( 735 ) 
Mathematics, — “The oscillation* about a position of equilibrium 
where a simple linear relation exists between the frequencies of 
the principal vibrations (Second part). By H. J. E. Beth. 
(Communicated by Prof. D. J. Korteweg.) 
(Communicated in the meeting of February 26, 1910). 
S = 4. 1 ) 
§ 14. In this case the ordinary expansions in series hold as 
long as ~ is great with respect to (see page 7 of the paper by 
Prof. Korteweg, mentioned above). The difficulty arises as soon as — 
has fallen to the order . The calculations not getting simpler 
with the absence of a residue of relation, we shall immediately 
assume a residue of relation of order A 2 . 
When the relation 
n 7 4 (> = 
exists and we proceed to investigate with a view to this which 
terms in (2) (page 620 of these Proceedings) become disturbing in 
the sense indicated in $ 3, we easily see that no terms of order A* 
appear among the disturbing ones. So when determining the first 
approximation we may omit the terms of order A’ in the equation 
of the surface, which terms agree with the just mentioned terms 
of order A*. It then becomes 
* = - (<V 
- W* 4 e i x * -f + e* x Y 4 «4 *3* 4 e s y*); 
for we need not take for the first approximation in the equations 
of movement any terms of higher order than A 8 . 
The abridged equations of motion, containing only terms of order 
A, still run as follows: 
X + 2c. x = 0 ,) 
y 4 2c, y = 0. 
= |/2c x , — i/2e, 
are the frequencies of the principal vibrations. 
*) For the case 5=3 see 1 st part, pages 619—635 of these Proceedings, 
