867 
the point, where, in the domain («), that function has its maximum 
modulus, coincides with a point where «, attains its upper limit value. 
6. We will now say something about the dependence between 
the quantities « and 2. The number « may vary from zero to an 
amount A, which is the upper limit of the radii of domains, in 
which the quantity a, determined by (6) is a limited function. The 
number A cannot in any case be greater than the radius of con- 
vergence of one of the functions a@,,(v), but it may be less, since, 
even when all those functions are regular in a certain domain (a), it is 
possible that the upper limit (6) has not a finite value in some point 
of (a). It might also occur that the limit (6) did exist in all the points 
of («), but was not bounded in that domain. On the other hand it 
may happen that the number A is infinite (e.g. if a, (a) = eon- 
stant = CE): 
From the fact that the quantity a has been defined as the upper 
limit of the function a, in the domain (a), it follows at once that 
a cannot decrease if « increases, in other words that « is a mono- 
tone function of «. Therefore according to (7), B is a monotonely 
increasing function of @, not smaller than «. (3 may-be equal to a, 
eraa (x) =— 1: m1, for in<that case. dir — Constant — 0). 
Let © be the value of # for «=O, and B the one for a= A; 
in many cases 4 will be infinite, but it need not be so. Every value 
3 may assume lies, as > is a monotonely increasing function of «, 
in the interval (6, B), and corresponds to only one value of a. The 
number 6, which, as a #-value, belongs to a—0O, may be zero, if 
Gro = 9. In that ease any function for which a, is an ordinary 
point, with arbitrarily small domain of regularity, has in. a, 
a transmuted determined by (1). If a, as a function of «, is in 
that case continuous in «=0, the series (1) produces for any 
function, with arbitrarily small domain of convergence, a transmuted 
in a certain domain of «a,. The transmuting series in that case 
is, according to a name introduced by PiINncHERLE, of the first kind. 
Chemistry. — “/n-, mono- and dwariant equilibria’. XU. By Prof. 
F. A. H. SCHREINEMAKERS. 
(Communicated in the meeting of December 21, 1916). 
21. ZLernary systems with two indifferent phases. 
In the previous communication we have deduced the four P,7- 
diagramtypes, which occur in ternary systems with two indifferent 
phases. Now we shall consider a case more in detail. 
