642 
Proceedings of the Royal Society 
electro-magnet, to record themselves alongside of the records of the 
beats of a clock, and thus very great precision may be obtained. 
When the stone is thrown obliquely, its path is curved ; were 
there no air, that path would be the well-known parabola, which 
forms the first ideal approximation. With speeds very small in 
comparison with the terminal velocity, the deviation from the para- 
bolic curve is slight ; but the deviation becomes excessive in the 
ordinary practice of gunnery. 
Beginning with the case of a body thrown obliquely upwards, 
we observe that its upward motion is resisted both by the air and 
by gravity ; when it has reached its highest point and is descend- 
ing, the downward motion is less rapidly accelerated than the up- 
ward motion had been retarded, so that the culminating point is 
reached earlier than the half-time of flight. On the other hand, the 
horizontal transference is retarded all along, wherefore the vertex of 
the curve is beyond the middle of the horizontal range, as is seen 
in fig. 3, which shows the positions of a projectile at equal inter- 
vals of time. 
Fig. 3 . 
After having passed the highest point at V, the projectile bends 
its motion more and more downwards, until ultimately the path 
almost coincides with the plumb-line ; the speed, at the same time, 
gradually approximates to the terminal velocity, and thus the curve 
in that direction is limited by a vertical asymptote. 
Beckoning backwards on the other side of the culminating point, 
the speed is rapidly augmented, and would become infinite within a 
definite time ; the inclination of the curve also tends to a definite 
limit, wherefore this branch, too, has a rectilineal asymptote ; so that 
the whole curve is continued in the angle formed by the two asymp- 
totes as represented in fig. 4. Thus the branch VB, unlimited in 
