March 17, 1892] 



NATURE 



475 



lating by the general tables the time of flight over the range 

 already obtained, and also the striking velocity, it is found that 

 the general tables may be used for elevations of the 4-inch gun 

 as high as 15°, or even more, with a muzzle velocity of 1900 f.s. 

 In this way the mere-it tyro may test my coefficients for his own 

 satisfaction by calculating the times of flight over ranges of two 

 or three miles given in any good range table for a high muzzle 

 velocity. The following are the results of such testing, using 

 the full extent of the range table of the 4-inch B. L. gun, chosen 

 by the authorities. Muzzle velocity, 1900 f.s. ; weight of ogival- 

 headed shot (two diameters), 25 pounds : — 



Here, as before, the calculated time is rather too short for 

 velocities 190x3 to 751 f.s. And if we allowed for a slight dimi- 

 nution of the density of the air for the higher elevations, as we 

 ought to do, the calculated would throughout fall very slightly 

 short of the experimental times of flight. Thus it is clear that 

 my coefficients of resistance give perfectly satisfactory results 

 when fairly tested by recent guns, chosen by Government, for 

 velocities 330 to 751 f.s., and from 751 to 1900 f.s., or from 

 330 to 19CX5 f.s. 



In the same way we may use the model range table, carefully 

 prepared for the 12-inch B. L. gun by Captain May, R.N., for a 

 muzzle velocity 1892 f.s., and weight of shot 714 pounds (Proc. 

 R.A. Inst., 1886, p. 356):— 



Here, again, the calculated times of flight, being a trifle too 

 short, show that my coefficients of resistance are very slightly 

 too low. 



When coefficients tested in this manner give calculated times 

 of flight accurately over ranges gradually increasing up to two or 

 three miles, those coefficients must be correct for all practical 

 purposes, and they will give correctly the striking velocity and 

 time of flight for any other reasonable distance from the gun. 



The tables of " Mayevski nach Siacci," printed by Krupp, 

 1S83, may be used to calculate the times of flight of the shot 

 fired from the 4-inch gun as above : — 



From this it is evident that the reduction of my coefficients pro- 

 posed by Mayevski on the strength of Krupp's experiments is 

 uncalled for. 



Again, it has been urged that my resistances ought to be 

 reduced in order to adapt them to recent guns, which, it is 

 assumed, impart an increased degree of steadiness to their pro- 

 jectiles. But that assumption requires proof. After most 

 carefully testing the admirable range tables of the 4-inch and 

 12-inch B.L. guns, I have failed to find any indication whatever 

 of increased steadiness in their projectiles. Besides, Admiral 

 Robert A, E. Scott wrote to the Morning Post (November 9, 

 1889), condemning the system of rifling the no-ton gun, which I 



NO. 



1 1 68, VOL. 4.5] 



he blamed for causing the projectiles to " issue from their guns 

 with a very unsteady motion." He then went on to notice the 

 large number of unsteady shot fired from the 9-2-inch gun in 

 1888. I would also remind my critics that my coefficients of 

 resistance for velocities 1000 to 1 700 f.s. were derived from ex- 

 periments made in 1867-68, while all those for velocities less 

 than 1000 f.s., and greater than 1700 f.s., were derived from ex- 

 periments in 1878-80, carried out with some of the newest and 

 best guns of the time. As conclusive evidence of the excellence 

 of the 3, 5, and 7-inch guns used in the early experiments, 

 reference may be made to the fact that, from the results of the 

 experiments of 1867-68, I was able to deduce the Newtonian 

 law of resistance for velocities 1350 to 1700 f.s. (Proc. R.A. Inst., 

 1871) ; and using the mean of the eight numerical coefficients 

 there given for velocities 1350, 1400, . . . 1700 f.s., the numerical 

 value of ,^ will be found to be 143*9. 



In 1879 experiments were made with a ne-M Armstrong 6-inch 

 B.L. gun, with velocities 1700 to 2250 f.s. (Reports, &c.. 

 Part ii., 48) ; and again, in 1880, further experiments were 

 carried out with a «^-£' Armstrong 8-inch B.L. gun, with veloci- 

 ties 2250 to 2800 f.s. (Final Report, 56). Combining these 

 three sets of experiments. Major Ingalls found that the New- 

 tonian law of resistance held good for velocities 1330 to 2800 f.s., 

 where k = 142-1 (Ext. Bal., 36). I also deduced the same law 

 for velocities 1300 to 2800 f.s., where k = 141-5 (Nature, 1886, 

 p. 606). And lastly, after a thorough revision of the reduction 

 of every round, I finally adopted the same law for velocities 

 above 1300 f.s., where k = ;'Vi-2. 



Hence it appears that the early experiments of 1867-68 were 

 so accurate that they gave a correct law of resistance for veloci- 

 ties 1350 to 1700 f.s., which has since been found to hold good 

 for velocities 1300 to 2800 f.s. ; and they also gave the co- 

 efficient /' = 143-9 (with studded %hoi) sufficiently accurate for all 

 practical purposes up to a velocity 2800 f.s. This is conclusive 

 evidence of the steadiness of the shot in the early experiments, 

 and of the accuracy of the method of reduction of those 

 experiments. 



But when those coefficients, which have been found correct 

 by the use of the general tables, are employed to calculate 

 trajectories of elongated shot moving with high velocities, the 

 calculated ranges and times of flight gradually fall more and 

 more below those quantities given in the range tables, as the 

 elevation increases beyond 4° or 5°. These defects are generally 

 only small when the variation in the density of the air is taken 

 into account ; but their presence indicates some slight disturbing 

 cause independent of the coefficients of resistance. We can 

 now make use of the exact method of calculating trajectories 

 given by modern analysis, which was first published by J. 

 tJernoulli. But this method applies with strictness only to the 

 motion of a spherical projectile, whose centre of gravity coincides 

 with its centre of figure. Many years ago Count St. Robert 

 remarked : " On doit en conclure que les formules ordinaires de 

 la balistique ne peuvent representer la trajectoire decrite par les 

 projectiles a//(7«^t.'f " (Balistique, p. 183). Also Mayevski has 

 published an elaborate paper, " De I'lnfluence du Mouvement de 

 Rotation sur la Trajectoire des Projectiles oblongs dans I'Air " 

 (Technologic Mil., 1866, pp. I-150), which, however, leads to no 

 useful result beyond showing that the author recognized the 

 effect of drift on the form of the trajectory. The chief cause 

 of the difficulty is this. For a .short time after a steady elongated 

 shot has left a rifled gun, the shot preserves the parallelism of 

 its axis, and in consequence of the action of gravity the point 

 of the shot gradually rises above its trajectory till the resistance 

 of the air causes the axis of the shot to begin to describe a 

 conical surface, with nearly constant vertical angle, about the 

 moving tangent to the trajectory. Consequently, soon after a 

 steady elongated shot leaves the muzzle of a rifled gun, the 

 resistance of the air acting on the inclined under side of the 

 shot, begins to raise the shot bodily, and continues to do so 

 until its axis has made one-fourth of a revolution about the 

 tangent to the trajectory. This vertical drift, near the gun, 

 causes the shot to move in its path as if it had been fired at a 

 slightly increased elevation. Consequently, the observed range 

 and time of flight are each somewhat greater than that due to 

 the elevation at which the gun was laid. 



Another difficulty, common, however, to both spherical and 

 elongated shot, is caused by the jump of the gun. In the range 

 tables of the 4 and 12-inch guns above considered, six minutes 

 were allowed for the effect of jump for all elevations. But 

 Major Ingalls remarks that " it varies in value from an angle 



