368 Mr. J. Prescott on the 



which is sure to be less than .-x, and might reasonably be as 



small as j^. Then our series of powers of: t is sufficiently 



convergent to be useful at any effective range of the rifle- 

 bullets. 



66. For any shot similar to a rifle-bullet, but of larger 

 riize, the drop of velocity for effective firing is smaller than 

 for the rifle-bullet ; this follows from equation (8), from 

 which it appears that, for the same fall of velocity, the value 

 of a is proportional to I, which is proportional to the dimen- 

 sions of the shot. Thus the angle through which the line of 

 flight of a three-inch solid shot, similar to a rifle-bullet, 

 would turn for a given fall of velocity, is about ten times 

 the angle that the line of flight of the bullet would turn 

 through for the same fall of velocity. For instance, 1 the 

 Mark VI. bullet has an angle of departure 28'*2, and an angle 

 of arrival 36' '0, that is, the line of flight turns through 

 1° 4'*2, in the 500 yards range, during which the velocity 

 falls from 2100 to" about 1180. The line of flight of a 

 similar three-inch shot would turn through 10° 42' for the 

 same fall of velocity ; but as X is also the product of I and a 

 function of the velocity, the range for the larger shot would 

 also be about ten times as great, that is, about 5000 yards 

 instead of 500 yards. For effective firing it is clear that u 

 cannot be a large angle with high- velocity projectiles, and 

 consequently the fall of velocity for the larger shot is 

 smaller at maximum effective ranges than it is for the 

 smaller shot. 



67. It follows from the preceding reasoning that, for 

 similar shot of different sizes, t is less at the longest effective 

 ranges for a large shot than for a small shot. Hence, since 

 the series in (93) is sufficiently convergent for a rifle-bullet, 

 it will be all the more satisfactory for any other spinning 

 shot. 



61. We have now got all the results which make up the 

 complete expression for (f> in equations (47), (73), and (93). 

 The value of $ given by (47) is 



A( 6 +^)=ig=^or^by(47a). . (99) 



The complete value of cf> is the sum of the three values in 

 (73), (93), and (99). The four constants E 1? E ? , F 2 , F 2 , 

 involved in (73) are required to make x, y, and their rates of 

 increase all zero at the muzzle. Four new constants are 

 required to suit the motion after the velocity has fallen 



