Permeability 
239 
where is the rate of passage of water out of the cell or tissue, 
£ is the permeability, P e and P are the osmotic pressures of the 
external solution and the cell sap respectively, T the turgor pressure 
(cf. Chapter IX), and A is the area of the cell surface through which 
water passes. 
Now the rate at which water is lost by the tissue is proportional 
to the rate of volume shrinkage, and if c.G.s. units are employed 
may be taken as equal to it. Let us suppose that in a small time 
interval St the length l of the cylinder contracts by a small length 
SI, and that in the same time the contraction in width is Sa, the 
original diameter of the cylinder being r and the thickness of the 
tissue a. Let us further suppose the water loss in the same time is 
Sw. The change in volume in the time St is then 
nal (2 r — a) — v {a — Sa) (l — SI) {2 ( r — 8 a) — (a — Sa)} 
or tt [{a (2 r — a) SI 4- 2rlSa) — (l 8 a + 2rSl + 8 a 81 ) 8a], 
Hence, using c.G.s. units, we have in the limit the rate of water 
loss by the tissue given by 
dw . dl , da 
- g = M (2 r-a) Jt +2nrl 7t . 
itt, v ( , . dl .da 
Whence ? - p -- p + T (zr - a) rl ^ 
Thus, if it is assumed (1) that differences in imbibitional swelling 
at different temperatures are negligible, (2) that the value of the 
exudation pressure P e — P + T at the same stage of shrinkage is 
practically constant for temperatures over the range 5 0 to 42 0 C., 
and (3) that the cylinders shrink only in a longitudinal direction, the 
above equation may be written 
where K is a constant since A is the same for all cylinders in the 
same stage of shrinkage. If, on the other hand, there should be a 
shrinkage in a transverse direction proportional to the longitudinal 
shrinkage, the equation connecting permeability and shrinkage may 
be written ,, 
^ Kp -lr 
K again being a constant. 
Now the first two assumptions are very reasonable, and it is 
probable that the actual shrinkage would approximate to one of the 
two cases considered under (3). In the first of these cases the 
