352 Mr. J. Prescott on the 



shot, mainly on the ratio of the length to the diameter, but 

 partly also on the sharpness of the nose. For a shot of 

 given form 9' is a definite function of 6 which is zero when 

 6 is zero. Let 



Expanding this in powers of 0, 



0'=/(O)+0/(O) + §V(O)+ 



To make 0' zero when 6 is zero /(0) must be zero. 

 Then, since 6 is always small in the actual effective flight, 



6' = Of (0) approximately. 

 We may write this 



V=fl, (37) 



where / is a constant greater than unity for any ordinary 

 shot. It is conceivable that /could be less than unity for 

 a flat-nosed shot. 



39. Our conclusion is now that the resistance makes an 

 angle f6 with the axis of the shot. Consequently, the 

 small component angles between the line of the resistance 

 and the axis OP are/(^ — e) and f(y — y + fi) instead of x 

 and (y + /3) as we assumed in deducing our equations of 

 motion. Our new conclusions change the couples from 

 'Rex and Bc(?/-f /8) to ~Rcf(% — e) and Hcf(y — y-i-ft), but they 

 do not affect the kinetic energy. Consequently, our cor- 

 rected equations are 



M + ~Bcoy-'Rcf(x-6)=Ax (38) 



Ay-Ba>x-ncf(y- v ) = -Bcod + ~Rcf/3. . (39) 



It is assumed, of course, that the angle between the axis 

 and the line of motion does not appreciably increase the 

 magnitude of the resistance. 



40. The resistance makes an angle with OP which has 

 components f{x — e) and/(y — rj + ft), while the true velocity 

 of makes an angle which has components (x — e) and 

 (y—v) w ^h the same line OP. Therefore, the resistance 

 makes an angle which has components (/— 1)(#— e) and 

 f{y—r]+fi)—{y — y) with the true velocity of 0; and thus 

 the resistance has components ~R(f—l)(x — e) and P{/(?/— rj 

 + /9)— (y— 77)} perpendicular to the velocity. These are 

 the small component forces that produce the angular de- 

 flexions e and rj. The component accelerations that accompany 



