118 AN EMPIRICAL STUDY OF GYRATING BODIES. 



whole of B b will be expended upon O, while A x will 

 exhaust itself in pulling in the same point. 



A yet more curious result follows when 2a is greater 

 than 90°, as in fig. 12. Cos 2a now becomes negative, 

 but its even powers are positive. Hence, for first revers- 

 al (see fig. 13), the new tangent tends to lift the instru- 

 ment ; the second tends to depress it (see fig. 14), etc. 



If 2a equals 180°, then cos a=0, and the whole series 

 disappears, as it ought. For, in this case, the whole 

 influence of gravity is expended on the supporting point, 

 which is then at the centre of gravity. 



In this formula, too, we find why the stability dimin- 

 ishes as the diameter of the wheel grows less, — the dis- 

 tance from O to P remaining unchanged ; why, for ex- 

 ample, the axis without any wheel, however fast it re- 

 volves, falls as if it did not revolve at all. For a and 2a 

 being almost zero, their cosine is very nearly unity. 

 Consequently, the series shows infinitely small diminu- 

 tion, and therefore only an infinitely small part of the 

 force of gravity is expended on the point of support. 



We also find here the reason wdiy a gyroscope whose 

 wheel has a heavy rim and a light centre, falls more 

 slowly than one with a wheel of uniform thickness ; and 

 this again, more slowly than one whose wheel has been 

 replaced by a sphere ; for the less the average distance 

 of the particles from the centre, the less the angle a be- 

 comes, and consequently the more rapid the fall. 



Thus far, I have limited myself to the consideration of 

 a single instant's effect of gravity, and have traced it 

 out to extinction. In the next instant another impulse 

 is given, and so each instant thereafter. 



To get the whole effect it is necessary to combine the 

 series. 



As the instants are exceedingly brief, I have spoken 

 of the action of gravity as an impulse. 



The series for each impulse will be as follows : 



102 



