53 



the direction of the arrow M. The moment of this force 

 has its minimum effect when acting at F in the direction 

 FR'. At any other point, d or d', the force is as gd' or g'd', 

 that is as the sine of Ad or Bd', the polar distance of the 

 point. 



When the particle which receives the impulse at A 

 arrives at B, it tends to move in the direction BR" parallel 

 to AR, and therefore tends to restore the axis, from its 

 position ab, to its previous position AB. 



Now if the particle when at F, tending to move in the 

 direction FR' by the impulse received at A, could move to 

 H through the semicircle FBH in a time infinitely short, 

 it would be equivalent to two equal forces at F and H, 

 at equal distances from the fulcrum C of the lever FH, 

 acting in lines FR', HR'", parallel to each other, and FH 

 would therefore remain at rest, that is, the sphere would 

 not rotate about the axis AB in the direction of the arrow 

 M. But the particle cannot be conveyed from F to H in 

 a time infinitely short ; therefore the particle, in moving 

 through the whole arc AFB, tends to make the sphere 

 rotate about the axis AB in the direction of the arrow M. 



Again, the impulse applied at A, in the direction AR, 

 causes B to move in the direction BQ ; when the particle 

 at B is carried by the rotation of the sphere to H, it 

 tends to move in the direction HQ', and therefore tends 

 to tvjra. the sphere about the axis AB in the direction of 

 the arrow M', the whole sphere will therefore revolve 

 about the axis AB in the direction of the arrows MM'. 



Now, if instead of a single impiilse acting at A, there is 

 a constant force acting in a fixed direction AR, the rota- 

 tion of the sphere about AB will continue until the axis 

 CL is brought to coincide with CH, the sphere will then 

 revolve about one axis HF. The force at A acting in the 

 direction AR, wliich produced its minimum effect on the 

 line of particles AB, when the sphere revolved about the 



