638 
Proceedings of the Royal Society 
at that epoch, the velocity must have been infinite, so that although 
the body may have come from an infinite distance, setting out 
therefrom with an infinite velocity, it must have begun that motion 
a finite time ago. 
If we represent time by distances measured along OF, one of the 
asymptotes of a hyperbola, the ordinates, such as P p, drawn parallel 
to the other asymptote, are proportional to the corresponding 
velocities. Thus the present velocity being P 'p, that at the future- 
time OA will be A a, while that at the former date OB had been 
B b ; and the areas B5pP, PpaA represent the distances passed 
over during the intervals of time BP and PA. The distance 
corresponding to the finite previous time OP is thus infinite, and 
so also must have been the velocity of projection at the date 0. 
When the motion is affected by some influence other than the 
resistance, the investigation becomes more intricate. The case of a 
constant gravitation in a fixed direction is the simplest of these 
complications, and the simplest case of this is when the directions 
of the motion and of gravitation coincide. 
If a stone be thrown straight upwards, its motion is impeded both 
by its weight and by the air’s resistance ; in the subsequent descent 
the motion is accelerated by gravity, but retarded by the air ; so 
that, for the ascent, the soliciting influence takes the form g + cv 2 , 
and for the descent becomes g - cv 1 . Now the change in the sign of 
the velocity from + v in the ascent to -v in the descent, is not 
accompanied by any change in the. sign of v 2 , and therefore both 
parts of the motion cannot be represented by any one algebraic 
formula. Accordingly we find the upward motion to be repre- 
sented by circular functions ; the downward motion by the corre- 
sponding catenarian ones. 
In fig. 2, the left hand row of dots represents the upward 
motion graduated to equal intervals of time. The stone is first 
shown at A as having come from some indefinite distance below ; 
its speed, rapidly diminished, is altogether extinguished at N. In 
order to avoid confusion, the descent is shown on the adjoining right 
hand column of dots. 
In descending, the acceleration due to gravity becomes less and 
less ; it would cease altogether if the velocity could become so great 
as to cause a resistance equal to the weight; the tendency, there- 
