THE DRIFT OF SERVICE FROJECTILBS. 
461 
Let ns now consider a curved trajectory (Fig. 2). Let 0 be the 
centre of gravity of the shot. With that centre and radius equal to the 
distance of the point from the 0 G, let us describe a sphere round the 
shot. Thus we see at starting that 0 P is the direction of the axis 
of the shot. At the end of the trajectory the axis will have a direc¬ 
tion something like 0 P' and at some time between, at the top 
of the trajectory the shot will have a horizontal direction, so that we 
can draw the direction at any time from the centre to the outside 
of the sphere, representing the direction of the axis at any given 
point of the trajectory. The direction of the resistance of the 
air, on the other hand, is the reverse of this. At starting its 
direction is P 0 and its direction cuts the sphere at P. At each 
succeeding point it cuts it at a lower and lower point, so that you 
may say that the direction of the resistance of the air cutting the 
sphere at a lower and lower point is in fact describing a vertical 
descending line on it from P to P'; at each successive instant you get 
a lower and lower point where the direction of the air cuts it. Simi¬ 
larly the point during its flight will come down until it gets to the posi¬ 
tion that it has at the end of the trajectory. Now, although the line 
that the air resistance cuts on the sphere will be part of a vertical 
great circle circular of the sphere, the point, as it describes a spiral 
round the trajectory will not be absolutely in that line, it will describe 
a kind of spiral round it as it comes down, sometimes being slightly to 
the right of it, sometimes on the left. 
Let G A at any time be the position of the shot and G H will be 
the tangent at that point; and let the point be in the same vertical 
plane of the trajectory and above it. There is no reason why we 
should begin here, but it is more convenient. During the next few 
moments of that shot's flight the point must come out of the vertical 
plane and try to get down to a level with the tangent to the trajectory 
and until it gets down on a level with it it must be going to the right 
and down. If the trajectory was a straight line by the time it got to 
a position, say G B, it would have got its point down to a level with 
the direction of the tangent G H, only of course the point would be 
out to the right; but, as the trajectory is curved at G B, the trajectory 
has changed its direction and its tangent, instead of being parallel to 
G H, will have got down to a position G' H'. The point, therefore, 
has further to go before it can catch it up. As the trajectory has been 
curving downwards, therefore, the point has got further to go before it 
gets down on to a level with the line of the trajectory than if the tan¬ 
gent were fixed in direction, the result being that the faster the 
trajectory curves the longer it will take the point to get down to it. 
Until it gets down on to a level with the tangent of the trajectory, the 
point and therefore the axis, must be going not only to the right but 
down, it is following it down and on the right of it. When it has 
once got down to a level with the tangent it is at its widest on the 
right and it then will be trying to get in underneath it; and that als o 
takes some comparatively considerable time, because it must move at 
right angles to the plane containing the line of motion at the time of 
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