72] 



ELLIPSOID OF GYRATION. 

 ' = r'a' = Apr'x = B*Ax, 



233 



76) y' 



from which we obtain 



77) x = 

 and by 72) 



78) 



' 



BR* 



Of" 



^f 



Accordingly the locus of P' is an ellipsoid, whose axes are 

 inversely proportional to those of the original ellipsoid. It is called 

 the inverse ellipsoid. If we take 



irp we have 



M 



79) 



_L + 



a 2 1 6 2 ~~ c 2 



1 



and the semi -axes of the inverse 

 ellipsoid are equal to the principal 

 radii of gyration a, fr, c. 



Since the two ellipsoids have the 

 directions of their principal axes coin- 

 cident (namely the directions in which p 

 and r coincide), the relations are 

 evidently reciprocal, and OP is per- 

 pendicular to the tangent plane at P'. 

 Let the length of the perpendicular in 

 this direction be p'. Then since the triangles OPQ, OP' Q' (Fig. 70) 

 are similar, 



p p' ' 

 Since the moment of inertia about OP is 



we have 



Fig. 70. 



and the property of the inverse ellipsoid is that the radius of gyration 

 about any line is equal to the part intercepted by a plane perpen- 

 dicular to it tangent to the inverse ellipsoid. The inverse ellipsoid 

 is accordingly called the ellipsoid of gyration. 



It is evident that the direct ellipsoid more nearly resembles the 

 given body in shape than the inverse ellipsoid, for if the body is 

 spread out much about any particular axis the inertia and radius of 



