Tin-: cvROF^coPK. 6i 



We may therefore deduce the rule that the free end of the axis 

 always moves in the opposite direction to that in which the 

 top of the wheel moves. 



Such is the action of the theoretical gyroscope under ordi- 

 nary circumstances. Before taking up the phenomena of the 

 practical instrument it is well to consider the effect of certain 

 external influences upon the theoretical instrument. 



F'irst : Let us consider the effect produced by an external 

 force tending constantly to accelerate or retard the orbital 

 velocity of the axis. Suppose the force to be accelerative. 

 The result will be that the applied force acting on the appa- 

 ratus combines with the force of gravitation so that the instru- 

 ment operates as though the gravitational force were directed 

 downward and slightly ahead. The axis tends to keep the 

 same average "level" with regard to this new force, or in 

 other words to describe an average path at right angles to the 

 force. The axis must therefore rise. The work done upon 

 the axis by the horizontally directed force appears in the 

 potential energy due to the more elevated average position of 

 the wheel. If the applied force is retarding, the circumstances 

 are all reversed and it follows necessarily that the wheel falls. 

 The potential energy due to the fall of the wheel now appears 

 as the work done by the apparatus on the external retarding 

 medium. 



Second : Consider the effect of starting the gyroscope with 

 an initial horizontal velocity and then leaving it to itself. 

 The idea at once suggests itself that if this initial velocity is 

 just sufficient to make the gyroscopic couple equal to the gra- 

 vitational couple that it is possible to have an orbital motion 

 free from nutation. This is true, although this condition of 

 things is essentially unstable (except under certain conditions 

 to be mentioned in treating of the practical gyroscope) and 

 upon the least disturbance the ordinary nutating motion will 

 supervene. The nutating motion is essentially stable, on the 

 other hand, and will reestablish itself after any disturbance. 



If the instrument is put under way with a velocity less 



