255-] ELLIPSOIDS OF INERTIA. 



(2) Elliptic area whose axes are 2 a, 2b, for a centroidal line 

 perpendicular to its plane : q> 2 =^(a 2 -\-& 2 ). 



(3) Solid ellipsoid whose axes are 2a, 2b, 2c, for its axes : 



A large number of special cases can be brought under this 

 rule, as will be seen from the following exercises. It should be 

 remembered that the radius of inertia of a homogeneous right 

 prism or cylinder for its axis is the same as that of its cross- 

 section. 



253. Exercises. Apply Routh's rule to find the radius of inertia in 

 the following cases : 



( i ) Solid sphere of radius a, for a diameter. 



(2) Right circular cylinder, for its axis. 



(3) Thin straight rod of length 2 a, for a perpendicular through its 

 middle point. 



(4) Rectangular disc whose sides are 2 a, 2 b, for a line in its plane 

 bisecting the sides 2 a. 



(5) Circular disc, for a diameter. 



2. ELLIPSOIDS OF INERTIA. 



254. The moments of inertia of a given mass for the different 

 lines of space are not independent of each other. Several 

 examples of this have already been given. It has been shown, 

 in particular (Art. 248), that if the moment of inertia be known 

 for any line, it can be found for any parallel line. It follows 

 that if the moments be known for all lines through any given 

 point, the moments for all lines of space can be found. We 

 now proceed to study the relations between the moments of 

 inertia for all the lines passing through any given point O. 



255. It will here be convenient to refer the given mass M to 

 a rectangular system of co-ordinates with the origin at the point 

 O. Let x, y, z be the co-ordinates of any particle m of the 



