4IO 



NATURE 



[August 28, 1890 



The conditions to be secured which Mr. Bashforth set 

 before himself were — 



(i) The time to be measured by a clock going uni- 

 formly. 



(2) The instrument to be capable of measuring the 

 times occupied by a cannon-ball in passing over at least 

 nine successive equal spaces. 



(3) The instrument to be capable of measuring the 

 longest known time of flight of a shot or shell. 



(4) Every beat of the clock to be recorded by the inter- 

 ruption of the same galvanic current, and under precisely 

 the same conditions. 



(5) The time of passing each screen to be recorded by 

 the momentary interruption of a second galvanic current, 

 and under precisely the same conditions. 



(6) Provision to be made for keeping the strings or 

 wires of the screens in a uniform state of tension, notwith- 

 standing the force of the wind and the blast accompanying 

 the ball. 



To secure these conditions practically, Mr. Bashforth 

 had to invent his own chronograph, for a detailed de- 

 scription of which the reader must refer to the book ; but 

 it consists essentially of a brass cylinder provided with a 

 heavy fly-wheel movable about a vertical axis ; and of 

 two markers tracing spiral lines on paper placed on the 

 cylinder, and giving an indication by a jerk on the spiral 

 corresponding to the cutting of one of the electric screens 

 by the shot, or to half-seconds of the clock. 



The fly-wheel being spun by hand, and the clock 

 making continual half-second records, the word is given 

 to fire the gun, and then the screen records are registered 

 on the paper by the screen-marker. When the paper is 

 full, after five or six rounds, the cylinder is transferred to 

 a micrometer instrument, and the records read off" with 

 a vernier and microscope as accurately as possible. 



We may take it that the average travel of the paper on 

 the cylinder is an inch for about a tenth of a second, so 

 that, with screens 150 feet apart, an average velocity of 

 1500 f.s. would give screen records at intervals of about 

 an inch. Readings of tenths of an inch will give hun- 

 dredths of a second, and of hundredths of an inch will 

 give thousandths of a second, which is about as far as can 

 be seen or measured with this instrument. But, by treat- 

 ing the last significant figure as indeterminate, and 

 smoothing down irregularities by differencing and inter- 

 polation, Mr. Bashforth is able to assign probable values to 

 the 4th and even 5th decimal of the second, in the instant 

 at which any screen is cut. 



Any improved instrument which would give a velocity 

 to the paper of ten times or one hundred times of Mr. 

 Bashforth's velocity would increase the recording ac- 

 curacy theoretically to the same extent ; but, as Mr. 

 Bashforth claims for his instrument, he has located the 

 shot at any instant to within about one foot of range, 

 an error comparable with inaccuracies in the measured 

 distance between the screens, inclination of the screens, 

 and bending or stretching of the screen wires before 

 breaking. 



The chronograph having given us the instants of time, 

 say /^, /g, <3v at which screens i, 2, 3,... at equal inter- 

 vals of / feet were cut by a shot, we have to employ the 

 methods of Finite Differences for converting these records 

 into expressions for the velocity and retardation at any point. 

 NO. 1087, VOL. 42] 



It will be noticed that the chronograph records give, 

 by interpolation, / as a function of s, not J as a function of 

 /, so that the velocity v is the reciprocal of dtjcis, while the 



. . . dH ^ 

 retardation is -^ z/3. ^nd if the shot weighs W pounds, the 



resistance of the air is W ^ v^ poundals, or W -, 2 "^^-^S 



pounds : the shot flying so fast that, practically, we may 

 take it as moving in a horizontal line. 



Formulas of the calculus of Finite Differences will give 

 us the values of dtjds and dHjds^ in terms of the succes- 

 sive differences A/, A2/,... of/; those employed by Mr. 

 Bashforth being— 



,dt 



ds 



ds^ 



■ A/ - iA2/ -f ^A-V 



= A2/- A3/ + 11 AV., 



To take a simple numerical illustration, sup pose it was 

 found by the chronograph that a shot weighing 70 pounds, 

 flying horizontally, cut three equidistant screens 150 feet 

 apart at instants of time 2-3439, 2*4325, 2-5221 seconds. 

 The time from the first to the third screen being 0-1782 

 second, the average velocity over this 300 feet is 

 300 — 0-1782 = 1684 f.s. ; and we may take this as being 

 the velocity at the middle screen — an assumption which 

 is accurately true if the resistance varies as the cube of 

 the velocity — that is, if d'^t/ds^ is constant. 



Again, A^/ = 2-3439 - 2 X 2-4325 -f 2*5221 = o-qoi ; 

 so that dHlds^ = 0-001 -h- (150)2; and therefore the re- 

 sistance of the air is 70 X (1684)2 x 0*001 -r- (150)2 

 = 14,850 poundals, or 464 pounds. 



Experiment confirmed the reasonable hypothesis that 

 the resistance of the air is proportional to the cross- 

 section, or to d'^, if ^ is the diameter in inches ; so that, 

 for similar projectiles, Bashforth introduces his coefficient 

 K, defined so as to make the resistance of the air at a 

 velocity v f.s. to a projectile d inches in diameter to be — 



^2 K Ly^ poundals, or d'^^i^^-^ pounds ; 

 \iooo/ '^ z Viooo/ ^ 



Df the shot 

 :ance is 



Kf-^Yceloes 



\IOOO/ 



while, if the weight of the shot is W pounds, the retarda- 

 tion due to the resistance is 



d^ 



W 

 and thus 



K 



.r.^dH _^ W, . ^V 

 ds^ d^ ds^ 



The coefficient K is now found experimentally to be 

 the same for all similar projectiles, whatever the weight, 

 W pounds, or diameter, d inches ; and the factor of 

 mechanical similitude W/^^, now called the ballistic co- 

 efficient and generally denoted by C, enables us to gene- 

 ralize the experiments made on one scale to projectiles of 

 all sizes. 



We now see the convenience of splitting up the resist- 

 ance of the air into two factors, one of them being the cube 

 of the velocity ; for in the retardation the other factor is 

 d'^tjds'^, which is given very simply in terms of A-/,... 



It is very often asserted that " Bashforth assumed the 

 resistance of the air to vary as the cube of the velocity " ; 

 whereas in reality Bashforth found it convenient to take 

 out the cube of the velocity as one factor of the resist- 



