266.] ELLIPSOIDS OF INERTIA. 



must be fulfilled. These reduce to 



and show that the line must pass through the centroid. And 

 as for a centroidal line these conditions are satisfied indepen- 

 dently of the value of a, it appears that a centroidal principal 

 axis is principal axis at every one of its points. Hence a line 

 cannot be principal axis at more than one of its points unless it 

 pass through the centroid ; in the latter case it is principal axis 

 at every one of its points. 



265. All those lines passing through a given point O for 

 which the moments of inertia have the same value 7 can be 

 shown to form a cone of the second order whose principal 

 diameters coincide with the axes of the momental ellipsoid 

 at O. This cone is called an equimomental cone. Its equation 

 is obtained by regarding 7 as constant in equation (12) and 

 introducing rectangular co-ordinates. Multiplying (12) by 

 ; = i, we find 



and multiplying by p 2 , we obtain the equation of the equi- 

 momental cone in the form 



266. A slightly different form of the equations (n), (12), (15) 

 is often more convenient ; it is obtained by introducing the 

 three principal radii of inertia q^ q^ q z defined by the relations 



The equation (11) of the momental ellipsoid at the point O then 

 assumes the form 



The expression of the radius of inertia q for any line (a, 0, 7) 

 through O becomes 



PART III 10 



