Mathematics. “The oscillations about a position of equilibrium 
where a simple linear relation exists between the frequencies 
of the vibrations”. (Third Part), By H. J. E. Bern. (Com- 
munieated by Prof. D. J. Korrewee). 
(Communicated in the meeting of December 24, 1910). 
§ 1. In my dissertation ©) were investigated the oscillations about 
a position of equilibrium of a mechanism with two degrees of 
freedom where a linear relation exists between the principal frequen- 
cies of vibration for which relation the sum of coefficients is NS 4, 
In what follows this investigation will be extended to a mechanism 
with an arbitrary number of degrees of freedom. 
In the first place we shall trace the influence of a relation between 
two of the frequencies of vibration. Then the relations shall be discussed 
which are possible between 3 or 4 of the frequencies of vibration. 
Relations of more than + of the frequencies of vibration are outside 
our consideration, as we have always to keep in mind NS 4, 
RELATIONS BETWEEN TWO OF THE PREQUENCIES OF VIBRATION, 
§ 2. We imagine a mechanism with / degrees of freedom. Between 
the frequencies of vibration 2, and nx, of the principal coordinates 
q, and q, exists the relation 
yn, =n, + Os 
where y=1, 2 or 3. The remaining 4—2 principal frequencies of 
vibration 7,1, . . . . # do not appear in the relation ; we suppose 
moreover that between the / frequencies or between some of them no 
exact or approximate relation exists except the just mentioned one. 
By the disturbing terms of the first kind in the equations of 
movement we shall understand terms which are always disturbing, 
also when no relation exists. When substituting the expressions for 
the coordinates by first approximation we find out of such a term 
a term having the same period and the same phase as the coordi- 
nate to which the equation, where the disturbing term appears, relates 
more in particular. These disturbing terms are of order /° or higher. 
By disturbing terms of the second kind we understand such as 
owe their disturbing property to the existing relations. When sub- 
stituting as above we find out of such a term a term corresponding 
to the coordinate in period but not in phase. These disturbing terms 
are of order ASL or higher. 
1) Amsterdam, 1910; also These Proceedings, page 618—635 and page 735— 
750 (1910); Archives Néerlandaises, Séries Il, Vol. XV, page 246—283 (1910). 
