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

336 731.] The following application, by Professor Tait, 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

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

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