68, 69, 70, 71] CENTRIFUGAL FORCES. 229 



momentum, not to be replaced by a single force placed at the center 

 of mass, for the couple is not equal to what its value would be in 



that case, unless = -|-- If the center of mass lies on the axis, 



although the centrifugal force R c vanishes, the centrifugal couple 8 C 

 does not, unless D = E = 0. 



The centrifugal forces then in general tend to change the instan- 

 taneous twist, unless the axis of the latter passes through the center 

 of mass, and for it D = E = 0. Such axes are called principal axes 

 of inertia of the body at the center of mass, and are characterized 

 by the property that if the body be moving with an instantaneous 

 twist about such an axis, it will remain twisting about it, unless 

 acted on by external forces. In order to examine the effect of the 

 distribution of mass of the body, we are led to interrupt the con- 

 sideration of dynamics in order to consider the purely geometrical 

 relations among moments and products of inertia. 



7O. Moments of Inertia. Parallel Axes. Consider the 

 moments of inertia of a body about two parallel axes. Let the 

 perpendicular distances from a point P 

 on the two axes be p and p 2 and let 

 the distance apart of the axes be d. 

 Let A and B (Fig. 69) be the inter- 

 sections of the axes with the plane of 

 p and p 2 . If we take AS for the 

 X-axis, A for origin, we have rig> 69 . 



P^ = Pi* + d 2 - ^d cos (p^x), 

 60) Zmp<? = Zmp^ + Md 2 - 2dZmp 1 cos (p x) 



The last term is equal to 2dMx and vanishes if the axis 1 passes 

 through the center of mass. Consequently the moment of inertia 

 about any axis is equal to the moment of inertia about a parallel- 

 axis through the center of mass plus the moment of inertia of a 

 particle of mass 'equal to that of the body placed at the center of 

 mass, about the original axis. Consequently of all moments of inertia 

 about parallel axes, that about an axis through the center of mass 

 is the least. In virtue of this theorem the study of moments of 

 inertia is reduced to the study of moments of inertia about axes in 

 different directions passing through the same point. 



71. Moments of Inertia at a Point. Ellipsoid of Inertia. 



Consider now moments of inertia about different axes all passing 

 through the same point 0. Let a, /?, y be the direction cosines of 



