I3 8 KINETICS OF A RIGID BODY. [252. 



(4) Solid sphere, for a diameter. 



(5) Area of ring bounded by concentric circles of radii a lt a 2 , for 

 a line through the centre perpendicular to the plane of the ring. For 

 a ring whose thickness a 2 a is infinitesimal, the result can also be 

 obtained by differentiation from Ex. (3) (b). 



(6) Spherical shell of infinitesimal thickness, for a diameter. 



(7) Right circular cylinder, of radius a and height 2h : (a) for 

 its axis ; (b) for a generating line ; (c) for a centroidal line in the mid- 

 dle cross-section. 



(8) Prove that, in a right prism or cylinder of any cross-section, 

 we have <? 2 = q* + ^ c 2 , where q is the radius of inertia of the prism or 

 cylinder for a line bisecting the axis at right angles, q a the radius of 

 inertia of the axis, q c that of the middle cross-section, for the same line. 



(9) Area of ellipse : (a} for the major axis ; (b) for the minor 

 axis ; (c) for the perpendicular to its plane through the centre. 



(10) Solid ellipsoid, for each of the three axes. 



(n) Area of the cross-section of a 1-iron, for a centroidal line par- 

 allel to the flange. (Compare Art. 244, Ex. (13).) 



(12) Area of the cross-section of a symmetrical double T-iron, 

 width of flanges b, thickness of flanges 8, height of web h, thickness 



of web 28; for the two axes of symmetry, and for a centroidal line per- 

 pendicular to its plane. 



(13) Wire bent into an equilateral triangle of side a, for a centroidal 

 line at right angles to the plane of the triangle. 



252. Bouth's Rule. In the case of homogeneous masses with 

 axes of symmetry, the radius of inertia for an axis of symmetry 

 can readily be derived by the following mnemonical rule : The 

 square of the radius of inertia is ^, \, or \ of the sum of the 

 squares of the perpendicular semi-axes, according as the mass is 

 rectangular, elliptic, or ellipsoidal. 



The proof rests on the following typical cases which are 

 easily proved directly (comp. Art. 251, Ex. (i), (9), (10)) : 



(i) Rectangular area whose sides are 2 a, 2b, for a centroidal 

 line perpendicular to its plane : g 2 = ^ 



