( 619 ) 
not sufficiently clearly state that the divergent result was only 
d*b 
founded on the assumption — = 0. I knew, however, that van der 
dx 1 
Waals in his Continuitat II p. 24 has already treated this question. 
Yet on theoretical considerations I abide by my opinion that in the 
neighbourhood of the limiting volume, so at very high pressures, 
d'b „ 
—- must be = 0. 
And now I think that I for my part, have sufficiently elucidated 
Mr. Kohnstamm’s Reply, so that further misunderstanding seems 
almost precluded. 
Baam, Pebr. 21, 1910. 
Mathematics. — “The oscillations about a position of equilibrium 
where a simple linear relation exists between the frequencies 
of the principal vibrations .” (1 st part). By Mr. H. J. E. Beth. 
(Communicated by Prof. Korteweg). 
Introduction. 
§ 1. In his paper 1 ): “On certain vibrations of higher order of 
abnormal intensity (vibrations of relation) for mechanisms .with more 
degrees of freedom” (Verhandelingen der Koninklijke Akademie van 
Wetenschappen, Vol. V. N°. 8, 1897; Archives Neerlandaises Vol. I, 
series II, pages 229—260) Prof. Korteweg has written down the 
expansions in series for the principal coordinates of an arbitrary 
mechanism with more degrees of freedom, performing small oscilla¬ 
tions about a position of stable equilibrium. From these expansions 
in series could be deduced that in a certain case it was possible for 
some vibrations of higher order, having in general a small intensity 
with respect to the principal vibrations, to obtain an abnormally 
great intensity; this is the case when between the frequencies 
n t n ;f etc. of the principal vibrations a relation exists of the form 
pn x + qn y -= Q ? 
where p, q etc. are positive or negative integers and q is with respect 
to n x , n y etc. a small quantity, called residue of relation. 
Furthermore however it became evident that, when $^4(5 is the 
sum of the absolute values of p, q etc.) and at the same time q = 0, 
J ) “Over zekere trillingen van hooger orde van abnormale intensiteit (relatie- 
trillingen) bij mechanismen met meerdere graden van vriiheid”. 
