KINETICS OF A RIGID BODY. [285. 



the origin), the former (TT O ) tangent to the fundamental ellipsoid 

 ^y02_|_yy2_|_2y2_. I? t h e latter (TT A ) tangent to any confocal surface 

 A or oi/(a* + A) + //( 2 + X) + z 2 /(^ + A) = i. The perpendiculars 

 ^ , ^x, let fall from the origin O on these tangent planes TT O , TT A , are given 

 by the relations (the proof being the same as in Art. 269). 



qf = <*<* + Pp + cV, (21) 



q*= (a 2 + AK + O* 2 + A)/? 2 + (^ + A)y 2 , (22) 



where a, /?, y are the direction-cosines of the common normal of the 

 planes 7r , TT A . Subtracting (21) from (22), we find, since a 2 -f/8 2 -|-y 2 = i, 



?A 2 -?o 2 = A; (23) 



i.e. the parameter A of any one of the confocal surfaces (20) is equal to 

 the difference of the squares of the perpendiculars let fall from the common 

 centre on any tangent plane to the surface A, and on the parallel tangent 

 plane to the fundamental ellipsoid \ = o. 



285. Let us now apply these results to the question of the distri- 

 bution of the principal axes throughout space. 



We take the centroid G of the given body as origin, and select as 

 fundamental ellipsoid of our confocal system the polar reciprocal of the 

 central ellipsoid, i.e. the ellipsoid (17) formed for the centroid, for 

 which the name " fundamental ellipsoid of the body" was introduced in 

 Art. 271. Its equation is 



if q-b q%, <? 3 are the principal radii of inertia of the body. 



The radius of inertia q Q for any centroidal line 4 can be constructed 

 (Art. 2 70) by laying a tangent plane to this ellipsoid perpendicular to 

 the line / ; if this line meets the tangent plane in Q (Fig. 33), then 

 ^ = GQo- Analytically, if a, ft, y be the direction-cosines of / , q is 

 given by formula (21) or (12'). 



286. To find the radius of inertia q for a line /, parallel to / , and 

 passing through any point P, we lay through P a plane TT A , perpendicular 

 to /, and a parallel plane TT O , tangent to the fundamental ellipsoid; let 

 <2 A , (?o be the intersections of these planes with the centroidal line / . 

 Then, putting <2o=?o, GQ^=q x , GP=r, PQ^ = d, we have, by 

 Art. 250, 



