938 



In onr concluding paper something will be said about the tempe- 

 rature influence, which will manifest itself l\v continual diminution 

 of bk- ho, at first slowly, then more rapidly, as the absolute zero is 

 iipproached. Descending from high to low temperatui-es one can 

 therefore pass through all the types. If the critical region of a substance 

 lies in the region of low temj)eratures, the critical quantities, and 

 also the isotherms in the neighbourhood of the critical point, will 

 present, as far as the course of h is concerned, the little variable 

 type with slight bt — b^ (y in the neighbourhood of 0,5). But these 

 same substances will of course show the same variability of b as 

 the "ordinary" substances at high temperatures. Reversely the ordinary 

 substances, considered at low temjieratures, will assume the Argon-, 

 Hydrogen- or Helium-type, with respect to the slight variability of 

 b at these temperatures. Etc. Etc. 



In this concluding paper 1 shall also communicate the />-values 

 for Ai-gon I have calculated: besides I shall venture to give some 

 theoretical considerations concerning the diminution of the factor 4 

 in b,, := 4:m with fall of the temperature. 



Fo7itanivent sur Chirens, Februai-y 1914. 



Mathematics. — "The envelope of the osculating ellipses, to hick are 

 described by the representative point of a vibrating mechanism 

 having tioo degrees of freedom of nearhj equal frequencies.'' 

 By H. .1. E. Beth. (Communicated by Professor D. J. Korteweg). 



(Communicated in the meeting of February 28, 1914). 



§ 1. In my paper on the small oscillations of mechanisms with 

 two degrees of freedom ^), Lissajous curves with their envelopes were 

 discussed, which envelopes form the boundaries of the domain of 

 motion. In a summarizing ti-eatment of a more general problem ^1 

 my furthei" inquiries as to these envelopes have also been included. 

 These inquiries were extended over a system of Lissajous curves, 

 more general than the system which is of importance for the dynamical 

 problem. However the envelopes were considered exclusively from 

 a dynamical point of view, so that purely geometrical properties 

 together with the shape of the curves outside the domain of motion 

 remained unknown. Moreover what came to light about the shape 

 of the envelope remained for the greater part restricted to simple 

 cases, e.g. in the case, formerly indicated by -.S-— 2, to the symmet- 

 rical case, as quoted, indicated by /> -)- r/ = , 1 = 0. 



""if These Proceedings pp. 619-635 and 735—750 (I9l0j. 

 2) Phil. Mag., sixth series Number 152 (1913). 



