1876.] 



of the Trajectories of Shot. 



19 



sions of shot, which could be dealt with more satisfactorily if there 

 existed an easy method of calculating ranges. Mr. Bashforth gives one 

 solution in his treatise, and the object of this paper is to give another. 



The expressions here proved depend upon three integrals, which may 

 be defined for ogival-headed shot as follows : — 



-J" 



(lOOOfdV 



(1000)3 tA' 



Of these integrals the two last have been already tabulated by Mr. Bash- 

 forth : the first is now given as low as ^?=900. The integrals are cal- 

 culated between every 10 feet, for which the values of K are given, the 

 arithmetical mean of K over the interval being taken. 



Let A B be a portion of the trajectory of a shot ; let the inclinations at 

 A and B be a and /3, and the horizontal components of the velocity at 

 the same points p and q. Then it is proved that the inclination <p of the 

 chord A B is approximately 



4- Enl " in the ascending branch, 

 2 b 



and — - — ^ - — - in the descending branch. 



2 ^) 6 



If it be assumed that the inclination of motion between A and B has 

 the mean value (p, the following four equations constitute the approximate 

 solution of the problem, and the limits of the integrals are such as to 

 make the results from the assumption approximate to the actual case : — 



- 



^2sec^-^p8ec^ = ^I> sec (^, (a) 



where D is the number of degrees in the difference between the inclina- 

 tions at A and B : 



