770 PROFESSOE SYLVESTER ON THE MOTION OE A RIGID BODY 



become each of them simply altered by the addition of the term — JJK, which may be 

 expressed by saying that tlic difference between the displacements at any mtjraent of 

 time of two bodies whose kinematical exponents are confocal ellipsoids, is equivalent to 

 a displacement round the invariable line proportional to the time elapsed since the 

 positions were coincident or parallel, as previously found by geometrical reasoning. 



Again, if we replace 1, ^, ^, M by 



^~i' ^~B' ^~C' ^^'-^' 



the equations (1), (2), (3) will remain unaltered, provided we write 180— a; 180— (3 ; 

 180— y in place of a, 3, y, and the equations (4) will receive an augmentation of Lx on 

 their right-hand sides, but remain otherwise unaltered, provided we substitute —5, — ;?, 

 — ^ for 1, 71, ^. Or again, we may state the same result without substituting for the angles 

 of inclinations their supplements, but leaving them unaltered if we change the sign of L ; 

 showing that if two bodies whose kinematical exponents or momental ellipsoids are 



so that using ( instead of r, the radical is 



^(i_a»Xi-x?n x==- 



w 



P-AV =I''+-(6-C«r')=i (Ba- ACar) 



y / 





So that the form is 



■where 



and thus 



L=-BY=P-?(a+CjSr')=l(-A6-BC/3r=) 



y y 



y \ Ab'Cji ) 

 _Aa Bo 5 aa, 



'^'W "'~~a3' " — p' 



aa 



so that the relation is the kno'n'ii one between the two forms 



\t,^. with reciprocal parameters." 



r ^L__ and r ^1 



