Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/368

731.] The following application, by Professor Tait^[1], of the principle of the Hodograph, enables us to investigate this kind of motion in a very simple manner by means of the equiangular spiral.

Let it be required to find the acceleration of a particle which describes a logarithmic or equiangular spiral with uniform angular velocity $\omega$ about the pole.

The property of this spiral is, that the tangent $PT$ makes with the radius vector $PS$ a constant angle $\alpha$.

If $v$ is the velocity at the point $P$, then

$v.\sin \alpha =\omega .SP$.

Hence, if we draw $SP^{\prime }$ parallel to $PT$ and equal to $SP$, the velocity at $P$ will be given both in magnitude and direction by

$v={\frac {\omega }{\sin \alpha }}SP^{\prime }$.

Fig. 58.

Hence $P^{\prime }$ will be a point in the hodograph. But $SP^{\prime }$ is $SP$ turned through a constant angle $\pi -\alpha$, so that the hodograph described by $P^{\prime }$ is the same as the original spiral turned about its pole through an angle $\pi -\alpha$.

The acceleration of $P$ is represented in magnitude and direction by the velocity of $P^{\prime }$ multiplied by the same factor, ${\frac {\omega }{\sin \alpha }}$.

Hence, if we perform on $SP^{\prime }$ the same operation of turning it

1. Proc. R. S. Edin., Dec. 16, 1867.