PHYSICS: E. B. WILSON 
589 
are the parametric equations of the path, v being any function of the 
time. The equations contain three constants of integration, which 
allow an arbitrary choice of axes, and one arbitrary function v. The 
accelerations along and perpendicular to the path are and 
the total acceleration is a/ v'^ + vv^\ and its inclination to the path, 
tan-i (vWVz/). 
A simple case may be had by taking v = e~^\ which represents a 
particle combing to rest. Equations (4) may be written 
^ e'"^ cos at dt, ^ = X° ^"""^ 
Or 
X = — ^ e-''^ cos (at-iir), y = e""' sin (at - f tt). 
Via V2 a 
And if 
6== at- 37r/4, r = -1- e"^"''^/' 
is the polar equation of the locus — showing an equiangular spiral with 
45° between radius and tangent. If s = x -\- iy, 
^-ail- i)t- a-jri/i 
The vector velocity dz/dt is 135° ahead of the radius z, and the ac- 
celeration is 270° ahead, which means a retarding acceleration decreas- 
ing the areal velocity and perpendicular to the radius. This type of 
acceleration is found in the straight-bore centrifugal gun. 
In (5) the magnitudes of the radius r, velocity v, acceleration v', and 
its rate are^n geometric progression with the ratios V2 a, the value 
of v' being V 2 ae'''^ If this acceleration be resolved along the radius 
and along the normal to the path, the respective components SiYe2\/2ae~^^ 
outward and 2 ae'""^ inward. A constant magnetic field perpendicular 
to the plane of motion is competent to furnish a component acceleration 
along the normal to the path and proportional to the velocity. The 
electric field in a plane perpendicular to the line joining two like charges 
at its middle point will act radially from the point of equilibrium upon 
a Hke charge with a force proportional to the distance. A proper 
slight adjustment of the intensity of the magnetic field will take care 
of the force (1). It is therefore possible easily to set up an electrical 
problem which is satisfied by the path (5). The general dynamical 
equation for the motion thus adjusted is 
