64 
on entropy and probability this has been shown for arbitrary 
observable parameters. The mean energy of deviation did not depend 
on the nature of the parameters, but on their number only; and 
also in the ease considered it is not the partial density in the elements 
but only the number of elements discernible for observation which 
plays a part. 
Groningen, April 1912. 
Mathematics. — “Calculus rationum.” (2nd Part). By Dr. G. DE 
Vries. (Communicated by Prof. JAN pr Vries.) 
(Communicated in the meeting of March 30, 1912). 
$ 16. If in the following remarkable root 
n(u) se I [= 0(u), > Mell 
nv) v 1 
we put v=uw, the left member assumes the form 1° apparently 
indefinite; the right member becomes *—1!(w)". Introducing the sign 
R for the ratio of two values of a variable lying infinitely close 
together, we can write: 
Ry | Ra ="—\(e)n for y= (z), 
This is a mutual root of two ratios lying infinitely close to unity. 
If it is now even obvious to introduce in agreement to the preceding 
a rational radix as measure for the field of ratio, then the signifi- 
cance of a mutual root of exponential numbers is strengthened by 
the fact that of the following forms 
is 
a 
lin— 3; lima bt; 
o br ao 
the latter has no sense, the former has. 
If for the comparison of two variables a third is introduced as 
independent variable and if we then put 
Be BI). Say el, 
then from this can be deduced: 
ef) = lim (1 — >) ; oF) = lim (: + ="). 
When joining these we find that Az disappears when one of the 
mutual roots is calculated. 
VIL eN ! A | 4 ALt A 
FD: = lim (1 He =!) (1 se =) ly LEO 
y & y 
Introducing for the rational radix the sign L/R 
