200 
(L1,) 
(12.) 
(18.) 
(14.) 
CONSIDERATIONS ON THE SUBJECT OF TRAJECTORIES. 
Above — 90° to “«#,” «.e. for all angles at which terminal 
velocity is not reached, the point of minimum velocity will 
be at infinite distance from A, but will, if such a thing be 
conceivable, be at a distance from 4 gradually decreasing 
from — 90° trajectory upwards. 
At a certain angle “y,”’ which it is thought may be 0°, lateral 
component of velocity of projection is at maximum and 
maximum lateral range, t.e. distance from BAB’ would be 
attained, i.e, vertical asymptote is attained at maximum 
distance from BA B’. 
The curve B'C is assumed to represent the points at which 
vertical asymptotes of trajectories are attained from — 90° 
to 66 ae” 
As elevation increases from y, lateral component of velocity 
of projection will decrease, and in an increasing ratio, hence 
greatest lateral distance of the trajectory from BAB’ be- 
fore attaining vertical asymptote will decrease, and each 
trajectory will cut every one (above y) below it in succession 
and on attaining verticle asymptote coincide with the 
asymptote of a trajectory below “ y.” 
For trajectories of elevation gravity has a retarding effect 
while projectile is travelling upwards and an accelerating 
effect when projectile commences to fall, hence minimum 
velocity above “a” will continue to decrease and, owing 
to angle at which gravity acts, trajectory in upward path 
of projectile will become gradually flatter until at + 90° it 
is absolutely flat, gravity has maximum retarding effect, 
range is at minimum, minimum velocity is nil, and there 
is no horizontal component of velocity of projection. 
The curve CB formed by the points at which each trajectory 
above “y” cuts that of the angle of elevation which is by 
a minimum below it, is the curve of maximum range, 1.e. 
any point in that curve is at the maximum attainable 
distance from A in that direction, and there is only one 
trajectory that will strike any point on line AB, curve BC, 
or vertical asymptote CD. 
Point of minimum velocity of + 90° trajectory being on 
curve BC at point B it appears possible that points of 
mummum velocity of angles above “a”? will be on that curve 
and increase from O to terminal along it. 
That angle # will be synonymous with “y” and that line of 
terminal velocity referred to in para. (4) will be the asymptote 
CD forming part of the envelope. 
It also appears that within the line BCD (except along 
AB) there are two trajectories and only two that will strike 
any point in the same, these two trajectories being the 
tivo that intersect at that point. 
