282 



Mr. W. D. Niven on the Calculation 



make a rough estimate of the loss of velocity over an arc of one degree ; 

 and we have seen that such an estimate must first be made if we are to 

 employ either of the foregoing methods. 



It is possible to make the preliminary guess more near the truth by 

 considering the approximate character of the curves. For example, the 

 curve for ogival-headed shot is approximately a straight line, the tangent 

 of whose inclination from 1700 down to 1250 is -5 ; and from 1250 down 

 to 1000, -4 ; below 1000 it„ is unsafe to assign a value. We can easily 

 prove for flat trajectories that the fall of velocity in one degree is either 



*\dD)\ '5 ) ° r d\cW)X -4 ) 

 between the values above assigned. 



Third Method. 



§ 13. Eeturning now to the fundamental equation 



du ~Rv 



d6 (j ' 



where E is the retardation. due to the resistance of the air, since E is 

 some function of v let us put it equal to f{v). Then, with our old nota- 

 tion still in use, we get by integration 



C P du n 



du n 

 = a— p. 



sec 6f(u sec 6) 



Now instead of taking some mean value of the quantity K, as was 

 done in the two previous methods, let us make the supposition that the 

 quantity 6 has its mean value — a supposition in this case by no means 

 extravagant, since for the greater part of the trajectory sec 6 will vary 

 very slowly. We then have, if D is the number of degrees in a — /3, 



18% I". . = 



X P 7 



du 



v J q u sec ^ f{u sec 6) 

 Now put u = V cos 0. The equation becomes 



i8o,r ec? =Dsec . 



np sec y 

 q sec <p 



