ACTED ON BY NO EXTERNAL F0B€E8. 



7.63 



Li fact it is easily seen that 



b^-\-(f = 3 — \- n ' 



cH«'=- 



a' ' a' 



b" + b^ 



Hence any uniform ellipsoid, with its centre fixed, compelled by friction to roll on a 

 rough horizontal plane will move precisely like a free body with properly assigned 

 moments of inertia acted on by no external forces, as was to be proved. We see from what 

 has been shown above that a uniform ellipsoid whose semiaxes are a, h, c, and which 

 rolls on a rough horizontal plane, will keep pace with the motion of a uniform free 



ellipsoid, provided that the moments of inertia of the latter are in the ratio of -^ : ^^ : -5? 



i. e. provided its axes are in the proportion of 







and thus the relative rate of motion of the rolling ellipsoid will not be aflfected if an 

 interior ellipsoid Avhose axes are in the proportions above written is entirely removed 

 or its density altered in any ratio. The internal ellipsoid wUl in fact move precisely 

 as if it were free and detached from the surrounding crust, and might be annihilated 

 without affecting the motion of the latter, in analogy with the well-known fact that any 

 weight at the centre of oscillation of a compound pendulum may be abstracted without 

 affecting its motion. 



The theories of the free body and of the ^'S- 2- 



ellipsoid constrained by pressure and fric- 

 tion to follow its path, and which has been 

 proved above to keep exact pace with it, 

 are so interwoven that it would be unsatis- 

 factory to leave the theory of the latter 

 incomplete in any point, and I shall there- 

 fore proceed to calculate the value of the 

 pressure and friction corresponding to any 

 position of the rolling body. On a sphere 

 described about the fixed point, let P and I denote the position of the instantaneous 

 axis of rotation, and the perpendicular to the fixed plane respectively. The pole of the 

 friction couple will be denoted by a point P' in the plane of PI distant by a quadrant 

 from P, for its plane passes through P and through Q the pole of PI, and the pole of 

 the pressure couple will obviously lie at Q itself. Let X, Y, Z mark in the sphere the 

 positions of the principal axes. 



