THE GEOMETRY OF MOTION 231 



is seen to be measured by the product of the square of 

 the speed and the curvature. 



S 1 6. — Fundamental Propositions in the Geometry 



of Motion 



We are now in a position, after restating our results, 

 to draw one or two important conclusions. 



Acceleration has spurt and shunt components. 



The spurt acceleration takes place in the direction of 

 motion, and is measured by the rate at which speed is 

 being increased (or, it may be, decreased). 



The shunt acceleration takes place perpendicular to 

 the direction of motion, and is measured by the product 

 of the curvature and the square of the speed. 



These two kinds of acceleration are usually spoken of 

 as speed acceleration and nor^nal acceleration. 



From these results we conclude that : — 



1. If a point be not accelerated it will describe, with 

 regard to the given frame of reference for which the 

 acceleration is measured, a straight line with uniform 

 speed. For there will be no spurt, and therefore the 

 speed must be uniform, and there will be no shunt, and 

 therefore the path must have zero curvature, but the only 

 path without bending is a straight line. Neither uniform 

 speed nor zero curvature alone denotes an absence of 

 acceleration. 



2. When a point is constrained to move in a given 

 path the normal acceleration may be determined in each 

 position from the speed and the form of the path, i.e. 

 from its curvature or bending. In this case the problem 

 is to find the speed from the speed acceleration. 



3. When a point is free to move in a given plane, 

 then its motion can be theoretically determined, if we 

 know its velocity in any one position, and its acceleration 

 for all positions. For from the normal acceleration and 

 the speed we can calculate the initial amount of bending 

 of the path ; thus the initial form of the path is known. 

 For a closely adjacent position on this initial form, we 



