199 
CONSIDERATIONS ON THE SUBJECT 
OF TRAJHCTORIES. 
BY 
CEA PUDEASION EY Sine Sia CelpHiehinVerhin, Ria Aer. 
Tue following considerations suggested themselves in connection with 
the conditions appertaining to gun practice with 7-pr. 200 lbs. and 2.5-in. 
guns in the Himalayas in 1884 and 1888. Whilst feeling that I have 
regarded the subject from a purely speculative point of view, which I 
am unable to do justice to or support mathematically, I hope that I 
may have suggested a few points which may be of interest as a subject 
of discussion to others. My best thanks are due to Mr. Greenhill, 
M.A., F.R.S. for his kind advice and assistance in dealing with the 
subject. 
(1.) Let A be the position of the gun capable of being fired at 
any angle in a vertical plane passing through it. The 
angles considered are angles of quadrant elevation and 
are numbered accordingly (v. Fig. 1), tangent elevation 
being disregarded. Muzzle velocity is assumed to be 
greater than terminal. 
(2.) In firing vertically downwards, 7.e. 90° depression, gravity 
has maximum accelerating effect and terminal velocity will 
tend to attainment at maximum distance from.A (muzzle 
velocity being greater than terminal, remaining velocity 
though continually decreasing towards terminal will never 
attain the same). 
(3.) As angle of depression decreases, horizontal component of 
velocity of projection will increase, but in decreasing ratio, 
and trajectory will become more and more curved owing 
to angle at which gravity acts. Vertical component of 
velocity of projection will decrease and vertical asymptote 
will be attained and terminal velocity tend to attainment 
at a gradually decreasing range from gun. 
(4.) At a certain angle “x” velocity will decrease and attain 
terminal velocity at which it will remain infinitely, 7.e. for 
this trajectory there will be a line and not a point of 
minimum velocity. 
(5.) For every trajectory above this angle “«” velocity will de- 
ercase to less than terminal (for + 90° decreasing to 0) 
and increase towards terminal which will never be at- 
tained, thus contaming a point of minimum velocity. 
6. VoL, XXIII. 
