52 



the three roots of the ordinary cubic (56) ; but by the man- 

 ner of dependence (53) of the characteristic a on <t and s, 

 and by the symbolic equation of cubic form (38) in o-, we 

 have, if s be any one of those three roots of (56), the relation 



(<t + s)(t'k = 0; (59) 



consequently the three vectors (58) are such that 



= (o- + S,)'l = (a- + 82)1-2 = ((T + S3)£3. (60) 



Each of the vectors, t„ £05 '35 is therefore, by (49), adapted to 

 become a permanent axis of rotation of the body ; while the 

 foregoing analysis shows that in general no other vector i, 

 which has not the direction of one of those three vectors (58), 

 or an exactly opposite direction, is fitted to become an axis of 

 such permanent rotation. And to prove that these three axes 

 are in general at right angles to each other, or that they 

 satisfy in general the three following equations of perpen- 

 dicularity, 



= S.lil2= S.l2l3= S.I3 li, (61) 



we may observe that, for any two vectors i, k, the form (36) 

 of the characteristic o- gives, 



S.K(Ti=^'2.7nS.KaS.ai= S.KTK, (62) 



and therefore, for any scalar s, 



S. k((t + s)i = S.i(<7 + s)k; ' (63) 



consequently the two first of the equations (60) give (by 

 changing i, k, s to ta, t,5 «i), 



(S1-S.2) S.iiio = 0; (64) 



and therefore they conduct to the first equation of perpendi- 

 cularity (61), or serve to show that the two axes, i^ and i^, are 

 mutually rectangular, at least in the general case, when the 

 two corresponding roots, 6', and 5.,, of the equation (56), are 

 unequal. The equations (48) and (32), namely, F. ly = 0, 

 S.ySi = 0, show also that these three rectangular axes of 

 inertia are in the directions of the axes of the ellipsoid (22), 



