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 about the pole.
The property of this spiral is, that the tangent makes with the radius vector a constant angle .
If is the velocity at the point , then
.
Hence, if we draw parallel to and equal to , the velocity at will be given both in magnitude and direction by
.
![](http://upload.wikimedia.org/wikipedia/commons/thumb/c/c4/A_Treatise_on_Electricity_and_Magnetism_-_Volume_2_Fig_58.png/250px-A_Treatise_on_Electricity_and_Magnetism_-_Volume_2_Fig_58.png)
Fig. 58.
Hence will be a point in the hodograph. But is turned through a constant angle , so that the hodograph described by is the same as the original spiral turned about its pole through an angle .
The acceleration of is represented in magnitude and direction by the velocity of multiplied by the same factor, .
Hence, if we perform on the same operation of turning it
- ↑ Proc. R. S. Edin., Dec. 16, 1867.