Motion of a Spinning Projectile. 341 



5, s^ so the values of 5 corresponding to ranges X, X + S, 

 X + 2S. Then, if all three ranges are described while 

 u > 1060, we get the following equations :•— 



re 



kX 



Hence 



s=p + qX + 



Sl =p + q(X + Z)+re k(X + d \ [• . . (24) 

 s 2 =p + q(X+28) + re k ( X + 2S >) 



Sl -s = qb + re* X (e M -l), \ 



S2 - Sl = q 8 + re* x e H (e^-l). J' ' ' (25) 



These last two equations give, on subtraction, 



s 2 -2s 1 + s = re /c \e M -iy .... (26) 



If we now take a similar set of three ranges in arith- 

 metical progression with the same common difference S, and 

 denote the new quantities by dashed letters, X' being the 

 smallest of the three ranges, then 



s 2 '-2 Sl ' + s'=re kx, (e M -iy. . . . (27) 



From equations (26) and (27) 



*s f -2*i / + *' = „*(X'-X) (28 ) 



s 2 — 2s 1 + s 



Since all the quantities in this last equation, except k } are 

 known from the range table, the equation determines k. 

 Then equation (26) gives r ; next, either of equations (25) 

 will give q ; and finally, any of the three equations in (24) 

 determines p. Thus we know how to get all the four 

 constants p, q, r, and k. 



22. In the preceding paragraph it was assumed that all 

 the ranges involved were described while the velocity was 

 greater than 1060 feet per sec. An exactly similar set of 

 equations will be true if all the ranges are described while 

 m<C1060, but in this case we shall determine the dashed 

 letters p\ q\ r\ V. 



Application to the Mark VII, bullet. — We must first get 

 some idea of the range when the velocity has been reduced 

 to 1060, so that we may know at what range the law of 

 resistance changes. According to Bashforth the resistance 

 is given by the equation 



