( 327 ) 
Although from a physical point of view r can only take positive 
values, in these considerations even the case x7,< 0 is not impos- 
sible; for the point pry, vry has only a mathematical meaning. 
In general, equation (13) may now be written thus: 
p=m, + m, Vv —err) + m, (v— vz)? + m, (v—07i)? +...., . (18) 
where m,, m, etc. are functions of « which can be developed in 
powers of z—w7,; for instance: 
Me, = Moo + m,, (L—A TE) + m,, (©c—ae7x)? +.... « « (18) 
The co-efficients m are functions of the temperature which is here 
considered constant; it will be obvious that mm, == pr» while m,, 
and m,,==0. By equalization of (18) with (13) we can express all 
the m’s in the 4’s, and in KAMERLINGH ONNES’ a’s and §’s; for we find: 
kn : 5 
Mao = kyo — De [na — (n + 1) B] (T — To) + .... 
Mar = — kno [na — (n + 19] — hai Tra — (n+ 1)katijo(a— B) vr 4 … ete. (19) 
so that to a first approximation : 
LAE eee PPT 9) hall oy an 
mM, = prB—k,, Tha, m,, =—-k,, The, m,, = -—k,, Tra-3k,,vi(a—B), etc. (19') 
HARTMAN *) has given a diagrammatical representation of the p, v, z 
diagram. This representation completely resembles a p, v, 7 diagram; 
but this resemblance is not necessary. It follows directly from the 
p,v, T diagram that £,, is positive, while %,, and k,, are negative; 
in the p,v,x diagram m,, is negative, but according to (19), m,, and 
m,, may be either positive or negative. The circumstance m,,<{0 does not 
indeed influence the general shape of the diagram; it indicates that 
the isothermals of the mixtures lie below those of the pure substance 
as is the case at the upper limit (7 = 1) of Harrman’s representation °). 
But while in the p,v, 7’ diagram the isothermals with maximum 
and minimum pressure occur under the critical, the opposite may be 
the case in the p,v,« diagram, if m,, and m,, have the same sign. 
The four cases which may now present themselves, leaving out very 
particular values of the coefficients, are given in the following table: 
1) Thesis for the doctorate, Leiden 1899, p. 6; Journ. of Phys. Chem., 5, 425, 1901 
2) From a mathematical point of view we may imagine the p, v, x diagram to be 
continued outside the limits =O and x= 1. It is also obvious that z, if diffe- 
ring little from 1, means the same as & infinitely small and that xz > 1 means 
the same as x < 0. 
22 
Proceedings Royal Acad. Amsterdam. Vol. V. 
