Motion of a Spinning Projectile. 339 



15. The constant I can be calculated from Bashforth's 

 tables, and the constant c involves only the muzzle velocity, 

 which is known from observations. But, as I believe the 

 range table is more accurate than any other data for a given 

 shot, I have used these tables for the Marks VI. and VII. 

 bullets to calculate I and c, and therefore u . 



16. So far we have only considered that part of the 

 trajectory which is described while the velocity is greater 

 than 1060 feet per second. In the other part of the trajec- 

 tory we must use l x instead of I and make the two portions 

 continuous at the junction. Indicating by a suffix 1 the 

 values of quantities at the point where u=1060 (except, of 

 course, Z 1? which -applies when w<1060), then starting from 

 equations such as (2) and (6), we get, by integration, the 

 following equations : — 



!_!._£=&, (is) 



X — x x 



u x " 



10^- = -^-^, (16) 



^-^ 2 -~ 2 ) .-.. (17) 



Y-Y I KX-X0(t a nX 1 -g 2 )-g- 2 {* '' -lj.(18) 



The last equation has the form 



2X 



Y=A + BX + (VT (19) 



For a shot fired horizontally we get the same form of 

 equation as this last one, and we may write the equation 

 thus : 



2X 



Y' = A + BX + C«\ (20) 



using Y', as before, for the ordinate of the trajectory in the 

 particular case of horizontal firing. Equations (13) and (20) 

 are the equations to different portions of the same trajectory, 

 the one being applicable while u is greater than 1060, and 

 the other, while u is less than 1060. 



17. If our assumed laws of resistance were absolutely 

 correct, then we should get the trajectory absolutely correct 

 also by making the ordinate and the slope of the trajectory 

 continuous at the point where the law changes, that is, 

 where u - 1060 feet per sec. ; but as the assumed resistance 



