202 203 1621.20 1630.70 1640.27 950 957 963 .47008 .47205 .47402 17 197 197 199 8.382 8.407 8.432 25 25' 25 207.07 208.49 209.93 142 '44 '45 .0703 .0707 .0710 4 3 4 TABLE I.— CONTIMJED. u S{u) Diff A{u) Diff /(») Piff r(«) Diff /?(,.) Diff jiy(») Diff 8io 809 S08 10842.5 10862.8 10883.: 203 204 204 1649.90 1659.60 1669.36 970 976 983 .47601 .4780c .48001 199 201 201 8.457 8.482 8-507 25 25 26 211.38 212.83 214.30 145 147 148 ■0714 .0718 .0722 4 4 4 807 806 S05 10903.G 1 0924. 1 10944.6 205 205 206 1679.19 1689.08 1699.04 989 99G 1003 .48202 .48404 .48608 202 204 204 8-533 8.558 S.584 25 26 26 215.78 217.27 218.77 149 150 '51 .0726 .0730 ■0734 4 4 4 804 803 802 10965.2 10985.8 1 1006.5 206 207 207 1709.07 1719.16 1729.32 I0O£ 1016 1023 .48812 .490 1 8 -49225 206 207 207 S.610 8.63s 8.661 25 26 26 220.28 221.81 22335 153 154 '55 •0738 •0742 .0746 4 4 4 801 800 799 1 1027.2 11048.0 1 1068.8 208 20S 20g 1739-55 1749.84 1760.21 1029 1037 1043 -49432 .49641 .49850 209 209 211 8.6S7 8.713 8.739 26 26 26 224.90 226.46 228.03 156 •57 '58 .0750 ■0754 .0758 4 4 4 798 797 796 1 1089.7 II 1 10.7 11131.7 210 210 210 1770.64 1781.15 1791.72 105) 1057 1065 .50061 -50273 .50486 21-2 2>3 214 S.765 8.701 8.818 26 27 26 229.61 231.21 232.82 160 161 .03 .0762 .0767 .0771 5 4 4 795 794 793 11152.7 11173.8 II 195.0 211 212 212 1802.37 1813.10 1823.89 1073 1079 1087 .50700 .50915 -5113' 215 216 217 8. 844 8.871 8.897 27 26 27 234-45 236.09 237.74 164 .65 .67 .0775 .0779 .0783 4 4 5 792 79' 79c 11216.2 • 1237.5 1 1258.8 213 213 215 1S34.76 iS45.>o 1856.71 1094 IIOI 1 1 16 -5134S .5i5''>6 .51786 218 220 222 8-924 8.951 8.978 27 27 27 239-41 241.09 242.79 168 170 171 .0788 .0792 .0796 4 4 4 78c 7«^ 7S7 11281-1.3 1 1 301.8 11323-4 215 216 216 1867.87 1879.08 1890.36 I 121 1128 1134 .5200S .52231 -52454 223 223 224 9.00s 9-032 9.060 27 28 27 244.50 246.23 247-97 •73 •74 •75 .oSoo .0804 .0S09 4 5 4 786 7fi5 7S4 113450 1 1 366-6 1 1 388.2 216 216 216 1901.70 1913.11 1924-57 1141 I 146 "53 .52678 .52904 -53130 226 226 227 9.087 9-114 9.142 27 28 28 249.72 251.48 253.26 176 178 178 .0813 .0S17 .0822 4 5 4 78 = ;8: 7S1 II 409.8 >i43'-5 11453-3 217 218 217 1936.10 1947-70 1959.36 1160 1166 1172 -53357 -53585 -53813 228 228 230 9.170 9-197 9.225 27 28 28 255.04 256.84 258.65 180 181 182 .0826 .0831 .0835 5 4 5 7SC 71^ 1 1475.0 M 496.8 11518.6 218 218 2l8j 1971.08 1982.87 1994.72 117c II 85 1 192 •54043 •54273 •54504 230 231 232 9-253 9.281 9-309 28 28 28 260.47 262.30 2C4.15 •83 185 186 .0S40 .0845 .0850 5 5 4 18 TABLE ].— CoxTiNUED. u S{u) ] Diff A{„) Diff m 1 Diff T{u) = Diff -B(«) Diff M{n) Diff 777 776 775 1 1540.4 1 1 562. 2 1 1 584.1 218 219 219 2006.64 2018.62 2030.68 1198 1206 1212 •54736 .54969 55203 233 234 23s 9-337 9-365 9-394 28 29 28 266.01 267.88 269.77 187 189 190 0854 0859 0864 774 773 772 1 1 606.0 1 1627.9 1 1649.9 219 220 220 2042.80 2054.98 2067.24 1218 1226 1232 •55438 •55674 •55911 236 237 237 9.422 9.450 9^479 28 •29 28 271.67 273^58 275-50 191 192 193 0869 .0874 .0878 771 770 769 11671.9 1 1693.9 11716.Q 220 221 220 2079.56 2091.95 2104.41 1239 1246 1253 .56148 •56387 .56626 239 239 241 9.507 9-536 9-565 29 29 28 277^43 279.3 8 281.34 195 196 197 0883 .0888 .0893 768 767 766 11738.0 1 1760. 1 1 1782.3 221 222 222 2116.94 2129.54 2142.21 1260 1267 1274 .56867 .5710S •5735^ 241 242 244 9-593 9.622 9.651 29 29 29 283.31 285.30 287.30 199 200 202 .089S .0902 .0907 765 764 763 11804.5 11826.7 11848.9 222 223 222 2i5'i-95 2167.76 2 1 80.64 1281 1288 1295 •57594 •5783S .58083 244 245 M7 9.680 9.709 9-738 29 29 29 289.32 291-35 293.39 203 204 206 .0912 .0917 .0922 762 761 760 11871.1 11893.4 11915.7 223 223 223 2193.59 2206.62 2219.71 1303 1309 i3'7 •58330 ■58577 •58825 247 248 249 9.767 9-797 9.826 30 29 29 295.45 297.52 299.60 207 208 210 .0927 .0932 .•0937 759 758 757 11938.0 1 1960.4 11982.8 224 224 225 2232.88 2216.12 2259.44 1324 1332 '339 •59074 •59324 •59575 250 251 252 9-85S 9.885 9914 30 29 30 301.7c 303-81 305-9^ 211 213 215 .0942 •0947 .0952 756 755 754 12005.3 12027.7 12050.2 224 225 226 2272.83 2286.30 2299.84 '347 1354 136. .59827 .60080 •60334 253 254 255 9-944 9-973 10.003 29 30 30 308.09 310.25 312.42 216 217 218 •0957 .0963 .0968 753 752 751 12072.8 12095.3 12117.9 225 226 226 23>345 2327-14 234091 1369 ^177 '3S4 .60589 .60S45 .61103 256 258 258 10.033 10.063 10.093 30 30 30 314.60 316.80 319.02 220 222 223 -0973 .097S .0984 750 749 748 1 2 140.5 12163.1 12185.8 226 227 227 235475 3368.67 2382.66 1392 1399 1408 .61361 .61620 .6188C 259 260 262 10.123 10.153 10.184 30 31 30 321.25 323-50 325-76 225 226 227 .09S9 -0994 .lOOC c 747 746 745 12208.5 12231.2 12253.C 227 227 22S 2396.74 2410.8c, 2425.12 1415 '423 1432 .62142 .62404 1.62667 t 262 263 265 9 10.214 10.24^ 10.275 30 31 3' 328.03 330-33 332-64 230 231 232 .1G05 .1011 AOlt 6 6 TxVBLE 1.- — Co.vTiNucn. a «("•) Din -1(") Difl I{U) Diff Tin) Dill B(u) Dili M{u] Diff 743 742 12276.; 12299.6 12322.4 229 228 229 2439.44 2453-83 2468.30 ■439 1447 1456 .62932 .63198 •.63464 266 266 26S 10.306 10.336 10.367 30 31 3' 334-9^ 337-3C 339-65 234 235 237 .1022 .1027 -1033 5 6 5 741 740 739 12345-3 12368.2 12391.1 229 229 230 24S2.86 2497.49 2512.21 1463 1472 14S0 .63732 .64001 .64271 27 i 10.398 10.425 10.46c 3' 31 3' 342.02 344-4' 346.81 239 240 242 .103S 6- .1044 6 .1050] 5 738 737 736 12414.1 12437-I 1 2460. 1 230 230 231 2527.01 2541.89 2556.86 1488 '497 1505 .64542 .64814 65087 272 273 274 10.491 10.522 10.554 31 32 3' 349.23 351-67 354. '.2 244 ?45 247 .1055 .1061 .1067 & 6 5 735 734 733 12483.2 12506.3 12529.4 231 231 232 2571.91 2587.04 2602.25 ■5'3 1521 1530 .6536. •65037 -65913 276 276 278 10.585 10616 10.648 31 32 31 356.59 359.0S 361.58 249 250 252 .1072 .1078 .1084 6 6 6 732 731 730 12552.6 12575-8 13599.0 232 232 233 2617.55 2632.94 2648.41 ■1539 1547 1556 .6S191 .66470 .66750 279 280 281 10679 10.71 1 ' 0.743 32 32 32 364.10 366.64 369.19 254 255 257 .1090 .1096 .U02 & 6 6 729 728 727 12622.3 12645.6 12668.9 233 233 234 2663.97 2679.61 2695.34 1561 '573 1582 .67031 •673 '3 .67596 282 283 285 ' 0.77.5 10.807 ' 0.839 32 32 32 371.76 374.35 376.96 25& 261 263 .1108 .1114 1 120 6 6 6 726 725 724 12692.3 127 1 5.6 12739.0 233 234 235 2711.16 2727.07 2743.07 1591 1600 i6og .67S81 .68167 .68454 286 287 288 10.871 10.903 10.936 32 33 32 379-59 382.23 384.89 264 266 268 .1126 •"33 -"39 7 6 6 723 722 721 12762.5 1278U.0 12809.5 235 235 236 2759.16 2775.33 2791.60 .617 1627 1636 .68742 .69031 .69322 289 291 292 10.968 1 1. 001 11.033 33 32 33 387--57 390.26 392-98 269 272 273 .1145 .1151 .1158 6 7 6 720 719 7.8 12833.1 1 2856.7 12880.3 236 236 236 2807.96 2824.41 2840.96 1645 1655 1664 .69614 .69907 .70201 293 294 295 1 1 .066 11.099 1 1.132 33 33 33 395-71 398.46 401.23 27s 277 279 .1164 .1170 .1177 6 7 6 717 716 71s 12903.9 12927.6 1 295 1. 3 237 23« 2857.60 2874.33 2891.15 ^^73 1682 1692 704.96 70793 .71091 297 298 299 [1.165 11.198 11.231 33 33 33 404.02 1-06.83 409,65 281 282 28s .1183 .1189 .1196 6 7 f 714 713 712 12975.1 12998.9 J 3022.7 238 238 238 1 2908.07 2925.08 2942.19 1701 1711 [720 -71390 71691 71993 301 302 303 1 1:264 11.297 "-330 33 33 34 Ti-2.50 tiS-37 M8.26 287 289 290 .1:02 1209 I2l6 7 7 6 TABLE I.— Continued. a S{u) Di£f A(u) Dift /(■«) Diff Tiu) Diff Biu) Die Diff 7" 710 709 13046.5 1 3070-4 13094-3 239 239 240 2959-39 2976.69 2994.09 1730 '740 1749 .72296 .72600 -72905 304 305 307 U.364 11.398 11.43a 34 34 33 421.16 424.09 427.04 293 29s 297 .1222 .1229 .1236 7 7 7 70S 707 706 13H8-3 13H2.3 13166.3 240 240 240 3011.58 3029.17 3046.86 '759 1769 1780 .73212 -73520 ■73830 308 310 3" 11.465 11-499 11.533 34 34 34 430.01 433-00 436.00 299 300 303 -'243 .1249 .1256 6 7 7 705 704 703 13190.3 1 32 '4-4 13238-5 241 241 242 3064.66 3082.55 3100.54 1789 '799 1810 .74141 -74453 .74766 3'2 313 315 11.567 1 1. 601 1 1.636 34 35 34 439.03 442.08 445.16 305 308 309 .1263 •V270 .1277 7 ■7 7 702 701 700 13262.7 13286.9 i33i'-i 242 242 242 3118.64 3136.84 3'5S-'4 182c 1830 1841 .75081 -75397 -757'5 3.6 3'8 319 1 1 .670 11.704 ".739 34 35 35 448.25 451.36 454.49 311 313 315 .1284 .1291 .1298 7 7 7 €99 698 697 13335-3 13359-6 13383-9 243 243 244 3173-55 3192.06 3210.67 1 85 1 1861 1872 .76034 -76354 .76675 320 321 323 11.774 11.809 11.844 35 35 35 457.64 460.82 464.02 318 320 322 .1305 .I3'2 -13'9 7 7 7 696 694 13408.3 '3432.; 1 3457- 1 244 244 245 3229.39 3248.22 3267.15 1883 '893 1904 .76998 ■77322 .77648 324 326 327 11.879 11.914 11.949 35 35 35 467.24 470.48 473-75 324 327 329 -1326 -1334 -1341 8 7 7 «93 692 691 1 348 1.6 13506.1 13530-6 245 245 246 3286.19 3305-33 3324-58 1914 1925 "937 •77975 -78304 -78634 329 330 332 11.984 12.020 12.055 36 35 36 477-04 480.35 483.68 331 333 336 .'348 .1355 .1363 7 8 7 690 689 €88 I35S5-2 '3579-8 1 3604.4 246 246 247 3343-95 3363-42 3383.00 1947 1958 1970 .78966 .79299 -79633 333 334 336 12.091 12.126 12.162 35 36 36 487.04 490.42 493.83 338 34' 343 .1370 .1378 •1385 8 7 8 687 686 685 13629.1 '3653-8 13678.6 247 248 248 3402.70 3422.50 344242 1980 1992 2003 -79969 .80306 .80645 337 339 340 12.198 12.234 12.270 36 36 36 497.26 500.71 504.19 345 348 350 •1393 .1400 .1408 7 8 3 684 683 682 13703-4 13728.2 '37S3-' 248 249 249 3462.45 3482.60 3502.86 2015 2026 2038 .80985 .81327 .81670 342 343 345 12.306 12.342 12.379 36 37 36 507.69 511.22 514-77 353 355 357 .1416 .1423 • 143' 7 8 8 681 680 679 13778.0 13802.9 13827.9 249 250 250 3523-24 3543-73 3564-34 2049 2061 2074 .82015 .82362 .82710 347 348 349 1 12.415 12.452 12.489 37 37 37 518-34 521.94 525.56 360 362 36s .1439 .1447 .1455 8 8 8 TABLE 1. — CoxTiKUED. u S{u) Difl A{n) DiQ H") Diff T{u) Diff £(«) Diff M(n) Diff 678 677 676 13852.9 •3877-9 13903.0 250 251 251 3585-07 3605 91 3626.88 2084 2097 2108 .83059 .83410 .83762 351 352 354 12.526 12.563 12.600 37 37 37 529.2! 532.88 536.58 367 370 373 ■1463 .1471 .1479 8 8 8 675 674 673 1 3928. 1 13953-3 '397«-5 252 252 252 3647.96 3669.17 3690.50 2121 2133 2144 .84116 -84472 .84829 356 357 359 12.637 12.67s 12.712 38 37 38 540.31 544.06 547.84 375 378 381 -1487 .1496 .1504 9 8 8 672 671 670 14003.7 14029.0 14054.3 253 253 253 3711.94 3733-51 3755-21 2157 2170 2182 .85188 -85549 .85911 3'^« 362 363 12.750 12.787 12.825 37 38 38 551.65 555-48 559-34 383 386 3S9 .1512 .1521 -1529 9 8 8 669 668 667 14079.6 14105.0 14130-4 254 254 255 3777-03 3798.98 3821.05 2195 2207 221C .86274 .86639 .87006 365 367 369 12.863 12.901 12.939 38 38 38 563.23 567.14 571.08 391 394 397 -1537 .1546 .1555 9 9 8 666 665 664 141559 14181.4 14206.9 255 255 256 3843-24 3865.57 3888.02 223.3 224=; 225F •87375 ■87745 .88117 370 372 373 12.977 13.015 13-053 38 38 39 575-05 579-05 583-08 400 403 405 -1563 .1572 .1581 9 9 8 663 662 661 14232.S 14258.1 14283.7 256 256 257 3910,60 3933-31 3956.16 2271 2285 2297 .88490 .88866 -89243 376 377 379 13.092 13-130 13.169 38 39 39 587-13 591.21 595-32 408 411 414 .1589 .1598 .1607 9 9 9 66c 659 658 14309.4 I4335-I 14360.9 257 258 258 3979- '3 4002.24 4025.48 2311 2324 2338 .89622 .90002 .90384 380 3S2 384 13.20S 13-247 13.286 39 39 40 599.46 603.63 607.82 417 419 423 .1616 .1625 .1634 9 9 9 657 656 655 14386.7 144 1 2.6 14438.5 259 259 259 4048.86 4072.37 4096.01 235> 2364 2378 .9076S -91153 -9«54i 385 388 389 13.326 13-365 13.404 39 39 40 612.05 616.31 620.60 426 429 432 .1643 .1652 .1661 9 9 10 654 65.^ 652 14464.4 14490.4 14516.4 260 260 260 4119.79 4i«-7i 4167.77 2392 2406 2419 .91930 .92321 .92715 391 394 395 13-444 13-484 13-524 40 40 40 624.92 629.27 633-65 435 438 441 .1671 .1680 .1689 9 9 10 651 650 649 14542.4 14568.5 14594.6 261 261 262 4191.96 4216.30 4240.78 2434 2448 2462 .93110 -93506 -93904 396 398 400 '3-564 13.604 13-644 40 40 40 638.06 642.50 646.97 444 447 45 • .1699 .1708 .1718 9 'O 9 648 647 646 14620.8 14647.0 14673 2 262 262 263 4265.40 4290.16 43 15-07 2476 2491 2505 .94304 .94706 .95110 2S 402 404 406 13-684 •3-725 13.766 41 41 40 651.48 656.02 660.60 454 J58 460 .1727 -1737 .1746 ID 9 10 TABLE I.— CoxTiNUED. S{u) Diff ^(«) Diff .I{u) Difi T{u) Diff B(u) Difl 31 {u) Diff 645 644 643 642 641 640 639 638 6i7 636 635 634 633 632 631 629 62S 627 626 625 t)24 623 622 621 620 619 618 617 616 615 614 613 14699.5 14725.9 14752-3 14778.7 14805 1483 1. 6 14858.1 14884.7 14911 14938.0 14964.7 14991.4 15018.2 15045.0 ■50? '.-9 15098. 15125. 15152. 15179.8 15206.9 15234.C 15261.2 15288.4 15315-7 15343-0 15370-3 15397-7 15425.1 15452.6 15480.1 264 264 264 264 265 265 266 266 267 267 267 268 268 269 269 270 270 270 271 271 72 272 273 273 273 274 374 275 275 276 15507.7 15535-3 15563.0 276 277 277 4340.12 4365-32 4390.67 4416.16 4441.81 •4467.60 4-493-55 4519.64 4545.89 4572.30 4598.86 4625.57 4652.44 4679.47 4706.65 4734.00 4761.51 4789.18 4817.02 4845.02 4873.J8 4901.51 4930.00 4958.67 4987.50 5016.51 5045.69 5075.04 5104.57 5134-27 5164.15 5194.21 5224.44 2520 2535 2549 2565 2579 2595 2609 2625 2641 2659 2671 2687 2703 2718 2735 2751 2767 2784 2800 2816 2833 2849 2867 883 2901 2918 293 s 2953 2970 2988 3006 3023 3042 •95516 -95923 -96333 .96745 .97158 -97574 -97991 .98410 .98831 .99254 .99680 1.00107 1.00536 1 .00967 •.01401 1.01837 1.02274 1.02713 1-03155 1.03598 1 .04044 1.04492 1.04943 1-05395 1.05850 1.06307 1.06766 407 410 412 413 416 4'7 41.9 421 423 426 427 429 431 434 436 437 439 442 443 446 448 451 452 455 457 459 461 1.07227 1 .07690 1.08 463 466 56468 1.08634 1.09095 1.09568 471 473 475 13.806 13-847 13.888 13.929 13-971 14.012 14-053 14.095 14-137 14.179 14.221 14.263 14-305 14.348 14.390 14-433 1.4-476 14.519 14.562 14.60 14648 14.692 14-735 14.779 14-S23 14.867 14.91 14.956 15. coo 15-045 15.090 15-135 15.1S0 41 41 41 42 41 41 42 42 42 42 42 42 43 42 43 43 43 43 43 43 44 43 44 44 44 44 45 44 45 45 45 45 45 665.20 669.84 674.51 679.21 683.95 688.72 693.52 C98.36 703-24 708.15 713-09 718.07 723-09 728.15 733-24 464 467 470 474 477 480 484 488 491 494 498 502 506 509 512 738.36516 743-52521 748.73 524 753-97 759-25 764-57 769.92 7.75-32 780.75 786.22 791.7 797.28 80: S08.51 814.19572 528 532 535 540 543 547 551 555 56c 563 819.91 825.67 831-47 576 580 585 1756 1766 .1776 .1786 1796 1806 1816 .1827 -1837 .1847 .1858 .1S68 .1879 1890 I goo 1911 .1922 19: ■1944 1955 1966 1977 19SK 1999 201 1 .2022 033 -2045 2057 .206S .20S0 092 .2104 TABLE I.— CosTiNUEi). S{,c) Di£f A{u) Diff J(«) Diff T(u) Di£f £{_u) Diff M{u) Diff 612 611 610 6og 608 607 606 605 604 603 602 601 600 599 598 597 596 595 594 593 .592 591 590 589 588 587 585 585 584 583 582 581 5 So 15590.7 15618.4 15646.2 15674.C •5701.9 15729.8 15757-8 15785.8 15813.9 15842.0 1 5 870. 1 15898.3 15926.6 15954.9 15983- 16011.6 1 6040. 1 16068.6 16097.1 16125 7 16154.4 16183.1 i62ii.€ 16240.6 16269.4 [6298.3 16327.2 16356.2 16385.2 16414-3 16443.4 16472.6 16501.8 277 278 278 279 279 280 280 281 281 281 282 283 283 283 284 285 285 285 286 287 287 87 288 288 289 289 290 290 291 291 292 292 293 5254.86 5285.46 5316.24 5347-21 5378.36 5409-7 5441.24 5.472-95 5504.86 3536.96 5569.26 5601.75 5634-43 667.31 5700.40 3329 3060 3078 3097 3115 3135 3153 3171 3191 3210 3230 3249 3268 3288 3309 1 10043 1. 10520 480 I.I 1000 477 480 482 I.I 1482 I.I 1966 486 1. 12452 484 486 489 3733-69 5767.18 5800.87 5834-76 5868.85 5903.16 5937-67 5972.39 6007.32 6042.47 6077.83 6113.41 6149.20 6185.22 6221.46 6257.9 6294.6 6331.52 3349 3369 3389 3409 3431 3451 3472 3493 3515 3536 3558 3579 3602 3624 3646 3669 3691 3714 1 . 1 294 1 f -13433 1. 1 3927 1. 1 4424 1.14923 1.15425 1.15929 1-16435 1 . 1 6944 1.17456 1.17970 1.18487 1.19006 1.19528 1.20053 1.20580 1.21 110 1.21643 1.22178 1.22716 23257 1.23801 1.24348 1.24897 1.25449 26004 26562 24 492 494 497 499 502 504 506 509 512 SH 517 519 522 525 527 530 533 535 38 54' 544 547 549 552 555 558 ;6i 15.22 15.270 i5-3«6 15.361 15.407 •S-453 15.499 15.546 15.592 15.638 5.68s 15-732 15-779 15.826 '5-873 15.921 5-968 16.016 16.064 16.113 16.161 ) 6.209 16.258 16.307 16.356 16.405 16.454 16.504 '6.553 16.603 16.653 16.704 16.754 46 46 47 46 46 47 47 47 47 47 48 47 48 48 49 ■48 48 49 49 49 49 49 50 49 50 50 51 50 5' 837-32 843.20 849.13 855.10 861.12 867 873-30 87945 885.65 891.90 898.20 904.54 910.92 917.35 923 588 593 597 602 607 611 6.5 620 625 630 634 638 643 649 8^653 930-37 936-95 943-58 950.26 957.00 963.78 970.61 977-49 984.42 991.41 998.46 1005.6 1012.7 1019.9 1027.2 '034.-5 1041.8 1049.2 658 663 668 674 678 683 688 693 699 705 709 71 72 73 73 71 74 75 2116 2128 .2140 2152 2165 .2177 2190 .2203 2215 .2228 .2241 .2254 .2267 .2280 2293 2307 2320 2334 -2347 .2361 2374 2388 .2402 .2416 2430 2444 2459 2473 2488 2502 2517 2532 2547 TABLE I.— CoxTiNUED. SCO Difl A{n) Dift /(«) Diff T{„) DM B{ii) Dift 31{ti) Diff 5;s S7'-: 57/ 57C 575 574 57 571 570 569 568 567 566 565 564 563 562 561 56c 555 558 557 55^ 555 554 553 552 55' 55c 54?' 547 16531.1 16-560.4 16589.8 16619.2 16648.7 16678.^ 16707.8 •6737-4 16767.1 293 294 Z94 295 295 296 296 297 298 16796.9 298 t6826.7 16856.6 168S6.5 1 69 1 6.4 1 6946.4 [6976.5 17006.6 17036.8 17067.0 17097.3 17127.6 17158.0 17188.4 17218.9 17249.4 17280.0 '73 1 07 173414 17372.2 17403.0 299 299 299 300 301 301 302 302 30 303 304 304 305 305 306 307 307 308 308 309 i 7433-9 17464.8 17495.8310 309 310 636S.66 6406.0 1 '^443-63 3481.46 5519.52 0557.82 6596.36 6635.14 6674116 67'i3-4 6752.93 6792.68 6832.68 6872.93 691 3-43 69.54.18 6995.19 7036.46 7077.99 7119.78 7161.8 7204.15 7246.73 7289.58 7332-7' 7376.11 4-19.78 7463.744423 3735 3762 3783 3806 3830 3854 3878 3902 3926 3951 3975 400c 4025 4050 4075 4101 4127 4153 4179 4205 4232 4258 4285 4313 4340 4367 4396 7';o7.97 7552.48 7597.28 7642.36 687-73 4451 44S0 4508 4537 i566| 1.27123 1.27687 1.28253 1.28823 1-29396 1.29971 1.30550 i.3«i3' 1.31716 1-32304 1.32895 '•33489 1.34086 1.346S6 1.35290 1-35897 1.36507 1.37120 '-37736 '-3835C 1.3S979 1.39606 1.40236 1 .40869 1.41506 1.42146 1.42789 1.43436 1 .44087 1. 4474 1 1.45 i99 1 .46060 1.46725 25 564 566 570 573 575 579 581 585 58S 59' 594 597 600 604 607 610 613 616 620 623 627 630 633 637 640 643 647 651 654 658 65 1 665 669 16.805 16.855 1 6.906 16.958 17.009 17.060 17.1 12 17.164 17.216 17.268 17.320 '7-373 17.425 17.478 17-53' 17.584 17.638 17.691 17.745 17.799 17-853 17.908 17.962 18.017 18.072 8.127 18.183 8.23S 18.294 '8.350 1 8.406 18.462 18.519 5, 53 S3 54 53 54 54 54 55 54 55 55 55 56 55 56 56 56 56 57 57 1056.7 1064.3 1071.9 1079.5 1087.2 1095.0 1 102.8 II 10.7 1 1 18.7 1126.7 1 134.8 1142.9 1151 1 159.4 1 167.7 176.1 II 84.6 1193.1 1 201. 7 1210.3 I2ig.i 1227.9 1236.7 1245.7 1254.7 1263.8 1272.9 1282. 1291.4 1300.8 1310.2 1319-7 1329-3 56 .2578 2593 .260S 2624 2640 -2655 .2671 .2687 2703 2719 2735 275' 2768 2784 .2801 .2818 835 .2852 2869 2886 2904 2921 2939 •2957 2975 2993 3011 3030 3048 3067 3085 3104 TABLE 1. CONTINDED. 546 545 544 543 542 54" 540 53<3 538 537 536 535 534 533 532 53" 530 529 528 527 526 525 5 523 52. 52 320 519 518 51; S16 5>5 514 S{u) Difl 17526.8 '7557-9 17589, 17620.3 17651.6 17682.9 17714-3 17745.8 17777-3 17808.9 17840.5 178^2.2 17903.9 17,935-7 17967.6 17999.5 18031 18063.5 18095.6 1S127.8 1 8 1 60.0 18192.3 1S224.7 18257 18289.6 18322.1 18354-7 18387.4 18420. 18452.9 18485.7 18518.6 18551.6 3" 312 312 313 313 314 315 315 316 316 317 317 318 319 319 320 320 321 322 32- 323 324 324 325 325 326 327 327 78 329 330 331 ^(«) 7733-,39 7779-34 7825.58 7872.12 79 1 8.96 7966. 1 2 8013.55 8061.30 8109.36 8157-73 8206.41 8255-41 8304.73 8354-36 8404.32 8454.61 8505.22 8556.16 8607.44 S659.06 S711.01 8763,30 8815.94 886S.92 8922.25 8975-93 9029.97 9084.36 9139.11 9194.23 9249.71 9305.56 9361.79 4595 4624 4654 4684 4716 4743 4775 4806 4837 Dift 4900 4932 4963 4996 5029 5061 5094 5128 5162 5195 5229 5264 5298 5333 5368 5404 5439 5475 5512 5548 5585 5623 5660 J(«) Di£f 1-47394 1.48066676 1.48742 672 676 680 1.49422 1-50106 1.50793 684 687 691 1.5 1484 695 1-5-179 1.52878 1-53581 1.54287 1.54998 '■55713 1.56431 i-57'5<4l727 1.57881 1.58612 '-59347 1 .60086 1.60830 1.6157S 1.62330 1.63086 1.63847 1. 646 1 2 1.6538 1.66155 1.66933 1.67716 1.68504 1.69296796 1.70092 802 1 .70S94 S06 26 699 703 706 71 7'5 718 723 73 735 739 744 74S 75 756 761 765 769 774 778 783 788 792 T(u) Diff 18.576 18.633 18.690 18.747 18.805 18.863 18.921 18.979 19,038 19.096 19.155 19.215 19.274 19-334 '9-394 19-454 19.514 '9-574 19-635 19.696 19-757 19.819 19.881 '9-943 20.005 20.067 20.130 20.193 20.256 20.319 20.383 20.447 20.51 1 B{v) Diff 339-0 348.7 358.6 368.5 378.4 388.5 398.6 408.9 419.2 429.5 440.0 450.6 467.2 472.0 483.8 493-7 504.7 5IS-7 526.9 538 549-5 561 572-5 584 595.9 6077 619.7 631..7 643.8 656.0 668, 680.8 1693.4 M(u) Diff -3123 3'42 .3161 .3181 .3200 3220 3240 .3260 .3280 .3301 -3321 -3342 •3363 -3384 -3405 -3427 -3448 -3470 ■349 •35'3 ■3535 ■355 .3580 .3602 •3^J2S 364S 367' 3694 37'8 3741 3765 ■3789 3813 TABLE I.— CoNnNUED; u S{u) Di« ^00 Difl- liu) Diff T{u) Difi £{u) Difi J/(«) Diff 513 512 511 18584.7 18617.8 1 8651.0 331 332 332 9418.39 9475-38 9532.74 569s 5775 1. 7 1 700 1.72510 1.73326 810 816 820 20.575 20.640 20.705 65 65 65 1706.0 1718.7 1731.6 127 129 130 .3838 .386a -3887 24 25 25 510 509 508 18684.2 1S717.5 18750.9 333 334 334 9590.49 9648.62 9707.15 58 13 5853 5891 1.74146 1.74971 1.75801 825 830 835 20.770 20.835 20.901 65 66 66 1744.C 1757.7 1770.9 131 132 133 • 3912 -3937 .3962 25 25 26 507 506 505 18784.3 18817.8 18851.4 335 33^5 336 9766.06 9S25.38 9885.09 5935 5971 6012 1.76636 1.77476 1.78321 84c 845 85c 20.967 21.033 21.099 66 66 67 17842 1797.6 1811. 1 134 '35 .36 .3988 .4014 .4040 26 26 26 S04 503 502 18885.0 18918.7 18952.5 337 338 338 ■9945.21 10005.74 10066.67 6053 6093 6134 1.79171 1.80026 1.80886 855 S6c 865 21.166 21.233 2i;300 67 67 67 1824.7 1838,5 1852.3 138 138 140 .4066 .4092 .4119 26 27 27 501 500 499 189S6.3 19020.2 19054.2 339 340 340 10128.01 10189.78 10251.9 6177 62-19 626 1.81751 1.82622 1.83498 871 876 88] 21.367 21.435 21.503 68 68 69 1866.3 1880.4 1 894.6 141 142 143 .4146 •4173 .4200 27 27 28 49S 497 496 19088.2 15122.3 19156.4 341 34 J 342 10314.5 10377.6 104.41.0 63. 634 639 1.84379 1.85265 1.S6157 886 S92 897 21.572 21.641 2 1.7 10 69 69 69 1908.9 1923.4 1938.0 145 146 '47 .4228 .4255 ■4283 27 28 29 495 494 493 19790.6 19224.9 19259-3 343 344 345 10504.9 10569.3 10634.1 64 1 648 652 1.87054 1.87957 1.S8865 903 908 913 21.779 21.848 21.918 69 70 70 1952.7 1967.5 1982.5 148 15c '5' ■43 '2 ■4340 •4369 28 29 29. 49^ 491 490 r9293.8 19328.3 19362.9 345 346 347 10699.3 10765.0 10831.1 657 661 665 1.89778 1 .90697 1.91622 919 925 930 21.988 22.058 22.128 70 70 71 1997.6 2012.8 2028.2 152 154 155 ■439S •4427 ■4456 29 29 30 489 488 487 19397.6 19432.3 19467.1 347 348 349 L0897.6 10964.7 Ti 032.2 671 675 679 1.92552 1.93488 1.94430 936 942 948 22.199 22.270 22.341 71 71 72 2043.7 2059.3 2075.1 156 '58 '59 .44S6 4515 ■4545 29 30 31 486 485 484 19502.0 19536.9 19572.0 349 351 35' 11 100. 1 II 168.6 "237-5 685 689 695 1.95378 1.96332 1.97292 954 960 966 22.413 22.485 22.557 72 72 73 2091.0 2107.1 2123.3 161 162 '63 .4576 .4606 ■4637 30 31 31 483 482 481 19607.1 19642.2 19677.5 351 353 353 1 1 307.0 11376.9 1 1447.2 699 703 709 1.98258 1.99230 2.00207 27 972 977 9S3 22.630 22.703 22.776 73 73 73 2139-6 2156.0 2172.6 164 166 168 .4668 .4700 ■4731 32 31 32 TABLE I.— CONTIMED. u S{,c) Diff A{h) Diff I(u) Diff T(u) Diff B{u) Diff 3Iin) Difl 480 479 478 19712.8 19748.2 19783.6 354 354 355 11518.1 11589.4 11661.3 713 719 724 2.01 190 2.02180 2.03176 990 996 1003 22.S49 22.923 22.997 74 74 74 2189.4 2206.3 2223.4 169 171 172 -4763 •4795 4827 32 3^ 33 477 476 475 1 98 19. 1 19854.7 19890.4 356 357 358 117337 1 1 S06.6 1 1880.0 729 73+ 739 2.04179 2.05188 2.06203 1009 1015 ^1022 23.071 23.146 23.221 75 75 75 2240.6 2257.9 2275.4 173 175 177 .4860 -4893 .4926 33 33 33 474 473 472 19926.2 19962.0 19997.9 358 359 j6o '1953-9 12028.4 12103.4 745 750 755 2.07225 2.0S253 2.09288 102S 1035 1041 23.296 23-372 23.448 76 76 76 2293.1 2311.0 2329.0 '79 180 ;8i ■4959 •4993 .5027 34 34 34 471 470 469 20033.9 20070.0 20106.2 362 362 12178.9 12254.9 I233I-5 760 766 77 i 2.10329 2.11376 2.12430 1047 1054 1061 23-524 73.601 23-678 77 77 77 2347- ' 2365-4 2383-9 '83 185 187 .50D1 .5096 ■S'31 35 35 35 46a 467 466 20142.4 20178.7 202 1 5.0 363 363 3<5S 12408.6 12486.3 12564.6 777 783 788 2.13491 2.14559 2.15635 1068 1076 1082 23-755 23-833 23.911 78 78 78 2402.6 2421.4 2440.4 188 100 192 .5166 .5202 -5237 36 35 36 465 464 463 20251.5 20288.0 20324.7 365 367 367 12643.4 12722.8 12802.7 794 799 S05 2.16717 2.17806 2. 1 8902 1089 1096 1104 23.989 24.06S 24.147 79 79 79 2459.6 2479.0 2498.5 194 195 197 ■5273 -5310 -5347 37 37 37 462 461 460 20361.4 20398.1 20435.0 367 369 369 12883.2 12964.3 1 3045.9 8ir S16 S22 2.20006 2.21116 2.22233 1110 1117 1124 24.226 24.306 24 386 80 80 80 2518.2 2538.0 2558.0 198 200 202 -5384 .5421 -5459 37 38 38 459 45S 457 20+71.9 2050S 9 20546.0 370 37' 371 13128.1 13211.0 '32944 829 834 841 2.23357 2.24489 2.25629 1132 1 140 1147 24.466 24-547 24.628 81 81 82 2578.2 2598.6 2619.2 204 206 208 -5497 ■5535 -5574 38 39 39 456 455 -154 20583.1 2062O.il 20657.7 373 373 374 13378.5 13463-3 13548.6 S48 853 859 2.26776 2.2793' 2.29094 "55 1 163 117! 24.710 24.792 24-S74 82 82 82 2640.0 2661.0 2682.2 210 212 213 .5613 -5653 .5692 40 39 40 453 452 451 20695.1 20732.6 20770.2 375 376 377 '3634-5 13721.1 13808.3 866 872 878 2.30265 2.3'443 2.32628 1178 118^ "93 24.956 25.039 25.122 83 83 84 2703.5 2725.1 2746.9 216 218 219 •5732 -5773 •5814 41 41 41 4S0 ^49 448 2OS07.9 20S45.6 208S3.4 377 378 380 13896.1 13984.6 1 4^7 3-7 885 891 89S 2.33821 2.35022 2.36232 a 120! I2I0 I21S 3 25.206 25.290 25-374 84 84 85 2768.8 27910 2813.4 222 224 226 •5855 .J896 •5938 41 43 43 TABLE I.— Continued. S{„) Did AW Dill !hi) Difl- T („) Dil B(„) Difl M{v) Difr 447 446 445 444 443 442 441 440 439 438 437 436 435 434 43 43 43 430 429 428 4^7 426 425 424 423 422 421 420 419 418 417 416 415 2og2 1 .4 20959.4 20997.4 035.6 073-9 1 12 150.7 189. 227. S 266.5 305-3 344-2 422.2 461.4 500.6 54Q-0 579.4 618.9 658. 698. 738.0 777-9 817.8 857-9 898.1 938.4 21978.7 22019. 1 22059.6 22100.2 22140.9 22181 3 So j8o 382 3S3 383 3S5 385 3S6 387 388 389 389 391 39 392 394 394 395 396 397 398 399 399 401 40: 403 403 404 405 406 407 409 409 4163-5 4254.0 4345-' 4437-0 4529-5 4622.7 4716.6 4811 4906.5 5002.5 5099-3 5196.8 5295.0 5394-0 5493-7 5594. 5695.4 5797-3 5900.0 6003.5 6107.9 6213.1 6319.1 6425.9 6533-5 6641.9 6751.2 6861.3 6972.2 7084. 1 7196.8 73'o.S 7425.01 905 911 919 925 932 939 946 95 960 968 975 990 997 1005 1012 1019 1027 1035 1044 1052 1060 1068 10; 1084 109^ 1 101 1 109 II 1127 "37 1145 1155 2-37450 2.38(376 2.39911 2.41154 2.42405 2.4366; 2.44933 2.46209 2.47494 3.4878S 2.50091 2.51404 2.52726 2.54057 2-55397 2.56746 2.58104 2.59471 2.60848 2.62235 2.63632 2.65039 2.66456 2.67883 2.69320 2.713/6" 2.72225 2.73692 .75169 2.76658 2.78158 2.79668 2.81 190 29 1226 123 1243 1251 i2bo 1268 1276 1285 1294 '303 13 1322 IJ3I 1340 '349 •358 '367 1377 1387 '39; 140; 141; 142; '43; '44; 145! 146; '47; I48< 150; .151C 152. .153. 25-459 25.544 25 629 25-715 25.801 25.888 1 25-975 26.062 26.150 26.238 26.327 26.416 26.505 26.59 26.68 26.776 26.867 26.959 -V-05 1 V-143 27.236 27.329 27-423 -'7-517 27.612 27.707 27-803 27.899 27-995 38.092 ;8.i89 18.287 28.385 2836.0 2858.8 2881.9 2905.1 2928.6 2952.3 2976.2 .5000.3 3024.6 3049.2 3074-J 3099- • 3 ' 24-4 3150.0 3'75-8 3201.9 1228. 2 3254-7 3281 5 3308.6 33J5-9 3363-6 339'-5 3419-6 3448.1 3476.8 3505-8 3535-0 3564-6 3594-4 3624.6 3655- 228 231 232 235 237 239 241 243 24b 249 250 253 25^ 25S 261 26^ 26I 26f- 271 273 277 279 281 285 87 290 292 302 305 307 3685.8I311 .5981 .6023 .6066 .61 'O .6154 6198 6243 6288 -6333 •6379 .6425 6472 •6519 6567 6615 6663 6712 676 .6812 .6862 6913 6965 .7017 7069 .712 .717 7229 7283 -7338 7393 7449 •7505 .7562 TABLE 1. — ( ;o.vTi\UED. S,„) Diir A{n) DLi I(u) Diff T(u) Dift £(u) Diff M(ii} Diff 414 413 412 41 410 409 408 407 406 40s 404 403 402 401 400 22222.7 22263.7 22304.8 2346. 2387-4 22428.8 22470.4 22512.0 325537419 22595.6 22637.5 22679.6 22721.8 22764.0 22S06.4 410 41 413 413 414 416 416 417 419 42 422 42 424 424 17540.5 17656.8 17774.1 17892.2 1801 1S13: 18252.4 '83744 18497.4 8621.4 18746.4 18872.3 18999.3 19127-3 19256.2 1 163 i'73 /181 1191 1200 I2tl 1220 1230 1240 1250 259 1270 1280 1289 1300 2.82723 2.84267 2.85822 2.87 ^8f 2.88965 2.90554 2.92155 2.93768 95393 2.97030 2.98679 3.00341 3.02015 3.03701 3-05399 SO 1544 155s 1566 1577 .589 160 16,3 1625 1637 1649 1662 1674 1686 1698 1710 28.484 28.583 28.683 2S.783 28.884 28.985 9.087 29.189 29.292 29-395 29.499 19.603 19.708 19.813 29.919 99 100 100 lOI 101 102 102 103 103 104 104 105 105 106 106 3716-9 3748-3 3780.0 3812.0 3844-3 3«76-9 3909.9 3943-3 39770 401 i.o 4045.4 4080. 1 41 1 5-2 4150.6 44 86.4 314 317 320 323 326 330 334 337 340 344 347 35' 354 358 361 .7619 .7677 -7736 7795 .785 79 IS 7976 .8037 .8099 .816, .8226 .8290 8355 .8420 8486 S8 59 59 60 60 61 61 62 63 64 64 65 65 66 66 TABLE I.- —Continued. — An-riUary A. 2 1200 J, J. 1250 J. A 1300 J. J. 1350 J. J„ 1400 J. 4. 400 5O0 6oo .0092 .0116 .0140 24 -4 25 7 8 10 .0085 .0108 .0130 23 22 23 6 8 9 .0079 .0100 .0121 21 21 21 6 7 9 .0073 .0093 .0112 20 19 20 ■5 7 8 .0068 .00S6 .0104 18 18 19 5 6 7 700 800 QOO .0165 .0191 .0217 26 26 26 12 14 16 •0153 .0177 .0201 24 24 24 1 1 13 '5 .0142 .0164 .0186 22 22 23 10 12 13 .0132 .0152 .0173 20 21 21 9 10 12 •0123 .0142 .oi6i 19 19 20 8 9 II 1000 1 100 1200 .0243 .0270 .0297 27 27 27 18 20 22 .0225 .0250 .0275 25 25 26 16 18 20 .0209 .0232 ■0255 23 23 24 '5 16 18 .0194 .0216 .0237 22 23 23 13 15 16 .0181 .0201 .0221 20 20 21 12 14 •5 1300 1400 1500 .0324 •0352 .0380 28 28 28 23 25 27- .0301 .0327 •0353 26 26 27 22 24 26 .0279 •0303 .0327 24 24 25 19 21 22 .0260 .0282 .0305 22 23 23 18 19 20 .0242 .0263 .0285 21 22 22 17 •9 20 1600 1700 1800 .0408 .0436 .0465 28 29 30 28 29 31 .0380 .0407 •0434 27 28 28 29 30 .0352 .0378 .0404 26 26 26 24 26 27 ■0328 •0352 •0377 24 25 25 21 24 25 .0307 .0329 .0352 22 23 23 21 22 24 1900 2000 JJIOO ■0495 .0525 ■0555 30 30 31 33 35 37 .0462 .0490 .0518 28 28 29 32 33 34 .0430 .0457 .0484 27 27 27 28 30' 32 .0402 .0427 .0452 25 25 26 27 29 30 ■0375 .0398 .0422 23 24 24 25 26 27 2200 2300 2400 .0586 .0617 .0648 31 31 32 39 41 42 •0547 .0576 .0606 29 30 30 36 38 40 .0511 .0538 .0566 27 28 29 33 34 35 .0478 .0504 •0531 26 27 27 32 33 34 .0446 .0471 .0497 25 26 25 28 30 32 2500 2600 2700 .0680 .0712 .0744 32 32 33 44 46 47 .of)36 .0666 .0697 30 31 31 41 43 45 .0595 .0623 .0652 28 29 30 37 3« 40 .0558 .0585 .0612 27 27 28 36 37 38 .0522 .0548 .0574 26 26 26 33 34 36 2800 2900 3000 .0777 .0810 .0843 33 33 33 49 50 52 .0728 .0760 .0791 32 31 32 46 48 49 .0682 .0712 .0742 30 30 30 42 43 44 .0640 .0669 .0698 29 29 29 40 42 43 .0600 .0627 .0655 27 28 28 37 38 40 3100 3200 3300 .0876 .0910 .0944 34 34 35 53 55 56 .0823 .0855 .0888 32 33 33 5> 52 53 .0772 .0803 .0835 31 32 32 45 47 49 .0727 .0756 .0786 29 30 30 44 45 47 .0683 .0711 •0739 28 28 29 41 43 44 3400 3500 3600 .0979 .1014 .1049 35 35 36 58 59 60 .0921 .0955 .0989 34 34 34 56 56 57 .0867 .0900 .0932 31 33 32 32 51 53 54 .0816 .0847 .0878 31 3' 31 48 51 52 .0768 .0797 .0826 29 29 30 45 47 48 TABLE ].- -Continued . AVLX il'mry A. z 1450 J. J." 1500 J, J, I55Q J. J, 1600 A 4. 1650 ^. 4; 400 500 600 .0063 .0080 .0097 17 18 3 5 6 .0060 .0075 .0Q91 15 16 16 4 5 6 .0056 .0070 .0085 14 14 15 4 4 S .0052 .00C6 .0080 14 14 14 3 4 5 .0049 .0062 .0075 13 13 14 ,'3 4 4 700 800 900 .0115 •0133 .0151 18 18 18 8 9 10 .0107 .0124 .0141 17 •7 »7 7 8 9 .0100 .0116 .0132 16 16 16 6 7 8 .0094 .0109 .0124 15 15 15 5 6 7 .0089 •01 03 .0117 14 14 14 5 6 7 1000 IIOO I2DO .0169 .0187 .0206 18 19 19 11 12 13 .0158 •0175 .0193 •7 18 •7 10 II 12 .0148 .0164 .0181 16 17 17 9 10 II .0139 .0154 .0170 '5 i6 16 8 9 10 ■01 3 1 .0145 .0160 14 '5 14 & 9 10 1300 .0225 19 15 .0210 18 12 .0198 17 12 .0186 16 12 .0174 16 10 1400 1500 .0244 .0265 21 21 16 18 .0228 .0247 19 19 13 15 .0215 .0232 17 17 •3 14 .0202 .0218 16 17 12 12 .0190 .0206 16 16 II 1600 1700 1800 .0286 .0307 .0328 21 21 22 20 21 2Z .0266 .0286 .0306 20 20 21 17 18 19 .0249 .0268 .0287 19 19 20 14 16 17 .0235 .0252 .0270 17 18 18 13 14 16 .0222 .0238 .0254 16 16 17 <3 14 •5 I goo 2000 2100 ■0350 .0372 .0395 22 23 23 23 24 26 .0327 .0348 .0369 21 21 22 20 22 23 .0307 .0326 .0346 19 20 21 19 20 21 .0288 .0306 .0325 18 19 19 17 18 19 .0271 .0288 .0306 17 18 18 16 17 18 2200 2300 2400 .0418 .0441 .0465 23 24 24 27 28 30 .0391 .0413 •043s 22 22 23 24 26 27 .0^67 .0387 .0408 20 21 22 23 23 24 •0344 .0364 .0384 20 20 20 20 22 23 .0324 •0342 .0361 18 '9 '9 19 20 21 2500 2600 2700 .0489 •0514 .0538 25 24 25 31 33 34, .0458 .0481 .0504 23 23 24 28 29 30 .0430 .0452 .0474 22 22 22 26 28 29 .0404 .0424 .0445 20 21 21 24 25 26 .0380 •0399 .0419 19 2» 20 22 23 24 2800 2900 3000 .0563 .0589 .0615 26 26 27 35 37 38 .0528 .0552 .0577 24 25 25 32 33 35 .0496 .0519 .0542 23 23 24 30 31 33 .0466 .0488 .0510 22 22 22 27 28 29 •3439 .0460 .0481 21 21 21 25 26 27 3100 3200 3300 .0642 .0668 .0695 26 27 28 40 41 42 .0602 .0627 ■0653 25 26 26 36 37 38 .0566 .0590 .0615 24 25 24 34 35 37 .0532 •0555 .0578 23 23 23 30 31 32 .0502 .0524 •0546 22 22 22 28 30 3! 3400 3500 3600 .0723 .0750 .0778 27 28 28 44 45 46 .0679 .0705 .0732 26 27 7 40 41 43 .0639 .0664 .0689 25 26 38 39 40 .0601 .0625 .0649 24 24 23 33 35 36 .0568 .0590 .0613 22 23 23 32 33 34 S3 TABLE I.- -CONTINHEr .— /lu;l iliary A. z 1700 J. ^. 1750 J. ■J„ 1800 A A 1850 k A 1900 4 4 400 500 600 .0046 .0058 .0071 12 13 13 3 3 4 .0043 .0055 .0067 12 12 12 2 3 4 .0041 .0052 .0063 II II 1 1 2 3 3 .0039 .0049 .0060 10 II 10 2 2 3 .0037 .0047 .0057 10 10 10 2 2 3 700 800 900 .0084 .0097 .0110 13 13 13 5 6 7 .0079 .0091 .0103 12 12 12 S S 6 .0074 .0086 .0097 12 I I 12 4 S 5 .0070 .0081 .0092 11 II II 3 4 S .0067 .0077 .0087 10 10 II 4 4 4 1000 1 100 1200 .0123 .0136 .0150 13 14 14 8 8 8 .0115 .0128 .0142 13 14 13 6 7 8 .0109 .0121 .0134 12 13 12 •6 6 7 .0103 .0115 .0127 12 12 12 5 6 7 .0098 .0109 .0120 II II II S S 6 1300 1400 1500 .0164 .0179 .0194 15 15 '5 9 10 12 •0155 .0169 .0182 14 13 14 9 10 10 .0146 .0159 .0172 13 14 7 8 9 .0139 .0151 .0163 12 12 13 8 8 8 .0131 .0143 .0155 12 12 12 6 7 8- 1600 1700 1800 .0209 .0224 .0239 '5 16 13 13 14 .0196 .021 1 .0225 15 14 15 10 12 12 .0186 .0199 .0213 13 14 14 10 10 II .0176 .0189 .0202 13 13 13 9 10 10 .0167 .0179 .0192 12 •3 12 9 9 10 I goo 2000 2100 .0255 .0271 .0288 16 17 IS 15 16 .0240 .0256 .0272 16 16 16 '3 14 15 .0227 .0242 .0257 15 •5 •5 12 13 14 .0215 .0229 .0243 14 14 15 II 12 13 .0204 .0217 .0230 13 13 14 10 II II 2300 2300 2400 .0305 .0322 .0340 17 18 18 17 18 19 .0288 .0304 .0321 16 '7 17 16. 17 18 .0272 .0287 .0303 15 16 l6 14 15 16. .0258 .0272 .0287 .14 •5 15 14 14 15 .0244 .0258 .0272 14 14 14 12 •3 14 2500 e6oo 270O •0358 .0376 •0395 18 19 •9 20 21 22 .0338 ■0355 •0373 17 18 18 19 20 21 .0319 •0335 .0352 16 17 17 17 18 19 .0302 .C317 •0333 15 16 16 i6 16 17 .0286 .0301 .0316 15 15 IS 14 IS 16 2800 2900 3000 .0414 •0434 .0454 20 20 20 23 25 27 ■0391 .0409 .0427 18 18 19 22 23 ■23 .0369 .03S6 .0401 17 18 18 20 20 21 •0349 .0366 .0383 17 17 17 18 19 20 •0331 .0346 .0362 15 16 16. 16 17 17 3100 3200 3300 .0474 .0494 .0515 20 21 21 28 28 29 .0446 .0466 .04S6 20 20 20 24 26 27 .0422 .0440 .0459 18 19 19 22 23 24 .0400 .0417 •0435 17 18 18 22' 22 23 .0378 •0395 .0412 17 17 17 18 20 21 3400 3500 3600 .0536 •05 s 7 ,0579 21 22 22 30 3> 32 .0506 .0526 •0547 20 I2I 28 29 3° .0478 .0497 .0517 83 19 20 20 25 26 28 •0453 .0471 ■ .0489 18 18 19 24 25 25 .0429 .0446 .0464 17 18 19 22 22 23 TABLE I. — GoxTiNUEi). — AuxU'mry A. 2 1950 J, 10 9 9 J, 2 3 2000 J, 9 8 9 J, 2 3 3 2050 J. 8 8 9 J. 2 2 2 2100 ■J. 8 8 8 J. I 2 2 2150 J, 7 8 7 ■Jr 400 500 600 .0035 .0045 .0054 .0034 .0043 .0051 .0332 .0040 .0048 .0030 .0038 .0046 .0029 .0036 .0044 I I 2 700 800 900 .C063 ,0073 .0083 10 10 10 3 i .0060 .0070 .0079 10 9 10 3 4 4 .0057 .0066 .0075 9 9 9 3 3 4 .0054 .0063 .0071 9 8 9 3 4 ■ 3 .0051 .0059 .006S S 9 8 2 2 3 1000 IIOO 1200 .0093 .0104 .0114 II 10 1 1 4 5 5 .CX389 .0099 .0109 10 10 10 5 5 6 .0084 .0094 .0103 10 .9 10 4 S s .0080 .0089 .0098 9 9 9 4 t .0076 ■.0085 .0094 9 9 9 3 4 4 1300 1400 1500 .0125 .0136 .0147 11 1 1 II 6 7 7 ,0119 .0129 .0140 10 II II 6 6 7 .0113 .0123 •0133 10 10 II 6 6 6 .0107 .0117 .0127 10 10 10 6 .0103 .0112 .or2i 9 9 10 5 S 5 i6oo 1700 1800 .0158 .0170 .0182 12 12 12 7 8 9 .0151 .0162 •0173 II II II 7 8 8 .0144 .0x54 ,0165 10 1 1 10 7 7 8 .0137 .0147 .0157 .10 10 10 6 7 7 .0131 .0140 .0150 9 10 10 6 6 7 lyoo 2000 2100 .0194 .0206 .0219 12 »3 13 10 10 ir .0184 .0196 .0208 12 12 12 9 10 10 .0175 .0186 .0198 II 12 12 8 8 9 .0167 .0178 .0189 II II 1 1 7 '8 9 .0160 .0170 .0180 10 10 II 8 8 8 2200 2300 2400 .0232 .0245 .025 S •3 13 14 12 12 13 .0220 .0233 .0246 13 13 12 10 II 12 .0210 .0222 .0234 12 12 12 10 II II .0200 .0211 .0223 II 12 1 1 9 10 1 1 .0191 .0201 .0212 10 II II 9 9 9 2500 2600 2700 .0272 .0286 .0300 14 14 15 14 15 15 .0258-13 .0271 ' 14 .0285 ! 14 1 12 '3 14 .0246 .0258 .0271 12 •3 13 12 12 13 .0234 .0246 .0258 12 12 13 1 1 II 12 .0223 •023s .0246 12 II 12 10 II I! 2800 2900 3000 •03 'S ■0330 •034s 15 '5 15 16 17 18 1 .0299 ■ 14 ■03 '3 14 .0327 14 1 I IS 15 •5 .0284 .0298 .0312 14 14 14 13' 14 IS .0271 .0284 .0297 13 '3 13 13 13 14 .0258 .0271 .0283 13 12 13 I I 13 3100 3200 3300 .0360 •0375 .0391 1-5 16 16 i9'.034i' 15 19 .0356 16 19 '.0372 16 1 t 15 16 17- .0326 .0340 ■035 s 14 IS IS 16 16 17 .0310 .0324 ■0338 14 14 14 14 15 16 .0296 .0309 .0322 13 13 14 14 14 15 3400 3500 3600 .0407 .0424 .0441 17 '9 21 •22 .0388 .0403 .0419 '16 >7 18 18 19 .0370 .0385 .0400 IS IS 16 18 19 19 .0352 .Q366 .0381 It 15 16 17 18 .0336 ■0349 .0363 IS 16 16 16 S4 TABLE l.—ComiKmJi.— Auxiliary A 3600 3700 3800 3900 4000 4100 4200 43DO 4400 4500 4600 4700 4800 4900 1049 1085 ,1121 .1157 .1194 36! 60 36 62 36 64 1250 38 67 .1232'. 38; 70 .0989 1023 .1057 37 65 .1092 35 62 5000 .1581 5100 .1270 .1307 .1345- .1384 -1.423 .1462 .1501. .1541 .1622 5300 5400 5500 5600 5700 5800 5900 ■6000 6100 •6200 6300 6400 6500 •6600 •6700 ^8cx) 5200 .1664 ■I 70s ■1747 .1789 .1832 •1875 .1919 .1963 !200S .2052 .2096 .3141 .2187 .2234 37, 38, 39 39 39 39 40 40 4 42 4 42 42 43 43 44 44 45 44 44 45 46 47 47 72 73 75 77- 79 80 .2281 46 .2327 47 .2374' 48 1 127 35 63 .1162 36 64 .1198 36 1234 36 .1270 37 1307 .1344 .1382 81 .1420 83 -1458 84 .1497 86 .1536 88 .1576 89 .1616 91 93 95 97 99 lOI 103 104 los 107 109 112 114 115 116 1656 .1696 •1737 .1778 .1820 .1862 .1905 .1948 .1991 .2034 .2078 .'2122 .2167 .2212 .2258 37 3S 38 38 39 39 40 40 40 40 41 41 42 42 43 43 43 43 44- 44 45 45 46 47 1300 .0932 .0964 .0997 .1030 34 .1064 34 .1098 34 66 68 69 71 72 74 75 76 78 80 82 84 85 86 9' 92 94 96 97 98 99 lOI .1132 .1166 .120 1236 36 1272 36 1308 37 4 ■ 1345 .1382 .1419 .1456 ■ 1-494 .1532 •157' .1610 .1649 1689 1729 104 105 .1770 41 ,1811 .1852 ,1894 ■ 1936 .1979 .2021 2003 2107 106 .2152 85 1,350 54 55 57 59 60 61 62 63 64 65 67 68 ,0878 .0909 ,0940 -0971 33 .1004 .1037 1070 33 1 103 34 •I 137 3+ 1171 1205 1240 70 .1275 72 .1310 73 -1346 74 76 77 79 80 81 83 84 86 87 88 90 92 94 95 96 98 100 1400 .0826 .0856 .0886 .0916 .0947 .0979 J.U 59 .1011 60 .1043 62 .1075 .1382 .1418 ■ 1455 .1492 .1530. .1568 .1606 .1645 1684 .1724 1764 . 1 804 .1844 .1885 .1926 .1967 .2009 .2052 34 35 35 35 36 36 36 37 37 38 38 38 39 39 40 40 40 40 41 41 41 42 43 43 50 51 52 53 55 57 32 58 33 59 72 73 74 75 77 78 79 81 82 84 86 87 88 90 9' 92 94 96 .1108 .1141 .1174 .1208 .1242 .1276 .1310 ■1345 .1381 .1417 •1453 .1490 .1527 .1564 .1602 .1640 .1678 .1717 .1756 •1795 .1835 .187s .1915 .1956 35 36- 36 36 37 37 37 38 38 38 39 39 39 40 40 40 41 61 62 63 65 66 67 68 69 71 72 73 75 77 7& 80 82 84 85 86 88 90 42 91 TABLE 1.- -Continued. — Auseiliary A. z 1450 4 A 1500 J. ^. 1550 ^. 4 1600 ^. ^. 1650 ^. ^. [3600 I3700 3800 .0778 .0806 •083s 28 29 29 46 47 48 .0732 .0759 .0787 27 28 28 43 44 45 .0689 •0715 .0742 26 27 27 40 41 43 .0649 •0674 .0699 25 25 26 36 38 39 •0613 x>636 .0660 23 24 24 34 35 37 13900 4000 4100 .0864 .0894 .0924 30 30 30 49 51 52 .0815 .0843 .0872 28 29 29 46 47 49 .0769 .0796 .0823 27 27 28 44 45 46 •0725 .0751- .0777 26 26 27 4« 42 43 .0684 .0709 ■0734 25 25 26 38 39 40 4200 4300 4400 .0954 .0985 .1016 31 3J 31 53 54 56 .0901 .0930 .0960 29 30 30 50 51 53 .0851 .0879 .0907 28 28 29 47 48 49 .0804 .0831 .0858 27 27 28 44 45 46 .0760 .0786 .0812 26 26 26 411 42 43 4500 4600 4700 • 1047 .1079 .iiii 32 32 32 57 59 60 .0990 .1020 .1051 30 31 31 54 11 .0936 .0965 .0995 29 30 30 SO 51 52 .0886 .0914 .0942 28 28 29 48 49 50 .0838 .0865 .0892 27 27 28 44- 45 4& 4800 4900 5000 •I 143 .1176 .1209 33 33 33 61 64 .1082 .1113 .1145 31 32 32 57 58 59 .1025 .1055 .1086 30 31 31 53 55 57 .0971 .1000 .1029 29 29 30 51 52 53 .0920 .0948 .0976 28 28 28 48 49 50 Sioo 5206 S300 iI242 .1276 .1310 34 34 35 65 66 67 .1177 .1210 .1243 •33 33 33 60 62 64 .1117 .1148 .1179 31 31 32 58 59 60 •1059 .1089 .1119 30 30 31 55 56 57 .1004 •1033 .1062. 29 29 30 51 52 53 5400 SSoo 5600 ■1345 .1380 .1415 35 35 35 69 70 71 .1276 .1310 •1344 34 34 34 65 67 68 .1211 .1243 .1276 32 33 33 61 62 63 .1150 .1181 .1213 31 32 32 58 61 .1092 .1122 .1152 30 30 31 55 56 57 5700 5800 S900 .1450 .i486 .1522 36 36 37 72 74 75 •1378 .1412 .1447 34 35 35 69 70 .1309 '1 343 •1377 34 34 34 64 66 68 .1245 .1277 .1309 32 32 33 62 63 64 .1183 .1214 .1245 31 31 32 58 59 60 6000 6100 6200 •1559 '.1596 ••633 37 37 38 77 78 79 .1482 .1518 •1554 36 36 37 71 73 74 .1411 .1445 .1480 34 35 35 69 70 71 .1343 •1375 .1409 33 34 34 65 66 67 .1277 .1309 •1342 32 33 33 61 62 64 6300 6400 6500 .1671 .1709 •1747 38 38 39 80 81 82 .1591 .1628 .1665 37 37 37 76 77 78 .1515 .155.1 ■1587 36 36 36 72 74 75 •1443 •1477 .1512 34' 35 35 68 69 71 •1375 .1408 .1441 33 33 34 65 66 66 6660 6700 6800 .1786 ;i825 ;i86s 39 40 40 84 85 86 .1702 .1740 .1779 38 39 38 79 81 83 .1623 .1659 .1696 36 37 37 76 76 77 •1547 •1583 .1619 36 36 35 72 74 75 •1475 .1509 .1544 34 35 35 67 70 36 TABLE I.- —Continued. — Auxiliary A. 2, 1700 ^. J, 1750 ^. J„ 1800 ^. ^. 1850 J. J„ 1900 J. J, 3600 3700 3800 .0579 .0601 .0623 22 22 23 32 33 33 .0547 .0568 .0590 21 22 22 30 31 32 .0517 ■0537 .0558 20 21 21 28 29 30 .0489 .0508 .0528 19 20 20 25 26 27 .0464 .0482 .0501 18 19 «9 23 25 25 3900 4000 4100 .0646 .0670 .0694 24 24 25 34 36 37 .0612 .0634 ■0657 22 23 23 33 34 35 .0579 .0600 .0622 21 22 22 31 32 33 .0548 .0568' •0589 20' 21 21 28 28 30 .0520 .0540 .0559 20 19 20 26 27 27 4200 4300 4-1 oo .0719 .0744 .0769 25 25 25 39 41 42 .0680 ,0703 .0727 23 24 24 36 37 38 .0644 .0666 .0689 22 23 22 34 34 35 .0610 .0632 .0654 22 22 22 31 32 33 .0579 .0600 .0621 21 21 21 28 30 31; 4SOO 4600 4700 .0794 .0820 .0846 26 26 26 43 44 45 .0751 .0776 .0801 25 25 25 40 42 42 .0711 •0734 .0759 23 25 25 35 36 38 .0676 .0698 .0721 22 23 23 34 35 36 .0642 .0663 .0685 21 22 22 32 32 33 4800 4900 5000 .0872 .0899 .0926 27 27 27 46 47 48 .0826 .0852 .0878 26 26 26 42 43 44 .0784 .0809 .0834 25 25 25 40 42 43 .0744 .0767 .079.1 23 24 25 37 37 38 .0707 .0730 •0753 23 23 23 34 35 36 5100 5200 5300 ■0953 .0981 .1009 28 28 28 49 40 51 .09b4 .0931 .0958 27 27 28 45 47 48 .0859 .0884 .0910 25 26 27 43 43 44 .0816 .0841 .0866 25 25 25 40 41 42 .0776 .0800 .0824 24 24 25 37 39 40 5400 5500 5600 .1037 .1066 .1095 29 29 30 SI 52 53 .0986 .1014 .1042 28 28 28 49 50 51 •0937 .0964 .0991 27 27 27 46 47 48 .0891 .0917 .0943 26 26 27 42 43 44 .0849 .0874 .0899 25 25 25 41 42 43 5700 5800 5900 .1125 .1155 .1185 30 30 31 55 56 57 .1070 .1099 .1128 29 29 30 52 53 54 .1018 .1046 .1074 28 28 29 48 49 50 .097a .0997 .1024 27 27 27 46 47 48 .0924 .0950 .0976 26 26 27 43 44 45 <)000 6100 6200 .1216 .1247 .1278 31 31 32 58 60 .1158 .1188 .1218 30 30 31 55 56 57 .1103 .1132 .1161 29 29 30 52 53 54 .1051 .1079 .1107 28 28 29 48 50 51 .1003 .1029 .1056 26 27 28 46 47 48 6300 6400 6500 .1310 • 1342 •1375 32 33 33 61 62 64 .1249 .1280 .1311 31 31 31 58 59 60 .1191 .1221 .1251 30 30 3' 57 .1136 .1165 .1194 29 29 30 52 53 54 .1084 .1112 .1140 28 28 29 49 50 50 66co 6700 6800 .1408 .1440 •1473 32 33 34 66 66 66 • 1342 •1374 .1407 32 33 32 60 61 63 .1282 .1313 .1344 3.1 3' 31 58 60 .1224 .1254 .1284 30 30 31 55 56 57 .1169 .1198 .1227 29 29 30 51, 52 53 87 TABLE I. — CoNTiKUED. — Auxiliary A. z 1950 'J. '/ 200a J. ^. 2050 J. J. 2100 J, 4 2150 '^- J, 3600 0441 iS 22 0419 17 20 0399 16 18 0381 IS 18 .0363 15 16 3700 0459 i; 23 .0436 17 21 .0415 16 19 .0396 '5 1.8 .0378 14 i» 3800 0476 i{ ! 23 ■0453 17 22 .0431 16 20 .0411 15 19 .0392 15 18 3900 .0494 IC ) 24 .0470 iS 23 .0447 17 21 .0426 16 19 .0407 15 19 4000 •05 '3 It ) 25 .0488 18 24 .0464 17 22 .0442 16 20 .0422 16 19 4100 .0532 IC J 26 .0506 18 25 .0481 18 23 .0458 »7 20 .0438 16 20 4200 .0551 i( P 27 .0524 18 26 .0499 17 24 .0475 17 21 .0454 16 21 4300 .0570 2( D 38 .0542 19 26 .0516 18 24 .0492 17 22 .0470 16 2? 4400 .0590 2( 3 28 .0561 19 27 •0534 •9 25 .0509 18 23 .0486 17 22 4500 .0610 2 I 30 .0580 20 27 ■0553 '9 26 .0527 18 24 .0503 17 25 4600 .0631 2 I 31 .0600 20 28 .0572 19 27 •0545 18 25 .0520 18 24 4701 .0652 2 I 32 .0620 20 29 .0591 19 28 .0563 19 25 •0538 18 25 4800 .0673 2 2 33 .0640 21 30 .0610 20 28 0582 '9 26 •0556 18 26- 4900 .0695 2 2 34 .0661 2/ 31 .0630 20 29 •.0601 19 27 .0574 19 27 5000 .0717 2 2 35 .0682 21 32 .0650 21 30 .0620 20 27 •0593 '9 28' 5100 .0739 2 2 36 .0703 22 32 .0671 21 31 .0640 20 28 .0612 19 29 5200 .0761 2 3 36 .0725 22 33 .0692 21 32 .0660 20 29 .0631 '9 29 5300 .0784 2 4 37 .0747 23 34 •0713 22 33 .0680 20 30 .0650 20 29 5400 .0808 2 4 38 .0770 23 35 ■0735 22. 34 .0701 21 31 .0670 20 30 5500 .0832 2 4 39 •0793 23 36 •0757 22 35 .0722 21 32 .0690 21 3» 5600 .0856 2 5 40 .0816 24 57 .0779 23 35 •0744 22 33 .071 1 22 31 5700 .0881 2 5 41 .0840 24 38 .0802 23 36 .0766 :2 33 ■0733 21 33 5800 .0906 2 5 4^ .0864 24 39 .0825 24 37 .0788 22 34 •0754 21 33 5900 .0931 2 6 43 .0888 25 39 .0849 24 39 .0810 23 35 .0775 22 33 6000 .0957 2 5 44 .0913 25 40 .0873 24 40 •0833 24 36 ■0797 23 34 6100 .0982 ; !6 45 .0938 25 41 .0897 24 40 .0857 24 37 .0820 23 36 6200 .1008 : '■7 45 .0963 26 42 .0921 25 40 .0881 24 38 .0S43 23 37 6300 .1035 ; 27 46 .0989 26 43 .0946 25 4' .0905 24 39 .0866 23 38 6400 .1062 . 28 47 .1015 26 44 .0971 25 42 .0929 24 40 .0889 23 33 65.00 .1090 28 49 .1041 27 45 .0996 26 43 •0953 25 41 .0912 24 38 6600 .1118 28 50 .1068 27 47 .1022 26 44 •0978 25 42 .0936 25 38 6700 .1146 28 51 .1095 28 47 .1048 27 ^5 ■ I 003 26 42 .0961 26 40 680c ■ I 174 29 51 .1123 28 48 .1075 27 46 .1029 26 42 .0987 25 42 TABLE I.— CosTiNUED.— 4aA-i7iaci/ B. z 1200 J, 7 7 1250 .0089 J. 23 J,. 7 1300 .00S2 J, 22 6 1350 .0076 J, 21 J. 5 1400 .0071 J, 20 400 0096 26 500 600 .0122 -7 10 .0112 25 8 .0104 23 7 .0097 22 6 .oogi 20 .0149 27 12 ■0137 26 10 .0127 24 8 .0119 22 8 .0111 21 700 .0176 28 »3 .0163 26 12 .0151 25 10 .014.1 23 9 .0132 21 800 .0204 29 15 .0189 27 13 .0176 26 12 .0164 24 II •0153 lo 900 •0233 30 17 .0216 28 14 .0202 26 14 .0188 25 13 .0175 24 1000 .0263 31 '9 .0244 30 16 .0228 28 '5 .0213 26 14 .0199 24 IIOO .0294 31 20 .0274 30 18 .02^6 29 17 .0239 11 16 .C223 24 26 1203 •0325 32 21 .0304 30 19 .02 8 5 28 19 .0266 27 19 .0247 1300 ■0357 33 23 •0334 31 2 I •0313 29 20 ,0203 27 20 .0273 26 1400 1500 ■0390 .C424 34 34 25 27 .0365 •0397 32 33 23 24 .0342 •0373 31 31 22 24 .0320 ■0349 29 30 21 23 .0399 .0326 27 28 lOoo .0458 35 28 .0430 33 26 .0404 32 25 •0379 30 25 •0354 29 1700 ■0453 36 30 .0463 34 27 .0436 32 27 .0409 31 26 ■03S3 30 1800 .0529 36 32 .0497 35 29 .04 68 33 28 .0440 32 27 •0413 30 1900 .0565 36 33 .0532 35 31 .0501 34 ^9 .0472 32 29 •0443 31 2000 .0601 37 34 .0567 36 32 •0535 35 31 .0504 34 30 .0474 33 2100 .0638 38 35 .0603 37 33 .9570 35 32 .0538 34 31 .0507 33 2200 .0676 39 36 .0640 37 35 .0605 3S 33 .0572 35 32 .0540 33 2300 3400 .0715 ■0/54 3<3 38 .0677 38 36 .0641 37 34 .0607 35 34 •OS73 34 40 39 0715 38 37 .0678 37 36 .0642 36 35 .0607 35 2500 .0794 40 41 •0753 39 38 •0715 38 37 .0678 37 36 38 .0642 .0677 35 36 2600 •0S34 41 42 .0792 40 39 •0753 38 38 •0715 37 2700 •0S75 42 43 .083^ 40 41 ■0791 39 39 .0752 38 39 .0713 37 2S0O .0917 43 45 .0872 41 42 .0830 4,0 40 .0790 39 40 .0750 .0788 .0826 38 38 2900 ,0960 43 47 .0913 42 43 .0870 40 41 .0829 39 41 3000 .1003 44 48 .0955 43 45 .0910 41 42 .0868 40 42 39 3100 .1047 45 49 .0998 43 47 .0951 42 43 .0908 ■40 43 .0865 39 3-03 33"0 .1092 46 51 .1041 44 48 •0993 42 45 .0948 41 44 .0904 40 .1138 46 53 .1085 44 50 •1035 43 46 .0989 42 45 .0944 40 3400 .1184 47 55 .1129 46 5' .107S 44 47 .1031 42 47 .0984 41 3500 .1231 48 56 .1175 47 53 .1122 45 49 .1073 43 48 .1025 .1067 42 3600 .1279 49 57 .1222 47 55 .1 167 89 45 5' .1116 44 49 43 TABLE 1.- -CONTISnEE . — Auxiliary B. a 1450 ^. ^, 1500 ^. J. 1550 A- A 1600 4 A, 1650 A, 4 !^oo 500 600 .0066 .0084 .0103 18 19 1.9 5 6 7 .0061 .0078 .0096 •7 18 •I 8 4 5 6 .0057 .0073 .0090 16 17 17 3 4 6 .0054 .0069 .0084 •5 15 16 3 4 5 .0051 .0065 .0079 14 14 '5 3 4 4 700 800 goo .0122 .0141 .0163 J9 22 23 8 10 12 .0114 •01 33 •0153 1.9 20 20 7 8 10 .0107 .0125 .0143 18 18 19 7 8 9 .0100 .0117 ■0134 »7 17 18 6 7 8 .0094 .0110 .0126 16 16 17 S 7 7 rooo 1100 1200 .0186 .0208 .0231 22 23 25 13 14 15 ■0173 .0194 .0216 21 22 23 II 12 14 .0162 .018:; .0202 20 20 22 10 II 12 .0152 .0171 .0190 19 19 20 9 II 12 .0143 .0160 .0178 'I 18 19 8 9 10 1300 1400 1500 .0256 .0281 .0306 25 25 27 17 18 19 .0239 .0263 .0287 24 •24 25 15 17 18 .0224 .0246 .0269 22 23 24 14 16 •7 .0210 .0230 .0252 20 22 22 »3 14 16 .0197 .0216 .0236 19 20 21 II 12 13 1600 1700 t8oo ■0333 .0360 .0388 27 28 29 21 22 24 .0312 ■0338 .0364 26 26 27 19 21 23 .0293 ■0317 .0341 24 24 26 >9 20 21 .0274 .0297 .0320 23 23 25 >7 18 19 .0257 .0279 .0301 22 22 23 15 16 '7 rooo 2000 2100 .0417 .0446 .0477 29 3' 32 26 27 29 .0391 .0419 .0448 28 29 30 24 25 27 .0367 ■0394 .0421 27 27 28 22 24 25 •0345 .0370 •0396 25 26 27 21 22 23 .0324 .0348 •0373 24 25 25 i8 20 22 2200 2300 2400 .0509 .0541 •0573 32 32 34 31 32 33 .0478 .0509 .0540 3' 31 32 29 31 32 .0449 .0478 .0508 29 30 3' 26 27 29 .0423 .0451 ■0479 28 28 29 25 26 27 .0398 .0425 .0432 27 27 27 23 25 26 2500 2600 2700 .0607 .0641 .0676 34 35 35 35 36 37 .0572 .0605 .0639 33 34 34 33 35 36 •0539 .0570 .0603 3« 33 33 31 32 34 .0508 ,0538 .0569 30 31 32 29 31 32 .0479 •0507 •0537 28 30 30 27 28 30 2800 2900 3000 .0711 .0748 .0785 37 37 38 38 40 41 .0673 .0708 .0744 35 36 3; 3? 38 39 .0636 .0670 .0705 34 35 35 35 36 38 .0601 •0634 .0667 33 33 34 34 36 37 .0567 .0598 .0630 31 32 33 3t 32 34 31 3200 3300 .0823 .0861 .0900 38 39 39 42 43 44 .0781 .0818 .0856 37 38 38 41 42 43 .0740 .0776 .0813 36 37 •37 39 40 41 .0701 .0736 .0772 35 36 36 38 39 40 .0663 .0697 .0732 34 35 35 35 37 38 3400 3500 3600 •0939 .0979 .1020 40 4' 41 45 46 47 .0894 •0933 •0973 39 40 40 44 45 46 .0S50 .0888 .0927 38 39 39 42 43 44 .0808 .084; .08S3 38 38 1 41 42 43 .0767 ;o8o3 .0840 36 37 37 39 40 42 40 TABLE I.- -Cc NTIXUED. — Auxiliary iJ. z 1700 J, J. 1750 J, J. 1800 J. 'J, 1850 J, J, 1900 ^. J, 400 500 600 .0048 .0061 .0075 13 14 3 3 4 .0045 .0058 .0071 >3 13 13 2 3 4 .0043 .0055 .0067 12 12 12 2 3 4 .0041 .0052 .0063 II 1 1 12 2 3 3 .0039 .0049 .0060 10 II II 1 J 700 800 900 .0089 .0103 .0119 14 16 16 5 6 7 .0084 .0097 .0112 13 IS '5 5 5 6 .0079 .0092 .0106 '3 14 14 4 5 6 .0075 .0087 .0100 12 •3 14 4 4 5 .0071 .0083 .0095 12 12 »3 4 5 1000 1 100 1 200 ■0135 .0151 .01 63 16 >7 18 8 8 9 .0127 .0143 .0159 16 16 17 7 8 9 .0120 •0135 .0150 15 15 16 6 7 8 .0114 .012S .0142 14 14 15 6 7 7 .0108 .0121 •0135 13 14 14 6 6 7 13CO 1400 1500 .01 86 .0204 .0223 18 19 10 II •3 .0176 .0193 .0210 17 17 18 10 I! I I .0166 .0182 .0199 16 >7 9 10 II .0157 .0172 .0188 15 16 •7 8 9 10 .0149 .0163 .0178 14 15 16 S 8 y 1600 1700 1800 .0242 .0263 .0284 21 21 22 14 15 16 .0228 .0248 .0268 20 20 21 12 14 15 .0216 ■0234 •0253 18 <9 20 II 12 14 .0205 .0222 .0239 17 17 19 II 12 12 .0194 .0210 .0227 16 17 18 10 10 II 1900 2000 2100 .0306 .0328 •0351 22 23 24 17 18 20 .0289 .03 1 & •0331 21 21 23 i6 17 l8 .0273 .0293 ■0313 20 20 22 15 16 •7 .0258 .0277 .0296 19 19 21 •3 14 15 .0245 .0263 .0281 18 18 19 13 14 14 2200 2300 2400 •0375 .0400 .0426 25 26 26 21 22 24 ■0354 •0378 .0402 24 24 25 19 21 22 •033 s ■0357 .0380 22 23 24 18 •9 20 .0317 ■0338 .0360 21 22 23 17 18 19 .0300 .0320 .0341 20 21 22 15 16 17 2500 2600 2700 .0452 .0479 .0507 27 28 29 25 26 28 .0427 •0453 .0479 26 26 28 23 25 25 .0404 .0428 .0454 24 26 26 21 22 24 ■0383 .0406 .0430 23 24 25 20 21 22 .0363 .0385 .0408 22 23 23 18 '9 21 2800 2900 3000 .0536 .0566 .0596 30 30 32 29 30 31 .0507 .0536 .0565 29 29 30 27 29 30 .0480 .0507 •0535 27 28 29 25 27 28 .0455 .04S0 .0507 25 27 27 24 25 26 ,0431 .0456 .0481 2s 25 26 22 23 24 3100 3200 3300 .0628 .0660 .0694 32 34 34 33 34 36 ■0595 .0626 .0658 3' 32 32 3« 32 34. .0564 .0594 .0624 30 30 31 30 31 32 ■0534 .0563 .0592 29 29 29 27 29 30 .0507 ■0534 .0561 27 27 29 25 27 28 3400 3500 3600 .0728 .0763 .0798 35 35 38 39 40 .0690 .0724 .0758 34 34 34 35 37 38 .0655 .06S7 .0720 41 32 ?3 54 34 35 3t> .0621 .0652 .0684 31 32 32 31 32 34 .0590 .0620 .0650 30 30 31 30 31 32 TABLE I; — Goxtixued. — Auxiliary B. 1950 400 500 600 700 800 900 1000 1 100 1200 .0037 .0047 .0057 Jr 1400 150,0 1600 1700' .0068 T 1 .0079 .OOOO .0102 .0115 .0128 .0141 .0155 .0169 .0184 .0200 800 .0216 1900 2000 2100 .0232 17 .0249 18 .0267 18 2200 .0285 2300 2400 2500 2600 2700 .0304 •0324 •0345 .0366 .0387 2800 .0409 2900 .0433 3000 .0457. 3100 3200 3300 3400 3500 3600 .0482 .0507 •0533 .0560 .0589 .0618 2000 .0036 .0046 .0056 16 9 16 10 i6' II .0065 10 .0075 \'( .0086 I '9 20 2 21 21 22 24 24 25 25 26 27 29 29 30 0097 .0109 .0121 .0134 .0147 .0161 .0175 .0190 ■0205 .0231 .0237 .0254 .0271 .0289 .0308 .0328 .0348 .4368 20 .0389 22 .0411 23 -0434 2050 J. 17 12 .0034. .0043 .0052 .0061 .0071 ■.0081 .0092 .0103 .OIJ5 .0127 .0140 •01.53 .0167 .0181 .0195 .0210 .0226 'J. 9 9 9 10 10 1 1 1 1 12 12 13 13 14 14 14 15 16 16 .0458 .0482 .0507 •0533 .0560 .0588 20 20 21 22 23 24 24 25 26 27 28 39 .0242 16 .02 5 8 .0275 .0293 .0314 •0331 .0370 .0391 •0413 .0436 .0459 .0483 .0508 •0534 .0560 43 2100 .0032 .0040 .0049 3 .0058 3 .006S J. Mi .0078 10 .0088 .0099 .0122 12 17 12 18 13 19 14 19 15 19 16 •0134 .0146 .0159 .0172 .0186 .0200 .0215 .0230 .0246 .0262 .0279 ,0350 20 16 .0297 18 .0315 19 •0334 19 21 22 23 34 25 26 •0353 •0373 0393 .0415 22 2150 U 4 11 5 12 5 9 10 16 10 •0437 .0460 .04S4 .0509 •0534 II 1-2 «3 14 14 15 16 17 18 19 20 2 23 !4 00311 .0038 .0047 .0055 .0064 .0074 .0084 .0094 .0105 .0116 -.0 1 27 .0139 .0151 .0164 .0177 .0191 .0205 .0220 ■02f35 .0250 .0266 8 9 8d 9 10 10 10 11 i I II 12 12 13 13 H 14 15 15 4 2 .0283 18 •0301 ■0319 •0337 ■0356 .0375 21 8 18 19 \9 .0396 .0417 •0439 .0461 ,048s 26 1 25 1.0509 4 4 5 !^ 5 6 6 7 7 8 9 10 II II 13 13 14 .15 15 16 16 18 «9 20 21 22 23 TABLE ).. — CoNTiNDED. — AuTtliary B. z 1200 M 3600 ■12/9 49 3700 .1328 49 3800 ■^m 50 J, I 1250 -I 3900 .1427 4000 .1477 41CX3 .1529 I » 4200 .1581 4300 .1634 4400 .I6S8 4500 4600 4*700 4800 4900 JOOO 5100 5200 5300 5 400 5500 5600 5700 5800 5900 6000 6100 6200 6300 6400 6500 6600 6700 6800 ■1743 .1798 .1855 .1912 .1970 .2029 .2089 .2149 .2210 .2272 ■2335 •2309 .2403 .2529 .2596 .26641 68 69 71 .2732 .2801 .2S72 •2943 .3016 .3089 •3'6 .3238177 57 59 61 63 65 67 69 70 72 75 76 78 80 82 84 86 88 90 92 94 96 99 loo 102 I OS 107 109 112 114 120 122 124 1222 .1269 .1316 .1364 .I4?2 .1462 .1512 .1564 .l6i6 .1668 .1722 ■1777 .18^2 .188S .1945 .2603 .2061 .2120 .2180 .2241 -•2303 .2366 .2429 .2494 ■2559 .2625 .2692 .2760 .2829 .2898 .2969 .3041 •3'J4 J, 55 57 5S 59 60 62 63 65 67 68 70 72 74 76 78 80 81 82 84 86 88 90 92 90 98 6g 102 69 104 71 106 108 1 10 1 12 1300 .1 167 .1212 .I25& •1305 •1352 .1400 .1449 .1499 .1549 .i6oO' .1652 .1705 .1758 .1812 .1867 .1923 .1980 .2038 .2096 ■2155 .2215 .2276 ■2337 .2400 .2463 .2527 =2592 .2658 .2725 .2792 .2861 .2931 .3002 43 J, 51 52 53 55 56 57 59 61 62 63 65 67 68 69 71 72 74 76 78 79 8t 83 85 87 90 92 93 95 99 lOI 104 1350 .n 16 .1 160 .1205 .1250 .1296 ■1343 .1390 .1438 .1487 •1537 .1587 .1638 . 1 690 •'743 .1796 .1851 .1906 .1962 .2018 .2076 •2134 .2193 .2252 •2313 •2375 •2437 .2500 • 2565 .2630 .2696 .2762 .2830 .289S 1400 067 no 153 197 242 2S7 333 380 427 475 524 574 625 676 728 781 834 943 999 .2056 .2113 .2171 .2230 .2290 •2351 .2412 •2475 •2538 .2602 .2667 •2733 •2799 43 43 44 45 45 46 47 47 48 49 50 51 51 52 53 53 54 55 56 57 57 58 59 60 61 61 63 63 64 65 66 66 68 TABLE I. — CoNTiNDED. — Auxiliary B. I4S0 3600 3700 13800 3900 4000 4100 4200 4300 4400 4500 4600 4700 4800 4900 5000 5100 5200 5300 5400 S500 5600 5700 5800 5900 6000 6100 6200 6300 6400 6500 .1020 .1061 .1103 .1146 1189 1233 1278 .1324 ,1370 .1416 .1463 .1512 .1561 .1611 .1661 4 41 42 43 43 44 45 46 46 46 47 49 49 50 50 52 i7'3 .1765 ,1818 .1871 .1926 .1981 .2037 .2693 .2151 .2209 .2268 .2328 .2389 •24ii .2513 47 48 49 50 SI 5.2 53 55 56 57 58 60 61 63 64 66 67 68 .1500 .0973 .1013 .1054 ,1096142 43 44 55 55 56 56 58 58 59 60 61 62 62 63 .1.138 .1181 .1225 .1269 ■ 1314 'I359 ,1405 ,1452 .1500 .1548 ■1597 .1647 .1698 1750 1802 .1855 ,1909 .1963 .2018 .2074 .2131 .2189 .2247 44 45 45 46 47 48 48 49 50 51 52 52 46 47 48 49 50 SJ 52 54 55 55 55 57 59 60 61 62 63 66 1550 .0927 .0966 .1006 .1047 .1088 • I 1 30 6600 .2576 64 6700 .2640 65 6800 1.2705 '66 .2307 .2367 .2428 .2489 2552 90I.2615 53 54 54 55 56 57 58 58 60 60 61 61 63 63 64 .1172 .1215 .1259 .1304 •1350 ■1395 .1441 .1488 .1536 •1585 •1635 .1684 4 39 40 41 41 42 42 43 44 45 46 45 46 47 48 49 50 49 51 1600 •1735 ..787 .1839 .1891 •1945 74 .21)00 .2056 .2113 .2"! 70 .2228 .2286 .23/16 .2406 .2466 8'7l.2S28 U 52 52 52 54 55 56 57 57 58 58 60 60 6q 62 63 44 45 46 47 48 49 50 51 52 S3 55 55 56 57 58 60 62 63 64 65 66 .0883 .0921 .0960 .1000 .1040 .1081 .1122 .1164 .1207 .1251 .1295 .1340 ■J385 .1431 .1478 .1525 •1573 .1621 .1671 "1722 1773 38 39 40 40 41 41 42 43 44 44 45 45 46 47 47 48 J 48 50 SI 51 52 1650 .0840 .0877 ■0915 ■C)9S4 39 47-0993 48-1033 .1825 ■1877 .1930 .1984 •2039 .2095 76 .2152 .2209 .2267 .2326 .2386 82 .2446 49 5° 51 S3 S4 55 56 ■57 58 59 60 60 62 64 65 66 67 68 69 70 72 57 58 59 60 60 61 .1073 .1114 .1156 .1198 .1241 .1285 .1329 •1374 .1420 .1466 •1513 .1561 .1609 .1658 .1708 -1759 1 .1810 .1862 •1915 .1969 .2023 40 40 41 42 42 43 44 44 45 46 46 47 48 48 49 50 51 51 52 53 54 54 55 42 43 44 45 46 47 48 49 SO 51 52 53 54 55 56 57 58 59 60 61 62 .2078 .2134 ■2191. .2248 .2307 .2366 56 57 57 59 59 60 63 64 65 -67 68 69 70 72 73 74 76 n TABLE I. — Continued. — Auxiliary B. 3600 3700 3800 3900 4000 4100 4200 4300 4^00 4500 4600 4700 4S00 4900 5000 5100 5200 S300 5400 5500 5600 5700 5800 5900 fiooo 6IC0 6200 6300 6400 6500 6600 6700 6800 1700 .0798 .0834 .0871 .0909 .0947 36 38 38 39 .0986 39 ,1025 ,1065 1 106 .1147 .1189 .1232 .1275 ■ 1364 .1409 • 1455 .1502 .1549 ■1597 .J 646 .1696 .1746 .1797 .1848 .1901 • 1954 .2008 .2062 .2118 •2 '74 ,2231 ,2289 42 43 43 44 45 45 46 47 47 48 49 SO 50 5' 51 53 53 54 54 56 56 57 58 59 40 42 43 44 44 .45 46 47 48 48 49 50 51 52 54 54 55 57 58 59 60 62 63 6j 64 66 67 68 69 71 72 73 75 1750 .0758 .0792 .0828 .0865 .0903 .0941 .0979 .1018 .1099 ,1140 ,1182 .1224 .1267 .1310 ■•355 .1400 •1445 .1491 •1538 .1586 •'634 .1683 •1733 .1784 •1835 .1887 .1940 •1993 .2047 .2102 .2158 .2214 J. -'. 34 38 36 38 i7 40 38 41 38 43 38 44 39 45 40 46 41 47 4' 48 42 49 42 50 43 51 43 52 45 52 45 54 45 55 46 55 47 56 48 57 48 58 49 59 50 60 5' 61 51 63 52 64 53 65 53 66 54 67 55 68 56 69 56 70 57 7' 1800 .0720 .0754 .0788 .0824 .0S60 .0897 •0934 .0972 .101 1 .1051 .1091 .1 132 • II73 .1215 .1258 .1301 •1345 .1390 •1435 .1481 .1528 ■1575 .1623 .1672 .1721 -J 77 1 .1822 .1874 .1926 .1979 •2033 .2088 .2143 45 40 46 41 47 41 1S50 .0684 .0716 .0749 .0783 .0818 .0854 .0891 .0928 .0966 .1005 .1044 .1084 .1124 .1165 .1207 .1250 •1293 ■1337 .1381 .1426 .1471 .1565 .1613 .1661 .1710 ■1759 .1810 .i86i •'913 .1966 .2020 .2074 J, J. 32 33 34 34 36 36 35 36 37 Z9 37 40 37 38 42 43 39 44 39 40 45 46 40 47 41 42 47 48 43 49 43 50 44 51 44 52 45 53 45 54 47 54 47 48 48 56 57 58 49 58 49 51 59 60 51 52 61 62 53 67, 54 54 55 64 66 67 1900 .0050 .0681 .0713 .0746 .0779 .0849 .0885 .0922 .0960 .0998 •1037 .1077 .U17 .1200 .1242 .1285 .1328 ,1372 ,1417 .1462 .1508 1555 1603 .1651 .1699 1749 ,1799 1850 ,1902 ,1954 ,2007 32 33 33 35 33 35 35 36 37 38 38 39 40 40 41 42 42 43 43 44 45 45 46 47 48 48 48 50 36 37 38 39 40 43 43 44 45 47 47 49 49 50 52 53 54 55 56 57 58 58 50 60 51 6i 52162 52164 53 165 54165 TABLE I. — Continued. — Auxiliary B. 1950 J, J. 2000 ^. J„ 2050 -/. ^. 2100 ^. ^. 2150 ^. ^. 3600 3700 .0618 30 30 .0588 29 28 .0560 27 26 ■0534 26 25 .0509 25 26 26 23 .0648 30 31 .0617 29 30 .0587 29 27 .0560 27 27 ■0534 .0560 24 3800 .0678 32 32 .0646 30 30 .0616 29 29 .0587- 27 27 25 3900 .0710 32 34 .0676 3' 31 .0645 20 31 .0014 29 28 .0586 28 25 4000 .0742 34 35 .0707 32 33 .0674 3'i 31 .0643 30 29 .0614 .0642 28 27 28 4100 .0776 34 37 •0739 33 34 .0705 3' 32 .0673 30 31 29 4200 .0810 35 38 .0772 34 36 .0736 33 33 .0703 31 32 .0671 30 29 4300 .0845 35 39 .0806 34 37 .0769 33 35 ■0734 32 33 .0701 31 31 4400 .0S80 37 40 .0840 36 38 .0802 34 36 .0766 33 34 ■0732 31 32 4500 .0917 37 41 .0876 36 40 .0836 35 37 .0799 34 36 .0763 33 32 4600 .0954 38 42 .0912 37 41 .C871 36 38 •0833 34 37 .0796 33 34 4700 .0992 38 43 .0949 37 42 .0907 37 40 .0867 36 38 .0829 34 35 4S00 .1030 40 44 .0986 39 42 .0944 37 41 .0903 37 40 .0863 36 36 4900 .1070 40 45 .1025 39 .44 .0981 38 41 .0940 37- 41 .0899 36 38 5000 .1 no 4' 46 .1064 40 45 .1019 39 42 .0977 38 42 ■0935 37 39 5100 .1151 42 47 .1 104 40 46 .1058 40 43 • 1015 38 43 .0972 38 41 5200 •i'93 42 49 .1144 41 46 .1098 40 45 ■ 1053 39 43 .1010 38 42 S300 •1235 42 50 .1185 42 47 .1138 41 46 .1092 40 44 .1048 39 42 5400 .1277 43 5° .1227 42 48 • I '79 41 47 .1132 4" 45 .1087 40 43 5500 .1320 44 51 .1269 43 49 .1220 42 47 ■ 1173 41 46 .1127 40 44 5600 .1364 44 52 .1312 44 SO- .1262 43 48 .1214 42 47 .1167 40 45 5700 .1408 45 52 •1356 44 51 • I 30s 44 49 .1256 43 49 .1207 42 4S 5800 •'453 46 53 .1400 45 51 • 1349 44 50 .1299 43 50 .1249 43 46 5900 .1499 47 54 .1445 46 52 ■1393 44 51 • 1342 44 49 .1292 43 47 6000 .1546 47 55 .1491 46 54 ■ 1437 45 51 .1386 45 50 • 1335 44 48 6100 •'593 48 56 • 1537 47 55 .1482 46 SI ■ 1431 45 52 ■1379 44 49 6206 .1641 48 57 .1584 48 56 .1528 47 52 .1476 46 S3 .1423 45 49 6300 .1689 49 57 .1632 48 57 ■ 1575 48 53 .1522 46 54 .1468 46 SO 6400 •1738 50 58 .1680 49 57 .1623 48 55 .1568 47 54 • 1514 48 SI 6500 .1788 50 59 .1729 50 58 .1671 49 56 .1615 48 S3 .1562 47 53 6600 .1838 SI 59 •1779 SO 59 .1720 SO 57 .1663 49 54 .1609 47 54 6700 .1889 53 60 .1829 SI 59 • 1770 50 58 .1712 49 56 .1656 47 54 6800 .1942 54 62 .1880 52 60 .1820 46 51 59 .1761 SO 58 •1703 48 54 TABLE 1. — Continued. — Auxiliary m. i 120U d. A 1250 A A 1300 J. A 1350 J. J„ 1400 A J, 400 500 600 .0188 .0238 .0289 SO 52 14 18 22 .0174 .0220 .0267 46 47 49 13 16 19 .0161 .0204 .0248 43 44 45 12 14 17 .0149 .0190 •0231 41 41 42 10 13 16 .0139 .0177 .0215 38 38 39 10 13 15 700 800 900 .0341 •0395 .0450 54 55 57 25 29 33 .0316 .0366 .0417 50 SI 52 23 26 29 .0293 •0340 .0388 47 48 49 20 24 27 .0273 .0316 .036 r 43 45 46 19 21 24 .0254 •0295 •0337 41 42 43 '7 21 24 1000 JIOO I2CX3 .0507 .0564 .0622 57 58 59 38 40 43 .0469 .0524 •0579 55 55 56 36 39 •0437 .0488 .0540 SI 52 52 30 33 i7 .0407 •0455 •0503 48 48 49 27 31 34 .0380 .0424 .0469 44 45 46 26 29 32 1300 1400 1500 .0681 .0742 .0804 61 62 62 46 SO 54 •0635 .0692 •0750 57 58 60 43 47 50 .0592 .0645 .0700 53 55 56 40 43 46 .0552 .0602 .0654 50 52 S3 37 40 43 •0515 .0562 .0611 47 49 50 34 37 40 i6oo J 700 1800 .0866 .0929 .0994 -53 65 66 56 59 63 .0810 .0870 .0931 60 61 54 56 59 .0756 .0814. .0872 58 58 59 49 52 55 .0707 .0762 .0817 55 55 57 46 50 55 .0661 .0712 .0764 SI 52 54 45 45 48 1900 2000 2100 .1060 .1126 ■ I 193 66 67 69 66 69 72 .0994 .1057 .1121 63 64 66 63 65 67 .0931 .0992 .1054 61 62 62 57 61 64 .0874 .0931 .0990 57 59 60 56 58 61 .0818 .0873 .0929 55 56 57 SI 55 57 2200 2300 2400 .1262 ■1332 .1402 70 70 72 75 81 .1187 ■1253 .1321 66 68 68 71 74 77 .1116 .1179 .1244 63 65 66 66 68 71 .1050 .iiii •1173 61 62 63 64 67 69 .0986 .1044 .1104 58 60 60 59 66 66 2500 2600 2700 .1474 .1546 .1619 72 73 75 85 88 90 ■1389 .1458 ■1529 69 71 71 79 82 86 • i3'0 .1376 •1443 66 67 69 74 76 79 .1236 .1300 .1364 64 64 66 72 75 77 .1164 .1225 .1287 61 62 63 68 70 73 2800 2900 3000 .1694 .1770 .1846 76 76 77 94 97 100 .1600 ■ 1673 .1746 73 73 75 88 91 94 .1512 .1582 .1652 70 70 71 82 84 86 .1430 .1498 .1566 68 68 69 80 83 86 •1350 .1415 .1480 65 65 67 76 78 80 3100 3200 3300 .1923 .2002 .2082 79 80 81 102 1 06 109 .1821 .1896 •1973 75 77 71 98 100 103 .1723 .1796 .1870 7i 74 76 88 92 95 •1635 .1704 •1775 69 71 72 88 89 92 •1547 .1615 .1683 68 68 69 83 86 88 3400 3500 3600 .2163 .2245 .2328 82 83 84 113 115 117 .2050 .2130 .2211 80 81 81 lOd. 108 1 12 .1946 .2022 .2099 47 76 77 77 99 102 105 .1847 .1920 .1994 73 74 75 95 98 101 •1752 .1822 .1893 70 71 73 90 93 95 TABLE I. — CoNTiNDED. — Auxiliary w. z I4S0 4 ^. 1500 ^. A 1550 J. A 1600 A A 1650 A A. 400 500 600 .0129 .0164 .020D 35 36 37 8 10 13 .0121 .0154 .0187 33 33 34 8 II 12 .0113 .0143 .0175 30 32 32 7 8 II .0106 •0135 .0164 29 29 31 6 8 10 .0100 .0127 .0154 27 27 29 .6 8 8 700 800 900 .0237 .0274 •0313 37 39 41 16 17 20 .0221 .0257 .0293 36 36 38 14 16 18 ,0207 .0241 .0275 34 34 35 12 'S 17 .0195 .0226 .0258 31 32 33 12 13 •5 .0183 .0213 .0243 30 30 31 10 13 14 1000 1100 1200 •0354 •039s •0437 41 42 44 23 26 28 •0331 .0369 .0409 38 40 40 21 23 26 .0310 .0346 •0383 36 37 39 19 21 23 .0291 .0325 .0360 34 '36 17 20 22 .0274 .0305 •0338 31 33 33 16 18 20 1306 1400 1500 .0481 .0525 .0571 44 46 33 34 37 .0449 .0491 'OS 34 42 43 44 27 30 33 .0422 .0461 .0501 39 40 41 26 29 31 .0396 .0432 .0470 36 38 39 25 26 28 .0371 .0406 .0442 35 36 37 21 25 25 1600. 17CX5 1800 .0619 .0667 .0716 48 49 51 41 43 46 .0578 .0624 .0670 46 46 48 36 39 42 .0542 .0585 .0628 43 43 46 33 36 38 .0509 .0549 .0590 40 41 43 30 32 35 .0479 .0517 •0555 38 38 40 28 32 1900 2000 2100 .0767 .0818 .0872 51 54 55 49 51 55 .0718 .0767 .0817 49 50 52 44 47 50 .0674 .0720 .0767 46 47 48 41 44 46 .0633 .0676 .0721 43 45 46 38 40 42 .0595 .0636 .0679 41 43 43 34 37 40 2200 2300 2400 .0927 .0982 .1038 55 56 58 58 60 63 .0869 .0922 ■Q975 53 53 55 54 57 59 .0815 .0865 .0916 50 51 53 48 50 53 .0767 .0815 .0863 48 48 49 45 48 50 .0722 .0767 .0813 45 46 46 42 45 47 2500 2600 2700 .1096 .1155 .1214 59 59 60 66 69 7' .1030 .1086 •I 143 56 57 58 61 64 66 .0969 .I022 .1077 S3 55 55 57 60 63 .0912 .0962 .1014 50 52 S3 53 56 58 .0859 .0906 .0956 50 49 51 54 2800 2900 3000 .1274 •1337 .1400 63 63 64 73 77 79 .1201 .1260 .1321 61 62 69 71 74 .1132 .1189 .1247 57 58 59 67 70 .1067 .1122 .1177 55 55 56 61 64 66 .1006 .1058 .1111 52 53 54 56 58 60 3100 3200 3300 .1464 .1529 ■•S9S 65 66 67 81 84 86 •1383 .1445 .1509 62 64 64 77 79 82 .1306 .1366 .1427 60 61 62 73 75 77 .1233 .1291 •1350 58 59 59 68 70 72 .1165 .1221 .1278 56 57 57 63 67 70 3400 3500 3600 .1662 .1729 .1798 69 89 91 93 •1573 .1638 .1705 65 67 67 84 86 89 .1489 .1552 .1616 48 63 64 65 80 82 84 .1409 .1470 ■ «532 61 62 63 74 77 79 •1335 ■1393 ■1453 58 60 60 71 73 76- TABLE I. — Continued.— -Auxiliary m. z 1700 J. J. 1750 J. ^. 1800 J. •J. 1850 J. J, 1900 J, J,. 400 500 600 .0094 ,oi 19 .0146 25 27 27 6 6 8 .0088 .0113 .0138 25 25 25 4 6 8 .0084 .0107 .0130 23 23 24 5 6 7 .0079 .0101 .0123 22 22 22 3 5 6 .0076 .0096 .0117 20 21 21 4 5 6 700 800 •0173 .0200 27 29 10 12 •01 63 .0188 25 27 9 10 ..)I54 .0178 24 25 9 10 .0145 .0168 23 24 7 8 .0138 .0160 22 22 7 8 900 .0229 29 '4 .0215 28 12 .0203 26 II .0192 25 10 .0182 24 10 1000 IIOO 1200 .0258 .0287 .0318 29 31 32 15 16 •7 •0243 .0271 .0301 28 30 30 14 15 17 .0229 .0256 .0284 27 28 28 12 13 15 .0217 .0243 .0269 26 26 26 u '3 '4 .0206 .0230 .0255 24 25 25 10 1 1 '3 1300 1400 1500 .0350 ■0383 .0417 33 34 34 19 21 24 •0331 .0362 •0393 31 31 32 19 21 22 .0312 .0341 .0371 29 30 31 17 20 .0295 .0323. •0351 28 28 30 '5 17 18 .0280 .0306 ■0333 26 27 28 '4 '5 •7 1600 1700 1800 .0451 .0487 .0523 36 36 38 26 28 30 .0425 .0459 •0493 34 34 36 23 26 27 .0402 •0433 .0466 31 33 34 21 22 25 •0381 .0411 .0441 30 30 32 20 22 22 .0361 .0389 .0419 28 30 30 18 19 22 1900 2000 2100 .0561 .0599 .0639 38 40 41 32 33 36 .0529 .0566 .0603 37 37 39 29 31 33 .0500 ■0535 •0570 35 35 36 27 29 30 •0473 .0506 .0540 33 34 35 24 27 29 .0449 .0479 .0511 30 32 33 23 24 25 2200 2300 2400 .0680 .0722 .0766 42 44 44 38 40 43 .0642 .0682 .0723 40 41 42 36 38 40 .0606 .0644 .0683 38 39 40 31' 34 36 •0575 .0610 .0647, 35 37 38 3' 32 34 .0544 .0578 .0613 34 35 36 27 29 31 2500 2600 2700 .0810 .0855 .0902 45 47 48 45 47 50 .0765 .0808 .0852 43 44 46 42 44 46 •0723 .0764 .0806 41 42 43 38 41 43 .0685 .0723 .0763 38 40 41 36 37 39 .0649 .0686 .0724 37 38 38 33 35 37 2800 2900 3000 .0950 .1000 ,1051 50 51 51 52 55 58 .0898 .0945 •0993 47 48 49 49 52 54 .0849 .0893 •0939 44 46 47 45 47 49 .0804 .0846 .0890 42 44 44 42 44 47 .0762 .0802 •0843 40 41 42 38 40 42 3100 3200 3300 .1102 .1154 .1208 52 54 56 00 62 64 .1042 .1092 •"44 50 52 52 56 58 61 .0986 .1034 ■ 1083 48 49 SO 52 54 56 •0934 .0980 .1027 46 47 47 49 51 53 .0885 .0929 •0973 44 44 46 44 47 49 3400 3500 3600 .1264 .1320 •'377 56 57 58 68 70 72 .1196 .1250 •1305 54 55 55 63 66 68 •1133 .1184 .1237 49 51 53 54 59 61 64 .1074 .1123 ••'73 49 50 5' 55 57 59 .1019 .1066 .1114 47 48 50 51 53 55 TABLE I.— CoKTiiKVW.— Auxiliary m. z 1950. 4 4 2000 d. 4 2050 4 4 2100 J, J. 2150 J. J. 400 .0072 19 2 .0070 19 5 .0065 »7 3 0062 16 3 .0059 16 3 500 .0091 20 2 .0089 18 7 .0082 18 4 .0078 17 3 .0075 16 4 6CX3 .0111 20 4 .0107 18 7 .0100 18 5 .0095 18 4 .0091 16 4 700 .0131 21 6 .0125 20 7 .0118 19 3 .0113 18 6 .0107 18 4 6 800 .0152 22 7 .0145 20 8 ■0137 20 6 .0131 19 6 .0125 18 900 .0174 22 9 .0165 21 8 ■0157 20 7 .0150 19 7 .0143 18 7 1000 .0196 23 10 .0186 22 P .0177 21 8 .0169 20 8 .0161 >9 8 1100 .0219 23 II .0208 22 10 .0198 21 9 .6189 20 9 .0180 19 9 1200 .0242 24 12 .0230 23 II .0219 22 10 .0209 21 10 .0199 20 9 1300 .0266 25 .'3 .0253 23 12 .0241 22 11 .0230 21 11 .0219 20 10 1400 .0291 25 15 .0276 25 13 .0263 23 12 .0251 22 12 .0239 22 II 1500 .0316 27 15 .0301. 25 IS .0286 24 '3 .0273 22 12 .0261 21 •3 1600 •0343 27 17 .0326 26 16 .0310 24 15 .0295 24 13 .0282 22 13 1700 .0370 27 18 .0352 26 18 •0334 25 15 .0319 24 15 ■0304 22 14 1800 •0397 29 19 .0378 27 19 ■0359 26 16 •0343 25 17 .0326 24 14 trjoo .0426 ^9 21 .0405 28 20 ■0385 27 17 .0368 25 18 .0350 24 15 16 2000 .0455 31 22 •0433 29 21 .0412 28 19 •0393 26 19 ■0374 25 2100 .0486 31 24 .C462 29 22 .0440 28 21 .0419 27 20 ■0399 26 "7 2200 .0517 32 26 .0491 31 23 .0468 29 22 .0446 28 21 .0425 26 19 2300 .0549 33 27 .0522 32 25 ■0-I97 29 23 .0474 28 23 ■045 > 27 20 2400 .05B2 34 28 .0554 32 28 .0526 31 24 .0502 29 24 .0478 28 21 2500 .0616 35 30 .0586 33 29 ■0557 32 26 .0531 30 25 .0506 29 22 2600 .0651 36 32 .0619 34 30 .0589 33 28 .0561 3' 26 •OS35 30 24 2700 .0687 37 34 .0653 35 31 .0622 33 30 ■0592 32 27 ■0565 30 25 3800 .0724 38 36 .0688 36 33 .0655 35 31 .0624 32 29 .0595 31 26 2900 .0762 39 38 .0724 37 34 .0690 35 34 .0656 34 30 .0626 32 27 3000 .0801 40 40 .0761 39 36 ■0725 37 35 .0690 35 32 .0658 34 29 3100 3200 .0841 .0S82 41 42 41 43 .0800 ■083.9 39 41 38 40 .0762 .0799 37 38 37 38 ■0725 .076? 36 37 33 35 .0692 .0726 34 35 31 33 3300 .0924 44 44 .0880 41 42 .0838 39 40 .0798 38 37 .0761 36 34 3400 .0968 45 47 .0921 43 44 .0877 41 41 .0836 39 39 .0797 37 36 3500 .1013 46 50 .0963 44 45 .0918 41 43 .0875 39 41 .0834 38|37 j6oo .1059 47 53 .1007 46 48 .0959 43 45 ■0914 41 42 .0872 39 '39 TABLE I. — Continued. — Auxiliary m. 3600 3700 3800 3900 4000 4100 4200 4300 4400 4500 4600 4700 4800 4900 5000 Sioo 5200 5300 5400 5SOO 5600 5700 5800 S900 6000 6100 6200 6300 .5013 .2328 .2412 .2497 .2584 .2671 .2761 .2851 .2941 •3033 •3127 .322 ■3317 •3413 •35' I .3610 •371 1 •3813 ■391S .4019 .4124 ■423 • ■4338 .4448 •4559 4671 .4784 4897 ^. 4 1250 6400 6500 6600 6700 6800 •5131 •5250 ■5370 ,5490 ,5612 84 85 87 87 qo 90 90 92 94 94 96 96 98 99 lOI 102 102 104 105 107 107 no III 112 "3 H3 .116 118 119 120 "7 120 124 128 •32 '37 141 •43 •47 •52 •55 158 160 164 168 172 176 •79 •83 •87 191 •94 •99 203 207 211 214 9 224 229 120234 122 237 1251239 ,2211 ,2292 •2373 ,2456 ■2539 ,2624 .2710 2798 ,2886 ,2975 ,3066 3^59 3253 3347 3442 3539 3637 3736 ■3836 •3937 .4040 •4144 .4249 •4356 .4464 •4573 .4683 •4794 .4907 .5021 •5 136 •5253 ■5373 1300 88 89 9^ 93 94 9+ 95 97 98 99 100 lOI 103 104 105 107 108 109 no in "3 114 ••5 117 120 120 112 116 118 121 123 126 129 •33 136 •38 142 146 150 153 •56 160 163 166 169 172 176 •79 •83 186 190 •94 •97 .2099 .2176 .2255 •2335 .2416 .2498 .258 .2665 .2750 .2837 .2924 •30^3 •3103 ■3^94 •3286 •3379 ■3474 ■3570 .3667 •3765 .3864 ■3965 4067 417c 4274 4379 4486 200 .4594 203 208 211 215 220 •4703 •4813 •1925 .5038 515] SI 77 79 80 81 82 83 84 85 87 87 89 90 9i 92 93 95 96 97 98 99 lOI 102 •03 104 105 107 108 109 no 112 ••3 '•5 117 4 •350 4 J, 105 107 no ••3 "5 118 121 124 126 129 •31 •34 •38 141 144 146 150 •53 156 •59 162 166 170 •73 •75 178 182 •85 188 19 •994 .2069 ,2145 .2301 .2380 .2460 .2541 .2624 .2708 •2793 .2879 •2965 •3053 .3142 •3233 •3324 ■34^7 •35" .3606 .3702 •3799 ■3897 •3997 •4099 .4201 •4304 100 102 102 •03 105 4409 106 45^5 .462 2j 108 107 :o3i.49So'ii lOI 103 106 109 112 114 116 118 122 125 128 •3' 132 •35 138 142 •45 148 •5 •54 156 159 162 165 169 172 •75 •78 182 185 191 •94 1400 J, .1893 .1966 .2039 .2113 .2189 .2266 •2344 .2423 .2502 .2583 .2665 .2748 .2833 .2918 ■ 3004 ,3091 •3179 .3269 .3360 •3452 .3546 .3640 •3735 •3832 3930 4029 4129 4231 •4333 •4437 •4542 .4648 .4756 73 73 74 76 77 78 79 79 95 99 101 103 :o6 1C9 •82 83 85 85 86 87 88 90 9' 92 94 94 95 97 99 lOO 102 102 104 105 106 108 109 114 116 120 •23 •25 129 •3^ •34 136 •38 141 •44 146 150 •53 156 •59 163 165 168 •73 177 180 •83 186 TABLE I.- — Continued. — Auxiliary m: z 1450 ^. ^. 1500 A. d. 1550 ^. ^, 1600 ^. 4 1650 • 1453 ••513 ••575 4 60 62 63 ^. I3600 13700 3800 .1798 .1867 •1938 69 7> 72 93 95 97 • 1705 .1772 .1841 67 69 89 9' 93 .1616 .1681 .1748 65 67 68 84 86 89 ••532 ■•595 .1659 el 64 66 79 82 84 76 78 81 3900 '4000 '4100 .2010 .2083 •2IS7 73 74 75 99 102 104 .1911 .1981 •2053 '70 72 73 95 97 100 .1816 .1884 •1953 68 69 70 ■91 93 95 .1725 .1791 .1858 66 67 68 87 89 9^ .1638 .1702 .1767 64 65 66 83 85 87 4200 !4300 4400 .2232 .2309 .2386 71 77 77 106 no 112 .2126 .2199 .2274 73 75 75 103 105 107 .2023 .2094 .2167 71 73 74 97 99 102 .1926 ••995 .2065 69 70 72 93 95 98 ••833 .1900 .1967 67 67 69 89 9« 92 4500 4600 4700 .2463 .2542 .2623 79 81 81 114 117 120 •2349 ■2425 ■2503 76 78 79 108 no "3 .2241 •2315 .2390 74 104 106 108 •2137 .2209 .2282 72 73 74 101 103 105 .2036 .2106 .2177 70 7^ 72 95 97 100 4800 4900 5000 .2704 .2787 .2870 •83 83 8S 122 126 128 .2582 .2661 .2742 79 81 82 116 118 120 .2466 ■2543 .2622 77 79 79 1 10 ill IIS .2356 •2431 .2507 75 76 77 •07 109 n2 .2249 .2322 •2395 73 73 75 102 104 105 5100 5200 5300 •295s .3041 .3128 86 87 88 13' 133 '35 .2824 .2908 •2993 84 85 85 123 126 129 .2701 .2782 .2864 81 82 83 117 120 123 .2584 .2662 .2741 78 79 80 ••4 116 n8 .2470 .2546 .2623 76 77J 78 108 no nz 5400 5500 5600 .3216 •3306 ■3396 90 90 91 138 141 143 ■3078 •3165 •3253 87 §8 88 131 13+ '37 ■2947 ■3031 .3116 84 85 85 126 128 130 .2821 .2903 .2986 82 83 83 120 •23 126 .2701 .2780 .2860 79 80 82 ••5 117 ••9 5700 5800 5.900 •3487 ■3579 •3673 92 94 95 146 149 152 ■3341 •3430 ■3521 89 9t 92 140 142 144 .3201 .3288 ■3377 87 89 90 132 134 138 .3069 ■3154 ■3239 85 85 87 127 130 132 .2942 .3024 •3'07 82 83 85 121 123 125 6000 6100 6200 3768 .3864 .3961 96 ■ 97 99 155 157 «S9 •3613 •3707 .3802 94 95 96 146 149 152 ■3467 •3558 •3650 91 92 93 141 144 146 •3326 •34^4 •3504 88 90 90 •34 •3<3 •39 .3192 .3278 •3365 86 87 88. 127 130 132 6300 6400 6500 .4060 .4160 .4260 100 100 102 162 165 167 .3898 ■3995 ■4093 97 98 99 155 158 161 •3743 •3837 •3932 94 95 96 149 151 •S3 •3594 .3686 ■3779 92 93 94 141 •44 •47 •3453 •3542 .3632 80 90 91 .36 •38 •39 6600 6700 6800 .4362 .4465 .4570 •03 105 106 170 »73 176 .4192 .4292 •4394 100 102 102 164 166 169 .4028 .4126 .4225 98 99 100 •55 •58 161 ■3873 •39-58 .4064 95 96 97 •50 •52 •55 •3723 .3816 •3909 93 93 95 141 •45 •47 62 TABLE I . — Continued.— -AuaMiary m. z 1700 J. ^. 1750 -1305 .1360 .1418 4 55 58 59 4 68 69 72 1800 •1237 .1291 .1346 54 55 56 64 67 69 1850 •1173 .1224 .C277 51 53 54 59 60 63 1900 .1114 .1164 .1214 50 50 52 ^., :7oo 3800 •'377 ■1435 .1494 58 59 61 72 75 76 55 58 59 3900 4CX30 •JIOO • 1555 .1617 .1680 62 63 64 80 82 •1477 •1537 •1598 60 6i 61 75 17 80 .1402 .1460 .1518 58 58 60 71 74 75 •1331 .1386 • 1443 55 57 58 65 67 70 .1266 •1319 •1373 53 54 56 61 63 66 12.00 4300 •MOO •1744 .1809 .1875 65 66 66 85 88 90 .1659 .1721 .1785 62 64 65 81 82 84 •1578 .1639 .1701 61 62 62 77 79 81 .1501 .1560 .1620 60 61 72 75 77 .1429 .1485 ■1543 56 58 59 69 70 73 4500 4600 4700 .1941 .2009 .2077 68 68 70 9> 93 94 .1850 .1916 .1983 66 67 67 87 90 92 •'763 .1826 .1891 63 65 66 82 84 86 .1681 • 1742 .1805 61 63 64 79 80 83 .1002 .1662 .1722 60 60 62 75 77 78 4800 4900 5000 .2147 .2218 .2290 71 72 72 97 99 102 .2050 .2119 .2188 69 69 71 93 95 96 •i9'7 .2024 .2092 67 68 68 88 91 93 .1869 •1933 .1999 64 66 67 85 86 88 .1784 • 1847 .1911 63 64 65 81 82 8+ 5100 ■5200 S300 .2362 .2436 .25 ri 74 75 75 103 •OS 108 .2259 •233' .2403 72 72 74 99 101 103 .2160 .2230 .2300 70 70 72 94 96 97 .2066 .2134 .2203 68 69 69 90 •92 94 .1976 .2'^42 .2109 66 67 68 86 88 90 5400 5500 5600 .2586 .2663 .2741 77 78 80 109 II I 114 •2477 •2552 .2627 75 75 77 105 107 108 .2372 .2445 .2519 71 74 74 100 102 104 .2272 ■2343 .2415 71 72 73 95 97 99 .2177 .2246 .2316 69 70 70 92 94 96 5700 5800 5900 .2821 .2901 .2982 80 81 82 "7 119 121 .2704 .2782 .2861 .78 79 80 II I "3 IIS •2593 .2669 .2746 76 77 78 105 108 no .2488 .2561 .2636 73 75 76 102 103 105 .2386 .2458 •2531 72 73 74 97 99 lOI 6000 6100 6200 .3064 •3147 •3232 83 85 85 123 124 127 .2941 •3023 •3105 82 82 84 117 120 122 .2824 .2903 .2983 79 80 81 112 114 116 .2712 .2789 .2867 77 78 79 107 no 112 .2605 .2679 •2755 74 76 78 103 104 106 6300 6400 6500 •3317 •3404 •3^ 87 89 128 131 13s .3189 •3273 •3358 84 85 87 125 126 128 .3064 •3147 .3230 83 83 85 118 121 123 .2946 .3026 .3107 80 81 83 113 115 117 •2833 .2911 .2990 78 79 80 109 III 112 6600 6700 6800 .3582 •3671 .3762 89 91 92 137 139 141 •3445 •3532 .3621 87 89 89 130 132 134 •3315 .3400 ■3487 85 87 87 (25 126 129 .3190 •3274 •3358 84 84 86 120 123 124 .3070 •315I •3234 81 83 84 114 116 118 63 TABLE 1. — Continued. — Auriliary m. z 1950 J. ^. 2000 A A 3050 J. A 2100 A J, 2150 J. '^, 3600 .1059 47 52 .1007 46 48 •0959 43 45 .0914- 41 42 .0872 39 39 3700 .1106 49 53 •1053 46 51 .1002 44 .47 .0955 42 44 .0911 40 4' 3800 .1155 50 56 ,1099 48 53 .1046 46 49 .0997 43 46 •0951 42 42 3900 .1205 51 58 .1147 48 55 -1092 46 52 .1040 45 47 •0993 43 44 46 48 4000 .1256 51 61 • 119s 49 57 .1138 48 531 .1085 46 49 .1036 44 4100 .1307 53 63 .1244 5< 58 .1186 49 55 .1131 47 5' .1080 45 4200 .1360 55 65 .1295 52 60 • 1235 50 S7 .1178 48 53 .1(25 46 50 4300 .1415 55 68 ••347 54 62 .1285 51 59 .1226 49 55 .1171 47 52 4400 .1470 57 69 .1401 55 65 •1336 53 61 .1275 50 57 .1218 48 54 4500 .1527 58 71 .1456 56 67 .1389 54 64 • 1325 52 59 .1266 .1316 • 1367 50 5& S8 60 4600 .1585 59 73 .1512 57 69 •1443 55 66 ■^377 54 61 51 4700 .1644 59 75 .1569 58 7> .1498 56 67 • 143' 54 64 52 4800 • '703 62 76 .1627 59 73 •1554 57 69 .1485 56 66 .1419 54 62 65 66 4900 .1765 62 79 .1686 60 75 .1611 58 70 .1541 56 68 • 1472 54 5000 .1827 63 8i .1746 62 77 .1669 60 . 72 ■1597 58 70 ■1527 56 5100 .1890 64 82 .1808 62 79 ■1729 61 74 .1655 58 72 •1583 57 6S 5200 ■.1954 65 84 .1870 63 80 .1790 61 77 ■ 1713 59 73 .1640 58 70 5300 .2019 66 86 •1933 64 82 .1851 63 79 .1772 61 74 .1698 59 71 5400 .2085 67 88 .1997 66 83 .1914 63 81 •1833 62 76 • 1757 60 73 5500 5600 .2IS2 68 89 .2063 66 86 •1977 64 82 .1895 63 78 .1817 6i 74 .2220 69 9> .2129 67 88 ,2041 66 83 .1958 63 80 .1878 62' 70 5700 .2289 70 93 .2196 68 89 .2107 67 86 .2021 65 81 .1940 63 77 5800 .2359 7> 95 .2264 69 90 .2174 67 88 .2086 66 83 .2003 65 79 5900 .2430 72 97 •2335 70 92 .2241 6g 89 .2152 67 84 .2068 65 81 Cjooo .2502 73 99 .2403 72 93 .2310 6g 9' .2219 68 86 •2133 66 67 68 83 6100 •2575 74 100 ■2475 72 96 •2379 7c 92 .2287 70 88 .2199 84 86 6200 .J649 75 102 ■2547 74 98 .2449 72 92 ■2357 70 9' .2266 6300 .2724 76 103 .2621 74 100 .2521 73 94 .2427 70 93 ■2334 69 88 89 6400 .2800 78 105 .2695 76 lOI .2594 73 97 .2497 71 94 .2403 71 6500 .2878 78 107 .2771 76 104 .2667 75 99 .2568 73 94 .2474 71 91 65oo .2956 79 109 .2847 78 105 .2742 76 lOI .2641 74 96 •2545 72 92 6700 6800 .3035 81 no .2925 78 107 .2818 77 103 .2715 75 98 .2617 73 93 .3116 82 113 .3003 8o 108 .2895 54 78 105 .2790 76 100 .2690 74 95 TABLE II. For Spherical Projeotiles. u Sin) Dm 25 A{u) Dift I I{u) Diff 40 T{u) Diff 2000 0.00 .00000 .000 12 1990 1980 25 49 24 25 0.0 1 0.02 I 2 .00040 .00080 40 41 .012 .025 13 12 1970 i960 1950 74 9P 124 25 25 26 0.04 0.08 0.13 4 5 5 .00121 .00163 .00205 42 42 43 •037 .050 .063 13 13 13 1940 1930 1920 150 175 201 25 26 25 0.18 0.25 0-33 7 8 9 .00248 .00292 .00336 44 44 45 .076 .089 .102 13 13 14 1910 1900 1890 226 252 278 26 26 26 0.42 0-53 0.65 II 12 13 .00381 .00427 .00473 46 46 47 .116 .129 •143 13 u 14 18S0 1870 i860 304 330 357 26 27 26 0.78 0.92 1.07 14 15 17 .00520 .00568 .00617 48 49 49 •157 .171 .185 14 14 14 '1850 1840 1830 383 409 436 26 27 27 1.24 1-43 1.63 19 20 21 .00666 .00716 ■ .00767 50 SI 52 •'99 .214 .228 •5 14 IS 1820 181O 1800 463 490 517 27 27 28 1.84 2.07 2.31 '23 24 26 .00819 .00872 .00926 53 54 55 •243 .258 •273 IS 15 15 1790 1780 1770 545 572 600 27 28 28 2.57 2.84 3-'4 27 30 31 .00981 .01036 .01093 55 57 57 .288 ■304 •319 16 15 16 1760 1750 1740 628 656 684 28 28 28 3-45 378 4-13 33 35 37 .01150 .01209 .01268 59 59 61 •335 •351 •367 16 16 16 1730 1720 I7IO 712 741 769 29 28 29 4.50 4.89 5-30 39 41 43 65 .01329 .01390 •01453 61 63 64 •383 .400 .416 17 16 17 TABLE II.— CoNTmuED u 798 827 856 Difif 29 29 30 A{u) Diff 45 47 50 /(«) Diff 65 66 67 T{u) Diff 1700 1690 1680 5-73 6.18 6.65 .01517 .01582 .01648 .433 .450 .468 •7 18 •7 1670 1660 1650 886 9'S 945 29 30 30 7-iS 7.67 8.21 52 54 56 .01715 .01783 .01853 68 70 71 .485 ■503 .521 18 18 18 1640 1630 1620 975 1005 10 ',6 30 31 30 8.77 9-35 9-97 58 62. 64 .01924 .01996 .02070 72 74 75 •539 •558. .576 •9 18 '9 I6I0 1600 1590 1066 1096 1127 30 31 31 io.6i 11.27 11.96 66 69 72 .02145 .02222 .02300 77 78 79 .595 .614 .633 '9 •9 20 1580 1570 1560 1158 1 189 1220 31 32 12.68 13-44 14.22 76 78 82 .02379 .02460 .02542 81 82 84 .653 .673 •693 20 20 20 1550 1540 1530 1252 1284 i3>6 32 32 32 15.04 15.90 16.78 86 88 92 .02626 .02712 .02799 86 87 89 .7^3 .734 •755 21 21 21 1520 I5IO 1500 1348 13S0 1413 32 33 33 17.70 18.65 19.63 95 98 100 .02888 >02979 .03072 91 93 94 .776 ■797 .819 21 22 22 1490 I4S0 1470 1446 1479 1512 33 33 34 20.63 21.68 22.77 105 109 114 .03166 .03262 .03360 96 98 101 .841 .863 .885 22 22 23 1460 1450 1440 1546 1580 1614 34 34 34 23.91 25.10 26-34 119 124 128 .03461 .03564 .03669 103 105 107 .908 •93' •955 23 24 24 '4.30 7420 I4I0 1648 1682 1717 34 35 35 27.62 28.95 30.33 •33 •38 '43 .03776 .03885 .03907 109 112 114 -979 1.003 1.028 24 25 25 1400 •390 1380 •752. 1787 1823 35 36 35 3>-76 33-25 34-79 •49 •54 160 56 .04111 .04227 .04346 116 119 122 '•053 1.079 1. 105 26 26 26 TABLE II.— Continued. X U S(f) Diff 36 37 36 A{u) Dift 164 170 175 /(«) Diff 124 127 129 T(u) Diffj' 1370 ij6o 1350 1858 1894 193' 36.39 38.03 39-73 .0446.8 .04592 .04719 1. 13' 1. 158 1. 185 27 27 27 1340 1330 1320 1967 2004 2041 37 37 37 41.48 43-29 45.14 i8i 185 191 .04848 .04981 .05117 133 136 139 1. 212 1.239 1.267 27 28 -27 1310 1300 1290 2078 2116 2154 38 38 38 47.05 49.01 51.04 196 203 212 .05256 .05398 .05542 142 144 148 1.294 1.322 I-35I 2? 29 39 1280 1270 1260 2192 2231 2269 39 38 39 53->6 55-37 57.67 221 230 240 .05690 .05842 .05998 152 156 160 1.381 1.411 ,1.442 30 31' 3t 1250 1240 1230 2308 2348 2388 40 40 40 60.07 62.56 65.14 249 258 267 .06158 .06323 .06492 165 169 174 1-473 1.505 '•538 32 33 33 1220 1210 120a 2428 2470 2512 42 42 22 67.81 70.59 73-54 278 295 156 .06666 .06846 ■07033 1 80 187 97_ 1.571 1.605 1.640 3? 35 iS 1195 1 190 .185 2534 2556 2578 22 22 22 75.10 76.70 78-32 160 162 16s .07130 .07229 .07329 99 100 102 1.658 1.676 1.694 f8 18 18 1 1 So 1 170 2600 2623 2646 23 23 23 79-97 81.66 83-39 169 173 177 .07431 .07535 .0764 1 104 106 108 1.712 1-731 1.751 T9 29 19 1 165 I f6o "55 2669 2692 2715 23 23 24 85.16 86 98 88.84 182 186 I go .07749 .07859 .07972 IIO 113 115 1.770 1.790 1.810 2b 20 21 1 150 1 145 1 140 2739 2763 2787 24 24 25, 90.74 92.69 94.68 195 '99 205 .08087 .08204 .08324 117 120 122 1. 831 1.852 1-873 2f 21 22 «J35 1130 1125 2812 2S37 2!s6j -5 24 25 9'5-73 98.82 00.97 209 215 221 .08446 .08!; 70 ' .0S697 124 127 130 1.895 1.917 1.940 22 23 ?3 G7 TABLE II.— CoxTiNCED. u S(u) 2886 2912 2938 Dift 26 26 26 A(u) Diff 226 233 239 /(«) Diff 132 135 «38 T(u) Diff 1 120 HIS IIIO 103.18 105.44 107.77 .08827 .08959 .09094 1.963 1.986 2.009 23 23 24 1 105 1 100 1095 2964 2991 3017 27 26 27 1 10.16 112.62 115-13 246 251 259 .09232 •09373 .09516 141 143 147 2-033 2.0S7 2.081 24 24- 25 1090 1085 1080 3044 3071 3099 27 28 28 117.72 120.38 123-13 266 275 283 .09663 .0981 2 .09965 149 '53 156 2.106 2.132 2.158 26 26 26 107s 1070 1065 3127 31SS 3184 28 29 125.96 128.87 J31.87 291 300 308 .10121 .10280 .10443 159 163 166 2.184 2.210 2.237 26 27 2S 1060 1055 1050 3213 3243 3273 3P 30 30 134-95 138.12 141-38 3'7 326 338 .10609 .10779 .10952 170 173 177 2.265 2.293 2.321 28 28 29 ro45 1040 '035 3303 3333 3364 30 31 31 14476 148.22 151.77 346 355 364 .UI29 .11310 .11495 181 185 189 2.350 2-379 2.409 29 30 3> 1030 1025 1020 3395 3427 3459 32 32 32 155-41 159-15 162.99 374 384 394 .11684 .11877 .12074 193 197 202 2.440 2.471 2.502 31 31 32 1015 lOIO 1005 3491 -3524 3557 33 33 34 166.93 170.99 175.17 406 418 430 .12276 .12482 .12693 206 211 215 2-534 2.566 2.599 32 33 33 1000 995 990 359' 3625 3660 34 35 35 179-47 183.90 188.46 443 456 470 .12908 .13128 •13354 220 226 231 2.632 2.665 2.699 33 34 35 985 980 975 369s 3731 3767 36 36 36 193 16 198.00 202.98 484 498 513 •13585 .13821 .14062 236 24 « 246 2-734 2.770 2.806 36 36 37 970 955 960 3803 3840 3877 37 37 38 208. 1 1 213.40 218.86 529 546 563 68 .14308 .14560 .14818 252 258 264 2.843 2.881 2.920 38 39 39 TABLE n.— C,o.\TiNUED. 95 S 950 945 940 935 930 925 920 9'5 910 905 900 895 890 880 875 870 865 860 855 850 845 840 835 830 825 S20 815 810 805 800 795 S{u) Dift 38 39 39 224.49 230.29 236.29 Diff 580 6oo 620 391S 3953 3992 4031 4070 41 10 39 40 40 242.49 248.86 255-43- 637 657 676 4151 4192 4234 41 42 43 262.19 269.17 276.37 698 720 743 4277 4320 4363 43 43 44 283.80 291.47 299.40 767 793 819 4407 4451 4496 44 45 46 307-59 316.04 32477 845 873 901 4542 4589 4636 47 47 48 33378 343-06 352-67 928 961 997 4684 4733 4781 48 49 49 362.64 372.96 383.60 1032 1064 1099 4830 4880 4931 50 51 52 394-59 405.96 417.71 1137 1 175 1216 4983 5036 5089 53 53 54 429.87 442.45 455-47 1258 1302 '347 5 143 5198 5253 55 55 56 468.94 482.89 497-33 1395 1444 •495, 5309 5366 5424 57 58 59 512.28 527.77 543-81 1549 1604 1661 I{u) .I50S2 •15352 .15628 .15911 .16201 .16498 .16802 •I71I3 -17432 •17759 . 1 8094 -18437 .18789 .19149 .19518 .19896 .20283 .20680 .21087 .21505 ■21933 .22372 .22823 -23285 .23761 .24248 .24746 .25257 -25783 -26323 .26876 -27444 .28031 Diff r(«) 270 276 283 2-959 2.999 3.040 290 297 304 3.082 3-125 3-168 3" 3'9 327 3.212 3-257 3-303 335 343 352 3-350 3-397 3-445 360 369 378 3-494 3-544 3-595 387 397 407 3-647 3-700 3754 418 428 439 3.809 3.865 3.922 451 462 476 3.980 4-039 4.100 487 498 511 4.161 4.224 4.288 526 540 553 4-354 4.421 4.489 568 587 601 4-559 4-630 4-702 Diff 40 41 42 43 43 44 45 46 47 47 48 49 50 51 52 53 54 55 56 57 58 59 61 61 63 64 66 67 68 /' 72 74 69 TABLE II.— CoNTuniED. u S(u) Difl 59 60 61 A{u) •Dift 1722 1784 1849 I{u) Diff _ 617 634 650 Tin) Diff 790 78s 780 5483. 5542 5602 560.42 577-64 595.48 .38632 : -29249 ' .29883 4.776 4.853 4.929 76 77 79 775 770 76s 5663 5725 5788 62 63 64 613.97 633-13 653.01 =• 1916 1988 2062 ^30533 .31203 .31891 670 •688 707 5.008 5.088 5.170 80 82 84 760 755 750 5852 5917 5983 65 66 .67 673-63 * 695.01 7'7-i9- 2138 2218 2303 ^32598. ,'•33325 i .34073 .727 748 770 5.254 5 -340 5-427, 86 87 90 745 740 735 6050 6118 6187 68 69 740.22 ■ 764.11 788.91 2389 2480 2574 •34845 . .35634 • .36448 ^37285 .38146 ■ .39033 •39945 .40885 .41853 791 814 837 5-5'7 5.608 5-701. 91 96 730 725 720 6256 6327 6399 71 72 73 814.65 -^ 841.38 869.14 ■ 2673 2776 2882 861 887 912 5-797 5.894 5-994 97 100 102' 715 710 705 6472 6546 6621 74 75 77 897.96 ' 927.92 959.07 2996 3115 3238 940 968 • 995 6.096 6.200 6.306 "104 106 109 700 695 690 6698 6776 6855 78 79. 80 991.45 1025.2 1060.2 3366 350 364 .42848 •43872 .44926 1024 1054 1089 6.415 6.526 6.640 111 114 116 685 680 675 6935 7016 7098 81 8z 84. 1096.6 1 134-4 1173.8 378 394 409 .46015 -47143 .48302 1128 1159 1192 6.756 6.875 6.997 119 122 125 670 665 66c 7182 7267 7354 85 87 88 1214.7 1257.4 1301.8 427 444 463 .49 '■94 .50722 .51989 1228 1267 1307 7.122 7.249 7.380 127 131 134 65s 650 645 7442 753' 7622 ' 89 9' 92 1 348-1 1396-3 1446.5 4S2 502 523 .53296 .54645 •56037 1349 1392 1436 7-514 7.651 7.-79 1 137 1. 10 •43 640 635 630 7714' 7808 7903. 94. 95 97 1498.8 JS534 1610.2 546 568 592 60 .57473 .58955 .60484 1482 1529 1579 7-934 8.081 8.231 147 150 •54 TABLE II.— GoNTiNUEn. u S{u) Diff 98 100 lOI A{u) Dift 618 644 673 /(.,) Diff 1633 1690 1737 T{ii) Diff 625 620 615 8000 8098 8198 1669.4 1731.2 1795.6 .62063 .63696 .65386 8.385 8-543 8.70s 158 162 166 1 610 60s 600 8299 8402 8507 103 105 107 1862.9 I933-1 2006.4 702 733 765 .67123 .68922 .70781 1799 1859 1923 8.871 9.041 9-215 170 174 179 595 590 585 8614 8722 8833 1 08 III 112 2082.9 21629 2246.5 800 836 872 .72704 .74692 -76747 1988 2055 2126 9-394 9-577 9.765 183 188 192 580 575 570 8945 9059 9»75 114 116 118 2333-7 2424.8 2520.2 911 954 998 .78873 .81072 .83348 2199 2276 2356 9-957 10.154 10.357 197 203 208 565 560 555 9293 9413 9535 120 122 124 2620.0 27M-3 2833.4 1043 1091 1142 .85704, .88144 .90670 2440 2526 2617 10.565 10.778 10.997 213 219 225 550 545 540 9659 9785 9914 126 129 131 2947.6 3067.2 3192-4 1196 1252 1312 .93287 .95998 .98808 2711 2810 2913 11.222 11-453 1 1.690 231 237 243 535 530 525 10045 10178 10313 133 135 138 3323-6 3461.0 3605.0 1374 1440 1509 1. 01 721 1.04740 1-07873 3019 3133 3247 11-933 12.183 1 2.440 250 . 257 264 520 515 510 1045 1 '°59' 10734 140 143 146 3755-9 39'4-i 4080.1 1582 1660 1743 1.1II20 1.14486 1.17981 3366 3495 3633 12.704 12.975 13-254 271 279 287 505 500 495 10880 11028 1 1 179 148 151 153 4254.4 4437-3 4629.3 1829 1920 2017 I.21614 1-25393 1. 29312 3779 3919 4070 13-541 13-836 14.138 295 30,2 312 490 485 480 11332 1 1488 11648 156 1 6a 162 483-' 5042.8 5265.4 2118 2226 2340 1-33382 1. 37614 1.42013 4232 4399 4575 14.450 14.770 15.100 320 330 340 475 470 463 11810 1197s 12M3 165 168 172 5499-4 ■5745-5 6004.3 2461 2588 2724 61 1.46588 1-51348 1.56301 4760 4953 5157 15.440 1,5-790 16.150 350 360 370 TABLE 11.— CONTINDED.- u S{u) Diff 175 178 A{u) Di£f 2868 3020 I{u) Diff 5368 5593 T{a) Diff 460 455 450 12315 12490 12668 6276.7 '^563.5 6865.5 1. 61458 1.66826 1. 72419 16.520 16.902 17.296^ 382 394 63 TABLE III. Valuesof-^f Of temperature and press'ire of atoiospAere two-thirds saturated with moisture. F 28 in. 29 in. 30 in. 3 1 ill- F 28° 29 30 28 in. 1.004 1.006 1.008 29 in. 0.969 0.971 0-973 30 in. 3 1 in. o° I 2 3 0.945 0.947 0.949 -0.951 0.912 0.914 0.916 0.918 0.882 0,884 0.886 0.888 0.853 0.853 0,857 0.859 0.937 0-939 0.941 0.907 0.909 ' 0.911 4 5 6 0.953 0.955 0.957 0.920 0,922 0.924 0.890 0.892 0.893 0.861 0.863 0:865 3t 32 33 I.OIO 1. 01 2 1.014 0.975 0.977 0-979 0.943 0-945 0.947 0.912 0.914 0.916 8 9 0.959 0.962 0.964 0.926 0.928 0.930 0.895 0.897 0,899 0.867 0869 0.S70 34 35 36 1.016 1. 01 8 1.021 0.981 0.983 0.986 0.949 0.951 0-953 0.918 0,920 0.922 lO n 12 0.966 0.968 0.970 0.932 0.935 0-937 0.901 0.905 0.905 0.872 0.874 0.876 37 3? 39 1.023 1.025 1.027 0.988 0.990 0.992 0-955 0.957 0.958 0.924 0.926 0.928 •3 H . J5 0.972 0.974 0.976 0.939 0.941 0.943 ■0.907 0.909 0.911 0.878 0.880 0.882 40 41 42 I 029 1.03 1 1-033 0.994 0.996 0.998 0.960 0.962 0.964 0.930 0.932 0-933 i6 17 i8 0.978 0.981 0.983 0.945 0.947 0.949 0.913 0915 0.917 0.884 0.8S6 0.888 43 44 45 1-035 1-037 1.040 1.000 1. 00? 1.004 0.966 0.968 0.970 0-935 0-937 0-939 •9 20 21 0.985 0.987 0.989 0.951 0.953 0-955 0.919 0.921 0.923 0.890 0.89 J 0.893 46 47 48 1.04.2 1.044 1.046 1.006 1.008 I.OlO 0.972 0.974 C.976 0.941 0-943 0.945 22 23 24 0.991 0-993 0.995 0.957 0.959 0.961 0.925 0.927 0.929 0.895 o.«97 0.S99 49^ 50 51 1.048 1.050 1.052 1.012 1.014 1 016 0.978 0.980 0.982 0-947 0-949 0-95 1 25 26 27 0.997 1. 000 1.002 0.963 0.965 0.967 0.931 0.933 0.935 0.901 0.903 0.905 52 53 54 1.054 1.036 1.058 1.018 1.020 1.022 0.984 0.986 0.988 0-953 0-954 0.936 TABLE III.— CONTINDED. F 28 in. 29 in. 30 in. 31 in. F 79° 80 81 28 In. 29 in. 1.073 1.075 1.077 30 in. 31 in. 55° 56 5; 1.06 1 1.0.63 1.065 1.024 1.026 1.028 0.990 0.992 0.994 0.958 0.960 0.962 I. Ill 1. 113 l.i 16 I -037 1-039 1.041 1.004 1.006 1.008 58 59 60 1.067 1.069 1.07 1 1.030 1.032 1.034 0.996' 0.998 1. 000 0.964 0.966 0.96S 82 83 84 1. 118 1. 120 1. 122 1.079 i.o8£ 1-083 1.043 1.045 1.047 I.OIO 1.012 1:014 61 62 63 1.073 1.075 1.078 I -037 1.039 1.041 1.002 1.004 1.006 0.970 0.972 0.974 85 86 87 1. 1 24 1. 126 1.128 1.085 1.088 1.090 1.049 1.051 I -OS 3 1.016 1.017 1.019 64 65 66 1.080 1.082 1.084 I -043 1.045 1.047 1.003 I.OIO 1.012 0.975 0.977 0.979 88 89 90 1-130 1. 132 I-I35 1.092 1.094 1.096 1.055 1.057 1.059 1.02 1 1.02} 1.025 fez- es 69 1.086 1.088 1.090 1.049 1.05 1 1-053 I.OI4 i.oi6 I.0I8 0.981 0.983 0.985 91 92 93 i-»37 1-139 1. 141 1.098 1. 100 1.102 1. 06 1 1.063 1.065 1.027 1.029 1.03 1 70 71 72 1.092 1.094 1.097 1.055 1.057 1.059 1.020 1.022 1.024 0.987 0.989 0.991 94 96 I-I43 1. 145 1. 147 1. 104 1. 106 1. 108 1.067 1.069 1. 07 1 1-033 I -03 5 1-037 73 74 75 1.099 I.IOI 1. 103 f.o6i 1.063 1.065 1.025 1.027 1.029 0-993 0.995 0.996 97 98 99 1. 149 1. 151 i.iSS I. no 1. 112 1. 114 1-073 1.075 1.077 1.038 1.040 I.O495 34 30 34 40 34 50. -73S08 •74330 •74854 522 524 528 .6872S .69157 .695S8 429 431 433 40 00 40 10 40 20 0.9 29 J 4 0.93563 D.94218 - 649 . 655 659 .83910 .84407 .84906 "497 ,499 ?S02 35-00 3S'io 35I20 •75382 .75912 .76447 530 535 537 .70021 •70455 .70891 434 436 438 40 30 40 40 40 50 0.94877 0.95541 0.96210 664 669 ■ 674 .85408 .85912 .86419 .'504- 1507 £510 35'^3q 35 40 .35 50 .76984 •77525 .78069 541 544 548 •71329 .71769 .72211 440 442 443 41 00 41 10 41 20 0.96884 0.97563 0.98248 679 685 689 .86929 .87441 •87955 :si2 I514 518 '36 00 36 10 36 20 .78617 .79168 79722 55' 554 558 .72654 .73100 ■73547 446 447 449 4t 30 41 40 41 50 0.98937 0-99633 1.00333 696 700, 706 ■88473 .88992 .89515 519 523 525 36 30 36*40 36 so .80280 .80842 .81408 562 566 569- •73996 •74447 -74900 451 453 •455 42 00 42 10 4:? 20 1.01039 1.01751 1.0246S 712 717 723 .90040 .90569 .91099 52^ 530 534 37' 00 37 10 3Z 20 .81977 .S2550 .83127 573 577 580 •75355 .75812 .70272 '457 '460 461 42 30 42 40 42 50 1.03191 1.03919 1.04654 728 735 741 •91633 .92170 .92709 537 539 543 37,30 37.40 37^50 •83707 .84292 .84880 585 58S 593 •76733 .77196 .7766. 463 465 468 43 00 43 10 43 20 ••OS 395 1.06141 1.06894 746 753 759 .93252 •93797 •94345 545 55' 3S 00 3S 10 3S 20 1 •85473 .86069 .76670 596 601 605 .78129 •78598' .79070 469 472 474 43 30 43 40 43 50 1.07653 1.0S41S 1.09190 765 772 778 .94S96 '95451 .96008 555 557- 561 TABLE IV.— COKTIKUED. («) Diff 785 792 79S Tiurl? Drff 564 567 570 e (ff) g-JiT—J J^ DTf Tan ti D.n 44°oo' 44 lo 44 20 1.09968 I-I07S3 '•"545 1 0.96569 0.97133 0.97700 49°3o' 49 40 49 SO 1.40001 1.41068 1.42147 1067 1079 1089 1.17085 1.17277 1.18474 692 697 701 44- 30 44 40 44 50 '•12343 1.13148 1. 1 3960 S05 812 819 0.9S270 0.9S843 0.99420 573 577 580 50 00 50 10 50 20 143236 '•44337 1.45450 1101 1113 1124 1.19175 1.19882 '•20593 707 7" 717 45 00 45- 10 45 20 '•14779 1.15606 '■16439 827 833 ■841 I .OOOOQ 1.00533 1.01170 583 587 591 50 30 50 40 50 50 1.46574 1.47710 1.48859 1136 "49 1161 1.21310 1.22031 1.22758 721 727 732 45 30 45 40 45 SO 1.172S0 I 18129 1. 1 8985 S49 856 864 I.OI75I 1.02355 1.02952 594 597 601 51 00 51 10 51 20 1.50020 '■51193 '•52379 "73 1186 1200 1.23490 1.24227 1.24969 737 742 748 46 00 46 10 46 20 1.19849 1. 20721 1. 2 1 600 •872 879 888 ' 03553 I.O4I58 1 .04766 605 608 612 5' 30 51 40 51 50 '•53579 '•5479' 1.56017 1212 1226 1240 1.25717 1.26471 1.27230 754 759 764 46 30 46 40 46 50 1.22488 1.23384 1.24288 896 904 913 1.05378 1.05994 1. 066 1 3 616 619 624 52 00 52 10 52 20 1.57257 1.5-8510 1.59778 '253 1268 1282 1.27994 1.28764 1.29541 770 777 782 47 03 47 10 47 20 1. 25201 1. 26123 1.27053 922 930 93S 1.07237 1.07864 1 .08496 627 632 63s 52 30 52 40 52 50 1.61060 1.62357 1.63668 1297 13" 1327 1.30323 1.31110 1.31904 787 794 800 47 30 47 40 47 50 1.27991 1.28939 1.29896 948 957 967 1.09131 1.09770 1. 10414 639 644 647 S3 00 53 10 S3 20 1.64995 1.66337 1 .67696 1342 1359 1374 1.32704 '.335" '.34323 807 812 "819 48 00 48 10 4S 20 1.30863 1.31838 1.32823 975 985 995 I.I 1061 1. 11713 1.12369 652 656 660 S3 30 53 40 53 50 1.69070 1.70460 1.71867 1390 1407 1424 '•35 '4? 1.35968 1.36800 826 832 838 48 30 48 40 48 SO 1. 33818 1.34823 '-35838 (005 1015 1025 1.13029 1.13694 '•I43'53 665 669 674 54 00 54 10 54 20 1.73291 '•74732 1.76191 1441 '459 1476 '.37638 1.38484 '.39336 846 852 859 49 OD 49 'o. 49 20 1.36863 1.37898 1.38944 '03 5 1046 1057 '•15037 1.15715 1.16398 678 683 687 54 30* 54 40 54 SO 1.77667 1. 79 1 62 1.80675 1495 1513 1532- '■40195 1.41061 I 41934 866 873 881 TABLE IV.— CosTiNUED. e 5S°oo' 55 10 55 20 55 30 55 40 55 50 56 00 56 10 ^6 20 56 30 56 40 56 50 57 00 57 10 57 20 57 30 57 40 57 50 58 00 58 10 58 20 58.30 58 40 58 50 59 00 59 10 59 20 59 .30 59 40 59 50 60 00 60 10 60 20 I {0) 1.82207 1.83758 1.85329 1. 869 1 9 1.88530 1.90162 J.91815 1.93489 1.95186 1.96905 1 .98646 2.00411 2.Q2199 1813 2.04012 1837 2.05849 1863 Diff 1551 1571 1590 161 1 1632 1653 1674 1697 1719 1741 176s 1788 2.07712 2.09600 2.11515 2.1345'J 2.15424 2.17421 2.19446 2.21500 2.23583 2.25697 2.27842 2.30018 2.32226 2.34468 2-36743 2-39053 2.41398 2-43779 1888 i'9'5 1941 1968 '997 2025 2054 2083 2114 2145 2176 2208 2242 2275 2310 2345 2381 2417 Tuntf .42815 •43703 .44598 .45501 .4641 1 -47330 .48256 .49190 -50133 .51084 .52043 .53010 .53986 .54972 .55966 .56969 .57981 .59002 .60033 .61074 .62125 .63185 .64256 .65337 ,66428 67530 68643 69766 7090! 72047 7320s 74375 75556 Diff 888 895 903 910 919 926 934 943 951 959 967 97^ 986 994 1003 1012 1021 1031 1041 1051 1060 1071 1081 1091 1 102 i"3 1 123 H35 1146 1158 1 170 1181 1 193 d 60° 30' 60 40 60 50 61 Ob 61 10 61 20 61 30 61 40 61 50 52 00 62 10 62 20 62 30 62 40 62 50 63 00 63 10 63 20 63 30 63 40 63 50 64 00 64 10 64 20 64 30 64 40 64 JO 65 00 65 10 65 20 65 30 65 40 65 50 ip) 2.46196 2.48651 2.51145 2.53678 2.56251 2.58865 2.61521 2.64220 2.66963 2.69752 2.72586 2.75468 2.78398 2.81378 2.84408 2.87490 2.90626 2.93816 2.97062 3-00366 3.03728 3-07150 3.10634 3.14182 3-17794 3-21474 3-25221 3-29039 3.32929 3-36894 3-40934 3-45052 3-49251 Dift 2455 2494 2533 2573 2614 2656 2699 2743 2789 2834 2882 2930 2980 3030 3082 3'36 3190 3246 3304 3362 3422 3484 3548 3612 3680 3747 3S18 3890 3965 4040 4118 4199 428 il Tan e 1.76749 1-77955 1.79174 1.80405 1.8 1 649 1 .82906 1.84177 1.85462 1.86760 1.88073 1 .89400 1.90741 1 .92098 1.93470 1.94858 1.96261 1 .97680 1.99116 2.00569 2.02039 2.03526 2.05030 2.06553 2.0S094 2.09654 2-.1 1233 2.12832 2.14451 2.16090 2.17749 2.19430 2.21132 2.22857 Diff 1206 1-219 1231 1244 1257 1271 1285 1298 1313 1327 1 341 1357 •372 'j88 1403 1419 '436 1453 1470 1487 1504 1523 1541 1560 1579 •599 1619 •639 1659 1681 1702 1725 1747 TABLE IV.— Continued. 6 («) Di£f 4366 4452 4543 Tan^ Dili 1770 ■793 1817 e (P) 4.12255 4.17849 4.23566 Diff 5594 5717 5844 Tan 6 Diff 66°oo' 66 lo 66 2o 3-53532 3.57898 3.62350 2.24604 2.26374 2.28167 68°oo' 68 10 68 20 2.47509 2.49597 2.51715 2088 2118 2150 66 30 66 40 66 50 3.66893 3.71527 3-76257 4634 4730 4826 2.29984 2.31826 2.33693 1842 1867 1892 68 30 68 40 68 50 4.29410 4.35385 4.41495 5975 6110 6249 2.53865 2.56046 2.58261 21S1 2215 224.S 67 00 67 10 67 20 3.81083 3.86010 3.91040 4927 5030 S137 2.35585 2.37504 2.39449 1919 '945 1973 69 00 69 10 69 20 4.47744 4.54137 4.60678 6393 6541 66p4 2.60509 2.62791 2.65109 2282 2318 2353 67 30 67 40 67 50 3.96177 4.01422 4.06781 5245 5359 5474 2.41422 2.43422 2.45451 2000 2029 2058 69 30 69 40 69 50 4.67372 4.74225 4.81241 6853 7016 7184 2.67462 2.69853 2.72281 2391 2428 2467 70 00 4.88425 7359 2.74748 2506 11 72 TABLE v.— FOR MORTAR-FIRING. TABLE V. FOR MORTAR-FIRING. 0=30°. Ko = o.i5X V X c D f D (0 D »» ^ D 300 310 320 2243 2385 2529 142 144 146 9-13 9-43 9.72 30 29 28 31° 53' 32 00 32 07 7 7 7 274 281 289 7 8 7 330 340 350 2675 2824 2977 149 153 156 10.00 10.29 10.58 29 29 29 32 14 32 22 32 29 8 7 8 296 303 310 7 7 7 360 370 380 3133 3292 3454 159 162 163 10.87 II. 16 11.44 29 28 28 32 37 32 45 32 53 8 8 8 317 324 330 7 6 7 390 400 410 3617 3782 3949 165 167 170 11.72 12.00 12.28 28 28 28 33 01 33 09 33 18 8 9 9 337 343 349 6 6 6 420 430 440 4119 4291 4466 172 175 177 12.56 12.84 13-11 28 27 28 33 27 33 36 33 45 9 9 8 355 361 367 6 6 6 450 460 470 4643 4821 5001 178 180 181 13-39 13-66 13-93 27 27 27 33 53 34 02 34 II 9 9 10 373 378 384 5 6 5 480 490 500 5182 5365 5549 183 184 187 14.20 14-47 14.74 27 27 27 34 21 34 30 34 39 9 9 :o 389 394 399 5 5 5 ti> D D 300 310 320 2514 2669 2828 155 159 163 11.69 12.06 12-43 37 57 37 42° 23' 42 32 42 41 9 9 9 273 280 288 7 8 7 330 340 350 2991 3156 3324 165 168 170 12.80 13-16 13-52 36 36 36 42 50 43 00 43 09 10 9 10 295 302 309 7 7 7 360 370 3S0 3494 3666 3841 172 175 177 13.88 14.24 14.60 36 36 36 43 19 43 29 43 40 10 11 10 316 322 129 6 7 6 390 400 410 4018 4198 4382 180 184 185 14.96 15-31 15-67 35 36 35 43 50 44 00 44 10 10 10 II 335 342 348 7 6 6 420 430 440 4567 4752 4939 185 187 188 16.02 16.36 16.71 34 35 34 44 21 44 32 44 43 II II II 354 360 366 6 6 6 450 460 470 5127 5318 5511 191 . 193 194 17.05 17.40 17-74 35 34 33 44 54 45 05 45 16 II II II 372 378 383 6 5 5 480 490 500 5705 5899 6094 194 195 197 18.07 18.40 18.73 33 33 32 45 27 45 38 45 50 II 12 II 388 393 398 5 5 5 1 = 45°. Ya = 0.27.3r. 300 310 320 2541 2698 2856 157 158 160 12.83 13.24 13-64 41 40 41 47° 47 47 28' 37 46 9 9 10 273 281 288 8 7 8 330 340 350 3016 3179 3348 163 169 172 14.05 14.45 14.84 40 39 40 47 48 48 56 06 16 10 10 10 296 303 310 7 7 7 360 370 380 3520 3693 3868 173 175 178 15.24 15-63 16.02 39 39 39 48 48 48 26 36 47 10 II 10 317 324 330 7 6 7 390 400 410 4046 4228 4410 182 182 182 16.41 16.79 17.18 38 39 38 •48 49 49 57 08 19 337 343 350 6 7 6 420 430 440 4592 4776 4963 184 187 188 17.56 17.94 18.32 38 38 37 49 49 49 30 41 52 356 362 368 6 6 6 450 460 470 5I5I 5343 5535 192 192 192 18.69 19.06 19-43 37 37 36 50 50 50 03 14 25 12 374 379 385 5 6 5 480 490 500 5727 5921 6116 194 195 196 19.79 20.16 20.52 37 36 36 50 50 51 37 48 00 II 12 12 390 396 401 6 5 5 74 TABLE v.— FOR MORTAR-FIRING. TABLE \ .—{Continued). D "0. D 300 310 320 2499 2649 2802 150 153 157 13.89 14-32 14-75 43 43 43 52° 28' 52 37 52 46 9 9 9 275 283 290 8 7 7 330 340 350 2959 3120 3284 161 164 166 15-18 15.61 16.04 43 43 43 52 55 53 05 53 14 10 9 10 297 304 311 7 7 7 360 370 380 3450 3617 . 3787 167 170 173 16.47 16.90 17-32 43 42 41 53 24 53 33 53 43 9 10 lO 318 325 332 7 7 7 390 400 410 3960 4136 4314 176 178 178 17-73 18.14 18.55 41 41 41 53 53 54 04 54 15 11 II n 339 345 352 6 7 6 420 430 440 4492 4671 4852 179 i8i 183 18.96 19.37 19.78 41 41 40 54 26 54 36 54 47 10 II 10 358 365 371 7 6 6 450 460 470 5035 5220 5406 185 186 186 20.18 20.58 20.98 40 40 39 54 57 55 08 55 19 II 11 II 377 383 389 6 6 5 480 490 500 5592 5779 5968 187 189 189 21.37 21.76 22.15 39 39 39 55 30 55 41 55 53 11 12 II 394 400 405 6 5 6

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