Permeability 175 
to the volume of water taken into the cell (Nageli and Schwendener, 
1877; Pfeffer, 1873; Thoday, 1918 b). Ursprung and Blum (1916 d) 
have shown, however, that such a relation only holds approximately. 
They point out that Sacerdote (1898) has shown that for a hollow 
cylinder with thin walls the increase in volume SF of the original 
volume F is given by the equation 
r=-** [y - 2<T > pi] • 
where R is the radius of the cylinder, e the thickness of the wall, 
p and p' the pressures on the inner and outer sides of the wall, F the 
volume and a and a are constants. 
When R/e is very large the first term on the right side of the 
equation may be neglected, and in this case the increase in volume 
will be proportional to the increase in pressure. But if the first term 
is not negligible in comparison with the second, the linear relation 
between wall pressure and volume is only approximate. Diagram¬ 
matic representations of this linear relation are given by Thoday 
(1918 b) and Hofler (1920). 
Assuming this linear relation, if V z is the volume of the cell when 
the turgor pressure is zero, but before plasmolysis has commenced 
(the state referred to by Ursprung and Blum as “ Grenzplasmolyse ”), 
and if the osmotic pressure is P Z) and if when the turgor pressure 
(or wall pressure) is T the volume of the cell is V and its osmotic 
pressure P, neglecting the volume of the protoplasm, and assuming 
constant temperature, the following simple relations hold: 
P Z V Z = PV 
V-V z 
V 
= aT, 
(Cf. Chapter VI), 
where a is a constant. The ratio V/V z Hofler denotes by the term 
“degree of turgor stretching 1 .” 
If the volume of the protoplasm of the cell is not negligible in 
comparison with the volume of the vacuole, these equations require 
modification. If p represents the fraction of the whole volume of 
the cell in the non-turgid but unplasmolysed condition occupied by 
the protoplasm, and if the latter undergoes no change in volume 
1 Hofler points out that the relation between osmotic pressure of the cell 
and turgor pressure is incorrectly given by Pantanelli (1904) and Hober (1914) 
following him. 
