Rotation of a Solid Body round a Fixed Point. 331 



III. To REPRESENT GEOMETRICALLY THE MpMENTS OF INERTIA 

 OF A BODY WITH RESPECT TO AXES DRAWN THROUGH A 

 FIXED POINT. 



The moment of inertia of a body with respect to any axis 



(a, /3, 7) is 



M = A cos'a + ff cos 2 j3 + 6" cos 2 ? - 2L' cos /3 cos 7 



- 2M f cos a cos 7 - 2W cos a cos j3 ; 



where 



A = l(y 2 + z 2 ) dm, L' = ji/zdm ; 



# = J(^ + g 2 ) dm, M' = J#2 dm ; 

 C' = l(x 2 + y 2 } dm, N' = fcydm. 



Assume M = -^ ; u. being the mass of the body, and / a distance 

 r 



measured on the line (a, /3, 7), and construct the ellipsoid whose 

 equation is 



-4V + ffy 2 + <7V - IL'yi - 2M'xz - 2N'xy = ^ ; (2) 



then it is evident that the moments of inertia of the body with 

 respect to axes passing through the fixed point are represented 

 by the squares of the reciprocals of the radii vectores of this 

 ellipsoid. Assume A = jua 2 , B = /*b*, C = pc 2 , and let the axes of 

 co-ordinates be the axes of the ellipsoid ; its equation will thus 

 become 



a 2 # 2 + &V + cV = 1; (3) 



and the equation of the reciprocal ellipsoid will be 

 *' y- z 2 



"a + T* + ~2 = L 4 



2 b l c 2 



This latter ellipsoid may be called the ellipsoid of gyration, as 

 the perpendiculars on its tangent planes represent the radii of 



