56 C. M. BROOMALL : 



rotations. The curved lines in the figure are meant to repre- 

 sent the paths of the particles as projected on the plane of the 

 paper. The arcs described in front of the plane of the paper 

 are shown in solid lines and those behind the plane of the 

 paper in dotted lines. The arrows show the direction of the 

 motion. In the figure the angular velocity of the wheel and 

 axis are assumed the same. 



As the various particles describe the curves centrifugal 

 forces are of course developed. These centrifugal forces at 

 all times except when the particles are in the plane of the 

 paper have moments around the line of intersection of the 

 plane of the wheel with the plane of the paper. A glance 

 shows that the particles above the plane of the paper and 

 likewise those behind the plane of the paper conjoin to make 

 the end a of the axis recede and the end b advance. 



Let us examine the matter more in detail. As will be seen 

 each particle in the figure is shown in three successive posi- 

 tions of its curved path. This together with the kuown velo- 

 city of the particle supplies the information necessary to deter- 

 mine the direction and amount of the centrifugal force of the 

 given particle at any moment. The centrifugal force of an}^ 

 particle, say when the wheel is in the position A' B\ is a func- 

 tion of the path of the particle in space as the wheel changes 

 from AR to A" B" . Considering the position A' B' therefore 

 we see that there is a perfect symmetry of motion of all the 

 particles in front of and behind the plane of the paper by 

 which all components of centrifugal force except those tend- 

 ing to move the axis perpendicular to the plane of the paper 

 are neutralized. In other words, the various particles pair off 

 with one another in such a way that the only unbalanced 

 forces left are those tending to push a perpendicularly back 

 from the plane of the paper and to draw b forward. This 

 constitutes what is known as the gyroscopic force. It is of 

 the nature of a couple the middle of whose lever arm is at the 

 centre of the wheel. It can of course only be balanced by 

 an equal couple. 



