1325 
inner particles are likewise deprived of their heat motion. Then they 
will take definite positions of equilibrium P, P’, P’’ ete. This con- 
figuration of the system may be considered as the result of a dilata- 
tion equal in all directions, starting from the configuration that would 
exist at the absolute zero and in the absence of external pressure. 
Now, always keeping fixed the outer particles, we may investigate 
the vibrations of the inner ones about their just mentioned positions 
of equilibrium -P, P’, P’’... This is a perfectly definite problem. It 
can be simplified in the well known way by the introduction of a 
certain number of normal coordinates q¢,,9,..-@s, Which we shall 
choose in such a manner that they are O in the position of equi- 
librium, so that they determine the deviation from that position. 
The corresponding velocities are q,,q,,---qs and if the values of 
the coordinates are not too great we have for the potential energy 
U and the kinetic energy 7’ expressions of the form 
fe ag, Te ee + as qs’) ’ : NE Ae (1) 
> C5952 H.+ Css") hem aak yey ye MA) 
with positive, constant coefficients. Further there are s fundamental 
modes of vibration. In the first of these all the normal coordinates 
except q, are 0, in the second all except q, and so on. The frequen- 
cies of these fundamental vibrations are determined by 
ds 
RE == 
Cs 
As to the deviation of a partiele from the position of equilibrium, 
its components for the first particle may be represented by 
U = 3 (4,¢,’ 
Ien 5 (e‚9,° 
S= Gt ende +. -4t- Os Js; \ 
n=8 A + B, 9, +. + Bs gs, 
by, ADL + Ys Js» 
for the second by (3) 
Bd de 94+ MRE | 
ted Gs ot Br Mak Wants B's qs 5 
S ik VN Teh aes le an Ys Qs 5 
and so on. 
Here the coefficients a, 3, 7 have definite constant values. As the 
number s of the normal coordinates is equal to that of the degrees 
of freedom of the system, viz. 
to three times the number of the 
particles that are assumed to be movable, all possible displacements 
af ue ö, B 1, aj 
Ju VED od Reade Ys 
may be represented by suitably chosen values of 
