286 
Froceedings of Royal Society of Edinburgh. [sess. 
But it is an obvious theorem of ordinary algebra, that, whatever be 
the quantities involved, the two expressions 
and 
(lx + my + nzY + {xyij - mf + yz((ni - nf + zx(n - Vf) 
(x + y + z)(px nd-y + n^z) 
are absolutely identical except in form. 
Hence the condition for real values of A is simply that 
Q + + caf + Q(a5 -^hc- caf + B( - + ca)‘^ 
shall be positive : — i.e. that its factors shall have the same sign. 
To compare with experiment, let us take r = l atm., c = 1 ; and 
find the relation between the values of p and q , the pressures when 
the volume is reduced to a = 0‘9, and & = 0*95, respectively. 
The factors of the above quantity are 
0-05 OT 0-05 
-^(0-95)2 ■*'2(o-9)2 (0-95)2(0'9)2 
, ^ 0-05(0-805)2 0-l(0-906)2 0-05(0'995)2 
(0-95)2 +1 (0-9)2 (0-95)2(0-9)2 
or, quite approximately enough for our purpose, 
-JP + 2-228^ -1*234 
and + 2*816g - 1*886 . 
In the latter form each has been divided by the (essentially 
positive) multiplier of p ; and, as p and q are each of the order 
1000 atm., the last terms may usually be disregarded. Thus it 
appears that the values of A cannot be real if pjq lie between the 
approximate limits 2*23 and 2*82. But from Amagat’s data we 
easily calculate the following sufficiently accurate values : — 
Ratio of Pressures at 0° C. for Volumes 0*9 and 0*95. 
Bisulphide Methylic Ethylic Chloride Propylic 
of Carbon. Alcohol. Alcohol. of Ethyl. Alcohol. 
Ether. 
2*51 2*61 2*65 2*65 2*69 2*71 2*73 
[The values of q range from 458 atm. in the case of ether to 1166 
atm. in that of water.] All of these ratios lie well within the limits 
