59. 



ON CERTAIN APPROXIMATE FORMULAE FOR CALCULATING THE 



TRAJECTORIES OF SHOT. 



[From the Proceedings of the Royal Society, Vol. xxvi. (1877) and Nature, Vol. XLI. (1890).] 



IN the postscript to a paper by Mr W. D. Niven, " On the Calculation 

 of the Trajectories of Shot," which is published in the Proceedings of the 

 Royal Society, Vol. xxvi. pp. 268 287, I have given, without demonstration, 

 some convenient and not inelegant formulae applicable to a limited arc of 

 a trajectory when the resistance is supposed to vary as the nth power of 

 the velocity. 



In these formulae, the angle between the chord of the arc and the 

 tangent at any point is supposed to be always small. The index n is 

 not restricted to integral values, but may take any value whatever. 



As the proof of these formulae is not altogether obvious, and a similar 

 method of treatment may be found useful in other problems, I think it 

 may not be unacceptable to your readers if I shew here how the formulae 

 may be demonstrated. 



Analysis. 



Investigation of formulae applicable to a small arc of a trajectory, when 

 the resistance varies as the nth power of the velocity. 



Let x and y denote the horizontal and vertical coordinates at time t, 

 u the horizontal velocity, and <f> the angle which the direction of motion 

 makes with the horizon at the same time. 



A. 60 



