244 
ME. J. LARMOR ON A DYNAMICAL THEORY OF 
sum of squares, which would be all necessarily positive if there were no gyrostatic 
influence. It may then be shown in the manner of the preceding analysis that 
is equal to a fraction of which the denominator is the period determinant of the 
molecule and the numerator is the minor of its leading term ; the numerator is there¬ 
fore the period determinant of the same molecule when its leading coordinate is 
prevented by constraint from varying. Thus we have the theorem 
yp, = Ar’ (i/ -«?) {f - «?). 
- 
wliere (a^, a^, . . . . a„)/27r are the natural frequencies of the molecule, while {a\, a\, 
.... a„)j2TT are its frequencies when it is subjected to that particular constraint 
('namely on a^) wliich would j^revent it from vibrating under the influence of the 
incident radiation ; also is the coefficient of inertia of that particular vibration, 
so that its kinetic energy is 
This constraint may be represented analytically by making the elastic coefficient rq 
infinite; we may therefore attempt to trace, by the examination of graphs of the 
separate terms involved, the effects on the free periods of continuously varying this 
constant. The behaviour of a gyrostatic system may be very different from what 
experience teaches as to vibrations about configuration of rest, for the mere imposition 
of constraint to limit the vibrations of one coordinate may upset the stability of 
others : thus if x represent the present period equation is of type ^ {x) cqi// {x) 
— 0, in which all the n roots of ^ (x) are real and positive, while the same may not 
be true of the n — 1 roots of i// [x). It follows, easily, however, by application of the 
princijfle of energy,'" that if the system be completely stable when all the gyrostatic 
motional forces are removed, then it will remain stable when these forces are 
restored ; and stability will therefore also be maintained when elastic connexions are 
strengthened or constraints are introduced.! In that case the roots of t// {x) will all 
be real and positive : and it is easy to deduce that they will separate those of cf) [x), 
and that in consequence g^, pg, . . . . will be all positive, so that Kundt’s law will 
hold good. The further conclusion is thus also somewhat probable that if the con¬ 
stitution of the gyrostatic molecule is thoroughly stable so that the imposition of 
mere constraint could not upset it, then Kundt’s law will hold. 
32. The specific refraction [jj? — + ‘•^) always increases along with the index 
jx : if the dispersion were controlled solely by powerful absorption bands in the ultra¬ 
violet, with positive g, the trend of the index would always be in the same direction 
as the frequency increases. Hence in the large class of substances wuth normal 
dispersion of visible light for wdiich K exceeds /x^, there must also be strong 
* Gf. Thomson and Tait’s ‘Nat. Phil.,’ Ed. 2, Part I., p. 409 ; the relations of the free periods of a 
gyrostatic system ax’e there discussed at length in pp. 370-415. 
t All the relations as to the march of the periods developed by Rayleigh (‘ Theory of Sound,’ Ed. 2, 
§92a), from Rourii’s analysis will then hold good. 
