Statics. 



towards the plane, for the rotatory motion about the axis ND 

 cannot cause any pressure upon this plane ; the resistance in 

 question, therefore, can never be equivalent to the force of grav- 

 ity, and there will always remain in the centre of gravity G, a 

 force tending to bring it to the plane. Hence, when there is no 

 friction, and the top has received at first no motion of rotation, 

 except about the axis of its figure, it must of necessity fall. 



252, It is not the same on the supposition of friction. In 

 this case, the resistance which takes place at .AT, acts not accord- 

 ing to the perpendicular JVA 7 , but according to the line NK'^ 

 which makes with the plane XZ an angle equal to the angle of fric- 

 tion, and passes through JV, one of the points of friction. What- 

 ever may be this angle and this point, the resistance which takes 

 place along the line NK', is equivalent to a force acting upon 

 the body in a contrary direction ; now as this direction does not 

 pass through the centre of gravity, it must produce in the body 

 137. a rotatory motion, that is, a variation in its actual motion of ro- 

 tation ; but it must also transmit itself entirely to the centre of 

 gravity. Let us suppose, therefore, that GL parallel to NK' is 

 this force ; if the vertical line GI represent the force of gravity, 

 and the parallelogram GLEI be completed, GE will be the actu- 

 al force which belongs to the centre of gravity G. 



Now the angle LGI and the force G7 remaining the same ; 

 the greater the force acting in the direction K'N, and consequent- 

 ly the greater the force GL, the more nearly will the line GE 

 approach the line GL; that is, the greater will be the tendency 

 of the point E to rise above G. It remains therefore to be seen, 

 whether from the nature of friction, together with the figure of 

 the body, and its motion of rotation, the ratio of the force in the 



\<r 13ft 



' direction NK' (or the force GL) to the force of gravity G7, can 

 be increased till the point E shall be above the point G ; in 

 which case it is clear that the centre of gravity may rise with 

 respect to the plane ; yet not so as to cause the point N to quit 

 it, because the motion of rotation which results from the force 

 in the direction K'N, will tend to bring this point toward the 

 the plane. Now (1.) As the body rests upon a point, it cannot 

 be denied that the parts of this point sink more deeply than if the 

 body rested upon a sensible surface- Regard being had to the 

 rotatory motion about JVD. and to the action of gravity, thepres- 



