Feb. 16, 1832.| 

 TRAJECTOR 



FOREST AND STREAM. 



51 



RAJECTORY CURVES OF SPORTING 

 RIFLES. 



THE UtWB GOVERNING THEM. 



1 WOULD like to correct a few typographical errors iu the 

 text to the table of "Trajectory Curves," published iu 

 your issue of Sept. 1st, 1881. Occasion will be taken, at the 

 :i;n time, to elaborate that text, in order to correct some 

 i - i. achiugs, on projectile science, frequently appear- 

 ing' in the sporting papers. 



These experiments were made for my own satisfaction, 

 and required more than a month's laborious care. The re- 

 sults were very satisfactory, and amply repaid me for the 

 I ii innl expense incurred," and were afterwards published to 

 counteract (be absurd claims of some riflemen, who. imagin- 

 I e laws of nature temporarily suspended for their 

 especial benefit, are still convinced* that their particular 

 ' nit " straight. " 



To the mass of persons using the rifle such experiments I 

 possess but little interest. They are at a loss to understand j 

 ft-hy any one should spend months among the mountains, un- | 

 uergoing what they consider hardships, except for the mere ; 

 killing and slaughter of game ; and then, to see one spending 

 Weeks, apparently busy and hard at work all the time, merely 

 ting through thin* paper screens; why, there is only one 

 thing about it, he must be "rattled." Using some standard 

 rifle, and the ammunition furnished with it, they arc entirely ! 

 Satisfied if the ammunition goes through the bore without 

 .: king and with reasonable" accuracy ; but as to the flight 

 their balls make through the air, or the effect of different 

 grades of powder or varying proportions of powder and ball 

 on that curve, they have not the remotest conception. Occa- 

 sionally making a hit at good long range, they are delighted, : 

 and do not: soon cease talking about it, though at the time 

 they may have been holding a foot from the point hit. A 

 miss is generally accounted for by the " deer or squirrel mov- 

 ing just as t pulled the trigger." If beaten at the target (usu- 

 ally 50 to 100 yards) there is always a reason; sometimes, 

 "Detu it, boys, I can't get down into my sights this morn- 

 ing. -: In the principles of projectiles they take but little in- j 

 terest, as they have but little knowledge of them; and this 

 remark will frequently apply to persons highly educated on 

 : i in ral subjects, and it is therefore the more pleasant to have 

 such flattering commendations on one's labors, from such a 

 eran rifleman as Major W. II. Merrell, as contained in 

 your issue of Sept. 8th; a rifleman who combines such cor- 

 rect theoretical knowledge with extended experience in the 

 field. I am sorry that the information about the "drop" of ; 

 tla- different balls cannot, at this time, be furnished, as the : 

 manner of making the experiments did not give that iufor- 

 jhation. That point could have been tested, approximately, 

 by measuring "the angle of projection" (the angle made by ; 

 line of sight with centre line of bore), but it was considered 

 unsatisfactory, because the best rifle does not always throw 

 its ball straight out from its bore, with proper allowance for 

 the action of gravity. With the ordinates for each curve i 

 known, however, the initial velocity and consequent drop of i 

 each ball can be calculated approximately, but my wander- 

 ing life and the stirring scenes of this season has, thus far, 

 pi-evented much study or deduction from these experiments. 

 The information will be furnished at the first leisure. 



A study of this table will show that its curves follow clearly 

 the known laws of projectiles hereafter alluded to, but, as a 

 rule, doubtless on account of imperfections of balls and rifles, . 

 the actual flight, through the air will not often be perfect j 

 theoretical curves. The cartridges used, however, under like | 

 conditions of barometer and atmosphere, will make substan- 

 tially the same curve, fired from any other good rifle, of same 

 length barrel. It must be borne in mind, however, that hav- 

 ing been made under an atmospheric pressure of about 25.00 

 inches of mercury they are flatter than if made under .like 

 conditions, under a pressure of 30:00 inches of mercury. 

 Take, for instance, Nos. 1 and 2, with initial velocities, as 

 stated by the Winchester Company, and Nos. 4 and 10, with 

 initial velocities, as given by the U*. S. Ordnance Department. 

 The highest points "of these curves, at 100 yards for a 200 

 yard range, will be: No. 1, 18.01 inches; No.*2, 13.41 inches; 

 no, 4, 12. -f 1 inches; No. 10, 14.37_ inches, and barometer 

 pressure, 30 inches mercury. At points intermediate in ele- 

 vation above sea level, these curves will vary in height, but 

 in the large game districts west of the Missouri they will ap- 

 ply, as given in the tables, very closely. 



Coming back, now, to typographical errors, and passing 

 over a few unimportant ones, that a careful reading will cor- 

 rect, a material error is in first line of third paragraph from 

 the end. "No. 3" should have been No, 4. as is evident; 

 and farther along, in same paragraph, "No. 12 is the Corn- 

 ing rifle among the buffalo hunters of the Lower Yellow- 

 stone;," should read, the "coming" rifle; that is, becoming 

 the popular rifle, about which I will have something to say 

 at another time. A material error is in fourth paragraph 

 from the end, in accounting for the great difference in the 

 curves of No. 1 and No. 4, the types say " probably " due to 

 the 8-inch longer barrel of No. 4. It should read,*" partial- 

 ly." instead of " probably," as the principal cause is really the 

 difference in weight of balls. It is a well settled law of pro- 

 jectiles that, the velocity being the same, and weights of ball 

 the same, the rcsist/mce of the atinmplwe retries as the squares 

 of the (loi oii-i: r of hot!, or, with the velocity constant and 

 varying wights and diameter of balls, the atmospheric- resist- 

 ance rura-.x U4 the square of the diameter m inches, divided by the 

 founds, or as d* In the case of No. i the 



height of Ml t 



w 



co-eificieiit of resistance is 0.77, and that of No. 4, 2.577, 

 B huh v iil account for much of the difference, A part of the 

 difference in these two curves is accounted for by the 8-inch' 

 longer barrel of No. 4. in accordance with the results of ex- 

 periments of the I'. S. Ordnance Department, as shown in 

 abstracts of ordnance notes given by myself in previous 

 numbers of Forest axd Stream. Major Farley's experi- 

 ments indicate that, with service ammunition (70-405 grs.) and 

 below 22 inch (carbine) length of barrel, the initial velocity 

 increased very rapidly up to that length. From 22 inch 

 length, with initial velocity of 1,210 f, s., there was a gradual 

 ; 'i irease up to rifle length' (32, 6 inches), initial velocity 1.820 

 f. s.; and from that to the 90-inch length a more gradual in- 

 i q 1,410 f. s. Difference of rifle length over carbine 

 pngth, 110 f. s. Powder used, musket powder; a slower 

 i>r,i uing powder than F. G. American powder, or the quicker 

 and si longer Curtis & Harvey brand. It is probable, that 

 with the latter quicker brands the maximum velocity would be 

 obtained, with a less length of barrel than 112 inches, but it 

 is doubtful whether the above difference of velocity would 

 he changed between 22 and 82.0 inch length of barrel. 



Another well-established law of projectiles is what is 

 known as "tbocufao law of resistance ;" (hat is, with diame- 

 ver and weight of bad constant, the atmospheric resistance- M- 



,■■■■>;: as the rube, of Iht celvoity. This law was carefully tested 

 by Professor Bashforth, who, from careful and exhaustive 

 experiments by means of his electric chronograph for five 

 years, whilst a member of the British Ordnance Committee, 

 "established that this law held good only at a velocity of 1,200 

 f . s. Belear, as well as ahore, that reloeity, the ati'uospheric 

 resistance was less than the cube of the velocity. For in- 

 stance: the co-efficient of resistance, established for the 

 ogival-headed and kindred forms of projectiles, were 108.9 

 for 1,200 f. s.; 75.0 for 1.000 f. s. ; 104.0 for 1.400 f. s. : and 

 83.9 for 1,700 f. s. With spherical projectiles. 153.4 for 

 1.2001 s.: 141.1 for 1,000 f. s. ; 141.3 for l,400f. ft.; 120.8 

 for 1.7001 s.; and 103.9 for 2,000 1 s. These figures indi- 

 cate the relative atmospheric resistance of the light (in propor- 

 tion to calibre) spherical ball and the heinier, elongated, coni- 

 cal ball, of same calibre. 



Another important law of projectiles is that governing the 

 drop of the ball front the action of gravity. A ball falling 

 freely through space, near sea level, falls 1.(3.09 feet during 

 the first second, and with an accelerated velocity, at the end 

 of each subsequent second, of 32,19 feet per second. This 

 holds good, whether the ball is suddenly released from a state 

 of rest, or whether it is propelled from the bore of the rifle. 

 Its drop will be exactly the same in, each case, iu the same 

 interval of time. Although this law is deduced from experi- 

 ments at such a slow velocity that its application is only theo- 

 retically correct in a vacuum, yet the velocity of a lead or 

 iron ball is so slight, during 'the first few seconds of its 

 "drop," that the resistance of the air can be disregarded. 



The drop of a ball " is as the squares of the times," which is 

 concisely expressed in the formula — 



h — * g t s 



in which h represents the drop; g = the gravity = 32.19 feet, 

 Greenwich standard; and t «= time in seconds. 



A great, deal of "bosh" has been lately printed about the 

 time required for gravity to overcome inertia. As this for- 

 mula, is based upon careful experiments, in which this ele- 

 ment of inertia entered, it of course includes that iiifnitmsimal 

 interval of time. 



But for the resistance of the air, a ball propelled from the 

 rifle in a horizontal direction, would pass over equal spaces in 

 equal times, and its curves, in that case, would be a para- 

 bola, and the ordinates, for instance, at 50 and 150 yards of 

 table, would be equal. But the atmospheric resistance act- 

 ing upon the ball, and constantly decreasing its velocity, 

 changes all this, and the descending branch of its flight is 

 considerably more curved than the ascending branch, the or- 

 dinates at 50 yards (see No. 1 — ordinate 9.70) and at 150 

 yards (see No. i — ordinate 11.25) being very unequal. 



Experiments pretty thoroughly show that with the same 

 calibre and length of barrel, ' ' as long as proportional weights 

 of powder ana ball are preserved, the initial velocities are 

 practically the same." This appears to hold good, even 

 with different calibres, where the calibre and length of bore 

 are so proportioned as to thoroughly utilize the powder and 

 ball; and to hold approximately ."even with the heaviest ord- 

 nance, a proportion of f powder in the .45 calibre, produc- 

 ing an initial velocity of about 2,000 f. s., and the same 

 powder proportion, in the largest rifle cannon, with the 

 700 lb. ball, producing a like velocity. 



Mr. T. S. Van Dyke has written a very readable and most 

 instructive book on the art of hunting deer, ' ' The Still Hun- 

 ter," which shows him to be a master in that art. He has, 

 however, embodied in that book and iu certain communica- 

 tions to the sporting papers, certain opinions and news in 

 projectile science at variance, iu my opinion, with the well- 

 developed laws heretofore alluded to, and which I propose 

 to point out. He says : "The line of flight of a ball or 

 ' trajectory,' as it is* generally called, may be very near pre- 

 dicted, a priori, or without experiment, by inevitable deduc- 

 tion from a few of the principles of natural philosophy." 

 Whoever attempts to trace the flight, of a ball in this manner, 

 will surely come to grief, as the author has naturally done. 

 The science of projectiles is an exact science, and Its laws 

 have been gradually developed by the most careful and ex- 

 haustive experiments for the past one hundred years, and the 

 results worked out into a practical form by the higher mathe- 

 matics. The formula;, embodying the elements controlling 

 the flight, of the rifle ball, contain certain elements represent- 

 ing the resistance of the air and the eirtion of 'gravity, which are 

 called constants, and those can alone be determined by careful 

 experiments. Without fixing a mathematical value to these 

 constants, these formulas are entirely useless in fixing this 

 curve ; and reasoning a priori would be expending thought 

 in vain, as far as any practical results arc concerned. 



The law governing the action of gravity has long been au- 

 thoritatively known; that governing the resist since of the air 

 was not so easily determined. As early as 1742 Kobins, and 

 during 1783-91, Hutton experimented "for this law with the 

 ballistic pendulum. In 1839 and 1840 MM. Piobert, Morin, 

 and Didion, under authority of the French government, made 

 similar experiments with an improved ballistic pendulum, 

 and the results oi» the two setts of experiments are said to 

 have agreed very closely. 



During the years 1865-1870, however, the most important 

 and elaborate experiments were made by Professor Francis 

 Bashforth (heretofore alluded to), by means of his electric 

 chronograph and a succession of screens placed at 50 yards 

 intervals. Extended experiments with different calibre's and 

 weights of balls fired through this system of screens, the. times 

 of passing each screen being accurately determined by the 

 chronograph, enabled him to affix correct values to the at- 

 mospheric resistance due to each kiud and calibre of ball, 

 Using these values in carefully devised formula-, he obtained 

 the co-efficients of atmospheric resistance for ogival-headed 

 and kindred forms of projectiles, for all velocities from 900 

 f. s. to 1,700 1 s., and for spherical projectiles, from 850 f. s. 

 to 2,15!) 1 s. Additional tables were also constructed, by 

 which the trajectory curves of all bulls, of varying calibres 

 and weights, hvvmg veiotiiiss of from J00 f s. tor '00 I' s, 

 for ogival-headed. projectiles, and from 850 f. s. to 1,900 f. s. 

 for spherical projectiles, might be calculated. The projectiles 

 used in these experiments were of 8, 5. 7 and 9-inch calibre, 

 and of from 6 lbs. to 250 lbs. weights. However, as a result 

 of the Field (Loudon) rifle trials, Nov. 1880. these tables were 

 used in calculating the curves made bv different express rifles, 

 and the calculated curves afterwards tested, by a. .system of 

 paper screens, and the calculated curves pronounced 'substan- 

 tially correct — so that these tables are applicable to the trajec- 

 tories of the small bore sporting rifles, with properly shaped 

 balls. The effect of the Field rifle trials of 1879 had a most 

 happy effect in dispelling the illusory claims of certain ex- 

 press rifle makers and sportsmen, who claimed that "their 

 rifles had a practically flat trajectory up to 200 yards; " 

 whereas the flattest trajectory developed during these trials, 

 had a rise of 10 inches for that range. Appended is a table, 

 showing the initial velocity, drop at 50 yards, 100 yards, 100 



yards and 200 yards of balls from those rifles developing the 

 flattest curves" during these trials. By way of comparison 

 and for future reference, is given the drop *Mr. Van Dyke 

 claims for his ,65 calibre, spherical ball, as will herenf ter ap- 

 pear, as well as the curve calculated for same ball. These 

 rifles all used as large, or larger, poWder proportions than 

 the .65 Calibre, and all, except one. used conical balls. And 

 yet he believes and informs us farther along that his .65 cali- 

 bre, spherical ball, with about the same initial velocity "has 

 no perceptible drop up to 120 yards," the most rapid of these 

 having a drop of 6i<$ inches at 100 yards : 



e'nuh™ Initial Ih-op Drop fyficm 



oaiio'e. Telocity. BOycvrds. 100 yttrds. 200 yaxdu. 



BOO, SOlld 1810 1.49 0,84 82.00 



BOO, hollow 171*0 1.56 0.S»7 35.17 



577, spherical 1050 1 .30 7.64 40.0 



577, hollow 1803 1.56 (5.83 34.07 



.450, hollow 1830 1.30 6.56 



.65, spherical 1815 1.00 20.00 



.65, calcinated 1815 1.04 8.12 47.52 



A study of the experiments with their results of those who 

 have gone before us, is sure to save one a world of doubt and 

 tribulation, into which one is siu-e to drift if he attempts to 

 arrive at practical results Iv reasoning a priori " with uo 

 such facts as a base. 



Mr. Van Dyke's first error is in stating the resistance 

 of the air to vary as the calibre (diameter), instead of as d* 



w 

 His next, false position is as to the action of gravity " in the- 

 first interval of time, after leaving the bore." My understand- 

 ing of his first position is that his .65 calibre rifle, with 150' 

 grains of powder, will drive a 437 1 2 ' grains spherical ball so. 

 rapidly that it will not drop up to 50 yards. A discussion! 

 has arisen, but it is my understanding, his position is the- 

 same, for in Chapter XXII of " Still Hunter," we find ; " So* 

 high a speed may be given to the hall, that during the first 

 small fraction of 'a second after the ball leaves the muzzle, in 

 which there is no doiratrard motion, (italics mine) it may be 

 driven as far as eighty (80) yards, before its fall is noticeable, 

 even to the most careful inspection, "meaning of course before 

 its "drop " can be measured on a target, as it is impossible to> 

 trace the ball with the eye. Again, a little farther on, we 

 find : " On the other hand, 150 grains behind the ounce round 

 ball, * * * * will drive the ball square through a twoi 

 inch bull's-eye at 100 yards, will have no perceptible fall even 

 at 120 yards* ; at 160 yards will not drop more than 6 or 8 : 

 inches, and at 200 yards, little more than 18 or 20 inches.'" 

 The evident meaning of this being, that up to 100 yards the 

 drop will not exceed (1) one inch, and slightly more at 120' 

 yards. This is claiming more than the letter causing the dis- 

 cussion. 



The theory of the instanto newts effect of gravity on the 

 ball as it leaves the muzzle is too well estaiiliehed at this day to- 

 bear discussion, and if discussed at all, can only be classed 

 with such questions as the following, that periodically con- 

 sume space in the Fobest and Stream— "Do deer bury 

 their horns 1" and "Is the spike, buck a scp; rate species of 

 the cervidse." All the arguments are on one side. 



My object in alluding to it at all, is to suggest to the author 

 that "he may be arguing from false premises. The only cor- 

 rect way of measuring the drop of the ball, for small inter- 

 vals of time, is either by an electric, chronograph, the most 

 certain, or next, by means of carefully arranged paper 

 screens, each of which methods require the. greatest care. My 

 inference, however, is that Mr. Van Dyke formed his opinion 

 by sighting the rifle for 25 yards very carefully, then loading 

 with exactly the same charge, and with the same sight shoot- 

 ing at the 50 or 100 yard target, the ball hitting so near the 

 point aimed at, that lie considered there was no drop to the 

 ball. A moment's reflection will show how erroneous deduc- 

 tions from such premises may be. In the first place, it would 

 be almost a miracle to obtain exactly the same velocity from 

 two different charges, however carefully weighed. In the 

 second place, it is almost impossible to hold the same, or to ' 

 sight the same, in two consecutive shots. An error of (.01 

 inch) one-hundredth of an inch in cither, would make an 

 error for 50 yards of about % inch, with a 28-inch barrel (24- 

 inch between front and rear sight). I have experimented in 

 this manner a great deal to obtain trajectory curves, and al- 

 ways found it very unsatisfactory and resulting merely in rude 

 approximations. This is the reason why the suggestion Is 

 made that false deductions may be made from erroneous pre- 

 mises ; from very unreliable experiment, especially, as he 

 speaks in same chapter of his want of facilities ' ' for making 

 correct measurements." I am convinced this is the case in 

 this instance, for it is evident to my mind from what follows 

 that the author is entirely "at sea," as to what trajectory 

 curve his ounce ball does make. Take the claim made in 

 quotation before given, that 150 grains powder will "drive 

 the ounce spherical ball square tlirough a 2-inch bull's-eye at 

 100 yards, will have no perceptible f all even at 120 yards," 

 etc., etc., and taking the most favorable view of it for his 

 theory, (that the drop is (1 inch) one inch for the first 100 yards) 

 and we figure out, an .- .-.- velocity for that range of 4,150 

 feet per second, the time required for passing that distance • 

 being (.0719 second), *ay seventy-two thousandths of a second. 

 This will require an initial velocity of about 4,500 feet per 

 second. The impossibility of accomplishing such a feat 

 with a powder proportion* of 1 to 2.91 will be the more 

 apparent when it is stated that the highest velocity attained 

 with the small bores is 2,150 feet per second, with powder 

 proportions of 1 to 1.75. 



Any ball with such a fearful velocity as to fall only one 

 inch m the first J00 yards, would almost pass over equal 

 spaces in equal times, and would not drop at end of the 200 

 yards range more than 6J*£ inches, instead of "18 or 20 inches." 

 Plotting this curve will convince any one that it is impossible 

 for a ball of even average velocity to travel such a path, as 

 there is too much of a sudden 'angle at 125 yards. An 

 idea, however, has just dawned, that may reconcile the 

 whole difficulty. If the ball, after starting, rises .2 or 2% 

 inches at 50 yards, "drives square through or above the 

 centre of bull's-eye, at 100 yards, about one inch below at 125 

 yards," and then'ce the balance of the curve, as the author 

 chalks out, it will make a properly proportioned curve, and 

 I believe that is the true state of the case. The author, in- 

 stead of holding the bore of the rifle perfectly horizontal, as he 

 thought (a very difficult task) " uubekuowiug " held it with 

 bore a very little elevated, and the ball takes the course 

 just indicated, mstra:| as he thought going almost pcifeetly 

 straight to the 2-inch bull's-eye. But as the bore of the rife 

 was elevated, this solution involves the result of a drop of 7 

 or 8 inches at 100 yards, and 40 to 45 inches at 200 yards, 

 which is believed to be the facts. I had already taken the 

 pains to calculate by Jjashi'orth's tallies the curve "of this ball, 

 with an assumed initial velocity of 1,615 f. s.. that being 

 about the velocity attained by express rifles (using the C. 

 and H. No. 6 powder) with powder proportion of 165 grains 

 to 4371-j grains. As the powder used in this case was only 



