105 
substances may belong to different types”. Of course it would be 
entirely premature now that we are still altogether ignorant about 
A . . a bed 
the causes that govern the value of the quantity — even for the 
a 
1 
best known systems, to pronounce an opinion about the question why 
for some systems the value is such that a very narrow region of 
unmixing appears, whereas for others there exists complete miscibility. 
Even quite apart from the fact that it does indeed follow from what 
- precedes and what follows that the experimental peculiarities found 
by Mr. Karz can all appear for perfectly normal substances, but 
that jt does not follow by any means, of course, that not all kinds 
of other circumstances might be found for the systems investigated 
by him, whieh do not affect the general character of the lines, but 
might have a very considerable influence on the numerical values 
of the quantities to be calculated. For this reason I have abstained 
from endeavours to find the numerical values of a’s and 0’s, and 
have confined myself to the general course of the investigated lines. 
§ 4d. The volume-contraction. Further on we shall return to the 
vapour-pressure lines, but for a reason which will soon become clear, 
we shall first speak about the volume contraction. According to 
Mr. Katz it may be represented by a hyperbola: 
a 
RE 
in which ce is the contraction in em? when 1 gr. of dry substance 
absorbs 7 gr. of water. What does the theory of the normal mixtures 
teach us about this quantity ? If we may assume that the tempera- 
ture has been chosen so low that we may put the limiting volume 
6 for the liquid volume, the increase of volume Av in consequence 
of the mixing of M,(l—e) gr. of water and M,xgr. of dry sub- 
stance becomes : 
Av=bz— b, (1 = «) —b,a = — w«(l—a)(b, + 6, — 26,,) . « (5) 
From this we must derive the relation between c and 7. Now 
evidently : 
Av 
B Ma 
follows from the definitions, cand Av taken for the same concentration. 
If we further mix 1 gr. of dry substance and 7 gr. of water resp. 
M, with M, 7 gr., the number of molecules are evidently in the 
; M,. 
ratio 1 : —4 so: 
M, 
