269.] 



ELLIPSOIDS OF INERTIA. 



147 



called the polar reciprocal of the former surface with regard 

 to the sphere. 



Let Q be the intersection of OP with 

 TT, and put OP = p, OQ = q\ then it 

 appears from the figure that 



pq = <?. (16) 



269, It is easy to see that the polar 

 reciprocal of the momental ellipsoid 

 (n') with respect to the sphere of 

 radius e is the ellipsoid 



oH o i o == * \*7/ 



Fig. 32. 



To prove this it is only necessary to show that the relation (16) 

 is fulfilled for p as radius vector of (i i'), and q as perpendicular 

 to the tangent plane of (17). Now this tangent plane has the 

 equation 



hence we have for the direction cosines , ft, y, and for the 

 length q, of the perpendicular to the tangent plane 



These relations give q l a=(x/q^)q y 

 whence 



For the radius vector p of (ii f ) whose direction cosines , y8, 7 

 are the same as those of q we have by (ii f ) : 



Hence / o 2 ^ 2 =e 4 ; and this is what we wished to prove. 



