Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/369

732.] through an angle $$\pi - \alpha$$ into the position $$SP^{\prime \prime}$$, the acceleration of $$P$$ will be equal in magnitude and direction to where $$SP^{\prime \prime}$$ is equal to $$SP$$ turned through an angle $$2\pi - 2\alpha$$.

If we draw $$PF$$ equal and parallel to $$SP^{\prime \prime}$$, the acceleration will be $$\frac{\omega^2}{\sin^2 \alpha} PF$$, which we may resolve into

The first of these components is a central force towards $$S$$ 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 $$\mu r$$, directed towards $$S$$, and proportional to the distance, and the second is $$-2kv$$, a resistance to the motion proportional to the velocity, where

If in these expressions we make $$\alpha = \frac{\pi}{2}$$, the orbit becomes a circle, and we have $$\mu_0 = \omega_0^2$$, and $$k = 0$$.

Hence, if the law of attraction remains the same, $$\mu = \mu_0$$, 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 $$P$$ on the horizontal line $$XY$$, we shall find that its distance from $$S$$ and its velocity are the horizontal components of those of $$P$$. Hence the acceleration of this point is also an attraction towards $$S$$, equal to $$\mu$$ times its distance from $$S$$, together with a retardation equal to $$k$$ 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 VOL. II.