150 KINETICS OF A RIGID BODY. [274. 



(4) Determine the momental ellipsoid and the principal axes at a 

 vertex of a cube whose edge is a. 



(5) Determine the radius of inertia of a thin wire bent into a circle, 

 for a line through the centre inclined at an angle a to the plane of the 

 circle. 



(6) A peg-top is composed of a cone of height H and radius a, and 

 a hemispherical cap of the same radius. The point, to a distance h 

 from the vertex of the cone, is made of a material three times as heavy 

 as the rest. Find the moment of inertia for the axis of rotation ; 

 specialize for h = a \ H. 



(7) Show that the principal axes at any point A, situated on one of 

 the principal axes of a body, are parallel to the centroidal principal axes, 

 and find their moments of inertia. 



(8) For a given body of mass M find the points at which the mo- 

 mental ellipsoid reduces to a sphere. 



(9) Determine a homogeneous ellipsoid having the same mass as a 

 given body, and such that its moment of inertia for every line shall be 

 the same as that of the given body. 



3. DISTRIBUTION OF PRINCIPAL AXES IN SPACE. 



274. It has been shown in the preceding articles how the principal 

 axes can be determined at any particular point. The distribution of 

 the principal axes throughout space and their position at the different 

 points is brought out very graphically by means of the theory of con- 

 focal quadrics. It can be shown that the directions of the principal 

 axes at any point are those of the principal diameters of the tangent 

 cone drawn from this point as vertex to the fundamental ellipsoid ; or, 

 what amounts to the same thing, they are the directions of the normals 

 of the three quadric surfaces passing through the point and confocal 

 to the fundamental ellipsoid. 



In order to explain and prove these propositions it will be necessary 

 to give a short sketch of the theory of confocal conies and quadrics. 



275. Two conic sections are said to be confocal when they have the 

 same foci. The directions of the axes of all conies having the same 

 two points S, S' as foci must evidently coincide, and the equation of 

 such conies can be written in the form 



