through an angle into the position , the acceleration of will be equal in magnitude and direction to
,
where is equal to turned through an angle . If we draw equal and parallel to , the acceleration will be , which we may resolve into
and .
The first of these components is a central force towards proportional to the distance.
The second is in a direction opposite to the velocity, and since
,
this force may be written
.
The acceleration of the particle is therefore compounded of two parts, the first of which is an attractive force , directed towards , and proportional to the distance, and the second is , a resistance to the motion proportional to the velocity, where
, and .
If in these expressions we make , the orbit becomes a circle, and we have , and .
Hence, if the law of attraction remains the same, , and
or the angular velocity in different spirals with the same law of attraction is proportional to the sine of the angle of the spiral.
732.] If we now consider the motion of a point which is the projection of the moving point on the horizontal line , we shall find that its distance from and its velocity are the horizontal components of those of . Hence the acceleration of this point is also an attraction towards , equal to times its distance from , together with a retardation equal to times its velocity.
We have therefore a complete construction for the rectilinear motion of a point, subject to an attraction proportional to the distance from a fixed point, and to a resistance proportional to the velocity. The motion of such a point is simply the horizontal