224 



SCIENCE 



[N. S. Vol. LI. No. 1314 



METERS PER 3£C0NO 



Fig. 1. 



the horizontal and uses as a resistance law 

 B = AnV"; the constants being taken from 

 BasMorth's experimental results. 



The method of Gauss, i. e., of using rectan- 

 gular coordinates, has been used by physicists, 

 to first order differences at any rate, for vari- 

 ous computations. In the case of a projec- 

 tile, if the retardation follows the square law 

 B = hv-, the equations of motion take the 

 well-known form 



6^ _ _ ds dx 

 dp ~ di ' at ' 



ds dy 



X = — kvi, 

 y = —g - kvy. 

 If we take as the law of retardation 



R = cv^B{vy) = vF(v-y) where F = 



G{v)H{y) 

 C 



the equations take the form 



X = — xF(v-y), 



y= —9 -Wiv, y). 

 The change in the retardation due to change 

 in density of the air with height y can be 

 taken account of in the function H{y). As a 

 result of many meteorological observations 

 niy) may be written 



H{y) = 10-<»««"', 

 J/ being measured in meters. 



In the notation introduced by Professor 

 Moulton G (v), = vB{v), is computed directly 

 from the French tables giving B(v) as a 

 function of v. The form of the function 

 B(v) plotted against v is shown in Fig. 1, and 

 will be called the B curve. 



IvTow if C the ballistic coefficient or pene- 

 tration coefficient, and the velocity and alti- 

 tude are known at any time, then x and y are 

 known. If these x and y retardations are 

 constant or nearly so, then the values of the 

 X and y velocities at any later time are known 

 if the time intervals be short. But the retar- 

 dation depends on the velocity, hence its value 

 for any interval will in general lie between the 

 retardations computed for the velocities at 

 the beginning and end of an interval. One is 

 soon able to approximate to the average — con- 

 sequently the values of the x and y velocities 

 at the end of the. first, and beginning of the 

 second, interval are known. Integration can 

 be performed to find the new x and y and the 

 process can be repeated for the next interval. 



After X and y and their first and second 

 derivatives are tabulated for the first four or 

 five short intervals (of i or i second), first 

 and second differences are tabulated and the 

 computation can proceed in longer time inter- 

 vals, usually one or two seconds. The for- 

 mulas for extrapolation are made use of for 

 extending the computation, and the results 



