Motion of a Spinning Projectile. 365 



Hence 



(/+l-/r)H 1 = l, (85) 



(4 + if-p-) H, = 2im (/+ 1)H, - 2im, 



=- 2 ™f,4=-v ■ ■ ■ <*> 



and for the rest 



{ n(n-l)+n(f+l)-fr}K n =2im(n+f-l)R n _ v (87) 



For an actual shot the number r is so large that small 

 integers are negligible in comparison with it. Conse- 

 quently the equations (85) and (86) give approximately the 

 following results : — 



Hl =-i, H 2 =^ (88) 



After this the relation between successive coefficients is 



2im(n+f—l) tt ™. 



R n=- f i.-f n -^ H -» " " ' (89) 



which, when n 2 is small compared with r, is approximately 

 the same as 



_ 2im(n+f-l) (90) 



But when n 2 is large compared with r the relation in (89) 

 is approximately 



In a region intermediate between the two we have just 

 considered, that is, in the region where n l is nearly equal to 

 r, the denominator o£ the fraction in (89) is very small 

 while the numerator is large, and if *J fr+±f 2 — if happens 

 to be exactly an integer the fraction becomes infinite when 

 n is equal to this integer. We need not, however, consider 

 the infinitely improbable case in which this number is 

 exactly an integer, but it should be noticed that, when n is 

 near this number, equation (89) shows that the coefficients 

 go on increasing as n increases until we reach the point where 

 (n 2 + fn—fr) is greater than 2m(n +/— 1). Yet the earlier 

 coefficients of the series, to which equation (90) applies, 

 diminish fairly quickly, and the corresponding terms of the 

 series diminish also because the ratio of wT to r is a very 



Phil. Mag. S. 6. Vol. 34. No. 202. Oct. 1917. 2 C 



