259-] ELLIPSOIDS OF INERTIA. ! 4I 



Thus the moment of inertia for any plane through the origin is 

 expressed as a homogeneous quadratic function of the direction 

 cosines of the normal of the plane. 



258. The moment of inertia f=2mq 2 for the line / can now 

 be found from equation (i), Art. 245, by substituting for 

 and 2;/z/ 2 their values from (7) and (8) : 



or, snce + / + 7= i, 



hence, finally, applying the relations of Art. 256, 



(9) 



The moment of inertia for any line through the origin is, 

 therefore, also a homogeneous quadratic function of the~direction 

 cosines of the line. 



259. These results suggest a geometrical interpretation. Im- 

 agine an arbitrary length OP=p laid off from the origin O on 

 the line / whose direction cosines are a, ft, 7 ; the co-ordinates 

 of the extremity P of this length will be x = pa, y = p/3, z = py. 

 Now, if equation (9) be multiplied by p 2 , it assumes the form 



which represents a quadratic surface provided that p be so 

 selected for the different lines through O as to make p 2 ! con- 

 stant, say/3 2 /=A; 2 Hence, if on every line 1 through the origin 

 a length OP = /o = /<:/VT be laid off, i.e. a length inversely pro- 

 portional to the square root of the moment of inertia I for this 

 line 1, the points P will lie on the quadric surface 



