28 9 .] 



PRINCIPAL AXES. 



155 



The figure gives the relation d z = r 1 q, which, in combination with 

 (23), reduces the expression for the radius of inertia for the line / to 

 the simple form : 



? 2 =^-A. (24) 



287. The value of r 2 X, and hence the value of q, remains the same 

 for the perpendiculars to all planes through P, tangent to the same 

 quadric surface X : these per- 



pendiculars form, therefore, 



an equimomental cone at P. 



By varying A. we thus obtain 



all the equimomental cones 



at P. The principal diame- 



ters of all these cones coin- 



cide in direction, since they 



coincide with the directions 



of the principal axes of the 



momental ellipsoid at P (see 



Art. 265); but they also coin- 



cide with the principal diam- 



eters of the cones enveloped Fig. 33. 



t>y the tangent planes TT A . It 



thus appears that the principal axes at the point P coincide in direction 



with the principal diameters of the tangent cone from P as vertex to 



the fundamental ellipsoid x 2 /q? +f/q? + #/$$ = i. 



288. Instead of the fundamental ellipsoid, we might have used any 

 quadric surface A. confocal to it. In particular, we may select the con- 

 focal surfaces A*, A. 2 , X 3 that pass through P. For each of these the cone 

 of the tangent planes collapses into a plane, viz. the tangent plane to 

 the surface at P, while the cone of the perpendiculars reduces to a single 

 line, viz. the normal to the surface at P. Thus we find that the prin- 

 cipal axes at any point P coincide in direction with the normals to the 

 three quadric surfaces, confocal to the fundamental ellipsoid and passing 

 through P. 



For the magnitudes of the principal radii q x , q y , q z at P, we evidently 

 have 



289. Exercise. 



(i) The principal radii q, q^ q% of a body being given, find the 

 equation of the momental ellipsoid at any point P, referred to axes 



