338 



Mr. J. Prescott on the 



13. Thus, if OSP is a portion of the curve given by 

 equation (12), then OQ is the range when the elevation of 



Fig. 3. 



the gun is \q. Moreover, if we write Y' for Y in (12) 

 to distinguish it from Y in (10), then 



Y=Y' + Xtan\ , 

 so that, if ON==X, we get 



Y=Y'-fMN 



= -SN + MN 

 =MS. 



Thus MS is the true height of the shot above the muzzle 

 of the gun when it is fired at elevation X . Consequently, 

 if MS were erected at N on the base OQ, then S would be 

 a point on the curve of the real trajectory with initial 

 elevation X . Since the angle \ is small, it will be seen 

 that, if the curve OSP were rotated about until P fell on 

 the line OQ, then M and P would nearly coincide with N 

 and Q respectively, and MS would be nearly vertical. The 

 curve thus rotated would differ very little from the true 

 trajectory in which the horizontal range is OP. 



14. Writing c for — the equation (12) becomes 



Y '=i x -£G 2f -i) • 



(13) 



The angle of elevation, X , which gives the range X, is 

 given by 



Xtan\ =-Y'= l ^T-l)-lx. t . (14) 



If we knew the constants I and c, we could make a range 

 table from the last equation. Conversely, given the range 

 table for a particular shot, we can find the values of the 

 constants I and c. 



