268 



Mr. W. D. Niven on the Calculation 



" On the Calculation of the Trajectories of Shot/' By W. 

 D. Niven, M.A., F.R.A.S. Communicated by J. Clerk 

 Maxwell, F.R.S., Professor of Experimental Physics in the 

 University of Cambridge. Received March 24, 1876 



In the present state of our knowledge of the resistance'of the air to 

 shot, the problem of integrating the equations of motion of the shot and 

 of plotting-out a representation of the curve described by it is peculiar, 

 because, according to the best experiments we possess, the law of the 

 retardation cannot be expressed by a single exact formula which is 

 available for the solution. We are therefore compelled to give a 

 solution adapted to Tables, the magnitudes of the retardation being set 

 down in those Tables for velocities which are common in practice. The 

 formulae given by Hutton and by Didion, even if they were true, apply 

 only to spherical shot ; and though they are very simple f ormulae, the 

 solutions obtained by means of them are not satisfactory — first, by 

 reason of their complexity, and next on account of the rough approxima- 

 tions which characterize the proofs. 



Prof. Helie, who gives an account of Didions method in his ' Traite de 

 Balistique,' says that it gives results which are not in accordance with 

 fact. The fault may probably be laid in a great measure to the charge of 

 the formula ; for there can be no doubt that Mr. Bashforth's method of 

 experimenting with his chronograph and screens gives more trustworthy 

 and more extensive information than the ballistic pendulum experiments 

 of Hutton and Didion ; and Hutton's formula, as well as Didion's, 

 agrees with Mr. Bashforth's Tables only for a limited range of velocities. 



Mr. Bashforth himself makes no attempt to condense his Tables into 

 concise formulae. Accordingly, as w r e shall presently see, he adopts a 

 solution which is capable of being employed in conjunction with his 

 Tables. He divides the trajectory into small arcs, and finds for every arc 

 the time and the horizontal and vertical distances from one end of the 

 arc to the other. The entire trajectory may then be plotted out, and the 

 whole time and range may be discovered, as well as the final inclination 

 of the direction of motion. The amount of labour, however, in calcu- 

 lating all the quantities for a single component arc, even with the aid of 

 copious tables, is so great that I was led to examine whether any thing 

 could be clone towards simplifying the solution and reducing the amount 

 of calculation. It will appear in the sequel that rules of comparatively 

 easy application can be employed, and that the tables necessary for their 

 use are already existing or can be easily formed. 



Meanwhile, without entering into details, it will be convenient to give 

 a brief account of the drift and scope of what is attempted in this paper. 



* Read May 11, 1876. See vol. xxr. p. 18. 



