March 5, 1920] 



SCIENCE 



223 



metliod as elaborated by Ingalls and Hamilton 

 has been the standard in American works on 

 ballistics. 



In Mayevski's law as given in American 

 texts 



'' c • 



R 



C is called the ballistic coefficient. Being the 

 reciprocal of a resistance it represents the 

 penetrating power or ability of a projectile to 

 continue in motion. It is assumed to be con- 

 stant for any definite projectile. But it was 

 found that when the angle of elevation was 

 changed, or even the muzzle velocity, in gen- 

 eral had to be changed to allow for the new 

 range. Attempts have repeatedly been made 

 to find a functional relation between and 

 these variables. At certain proving grounds 

 in the United States a relation was supposed 

 to have been established but we find that the 

 law adopted does not agree with data which 

 we have secured from Aberdeen. It follows 

 that, though the mathematical computations 

 have been carried through with great rigor 

 and accuracy, actual firings for various eleva- 

 tions have to be made in order, from the 

 ranges observed, to compute the ballistic co- 

 efficient for those elevations. In other words, 

 the ballistic coefficient always contains in it 

 a factor which represents the amount by 

 which the theoretical range has to be multi- 

 plied in order to obtain the actual range. If 

 range and time be the only quantities required 

 these can be found by actual firings and al- 

 most any approximate law of air resistance 

 will satisfy. But it costs money to range-fire 

 guns. For example, this cost for a 12-inch 

 gun is of the order of $12,000 and for a 14- 

 inch naval gun $20,000. These amounts are 

 apt to be exceeded. 



It would be a very great saving in time and 

 money if the range and trajectory of a 

 projectile could be determined with a known 

 powder charge without range firing. This 

 can only be done when the complete law of 

 air resistance is known. The modern prob- 

 lems connected with antiaircraft warfare and 

 with accurate barrage firing absolutely re- 

 quire such a law. 



ISTotwithstanding the fact that the law of 

 air resistance for modem projectiles is un- 

 known and that the ballistic coefficient merely 

 represents an approximate relation between 

 the theoretical and actual ranges, great con- 

 fidence has been placed in so-called experi- 

 mental determinations of this quantity. For 

 example, in the official manual for the U. S. 

 Eifle the value of the ballistic coefficient of 

 the ordinary service rifle bullet (.30-inch 

 caliber) is given as 0.3894075 " as determined 

 experimentally at the Frankford Arsenal." 

 The experimental skill which can determine 

 to an accuracy indicated by seven places of 

 decimals a quantity as highly capricious as 

 the so-called ballistic coefficient, is of rather 

 questionable value. 



Going back to the law of air resistance, it 

 is evident that Mayevski's law is not satis- 

 factory either to mathematicians or to phys- 

 icists. There are abrupt changes when the 

 index n is changed. The mathematician can 

 not differentiate at these corners, the physicist 

 can not see the necessity for their existence. 

 The law as laid down by the Gavre Com- 

 mission which is ordinarily written in the 

 form It = cv-Ti(v), where B{v) is a function 

 of t), is satisfactory in that it has no discon- 

 tinuities. But though it is satisfactory in 

 this respect it may still be incomplete. 



The Gavre law or any other smooth law 

 lends itself to numerical integration by the 

 method of Gauss, who developed it one hun- 

 dred years ago. He used this method in the 

 problem of special perturbations in celestial 

 mechanics. It has since been presented in 

 some text-books in theoretical astronomy. An 

 early application to physics curiously enough 

 was made by an astronomer, John Couch 

 Adams, in the integration of an equation 

 occurring in the theory of capillarity. But 

 though Adams was thoroughly acquainted 

 with this method he apparently did not feel 

 that it was as satisfactory for computing a 

 trajectory as that of Euler. For in an article 

 on " Certain Approximate Solutions for Cal- 

 culating the Trajectory of a Shot" (Collected 

 Works), he refers the motion to the angle 

 that the tangent to the trajectory makes with 



