216 
MR. G. H. DARWIN ON THE STRESSES 
The process is however so closely analogous, that it presents no difficulty, and may he 
dismissed shortly. I shall accordingly follow closely the processes of §§ 1 and 2. 
If a solid be very compressible it takes a comparatively small hydrostatic pressure 
to produce a given amount of compression ; that is to say, although the compressibility 
is large, the modulus of compressibility is small compared with that of rigidity. The 
modulus of compressibility I shall call the incompressibility. In the preceding investi¬ 
gation the converse was the case, for the incompressibility was infinite compared with 
the rigidity. By the definitions of w and u in § 1, the incompressibility 
k—O) - jjrV. 
It will be found convenient to use k and to as the two moduli, which define the 
nature of the elastic solid. 
In the denominators in E h F h G b of (1), the expression (2(f-f 1)~-f l)w — (2 i +1) u 
occurs, this I shall call K , in analogy with I. 
Then 
K=2i(i- l)o)-b 3(2'i~b l)& 
If we develop the last differential coefficient in the expression (l) for a, we find 
Also 
and 
2 vKot= 3 kl -r—rO ?+ ico 
W —1 / dx 
Lilt 
(44) 
KB=iW 
P = (3&—2cu)S+ 
i 
J 
Differentiating (44) with regard to x, and substituting in (46) we find 
KV=3k 
a—1 
a-—r 
0 dW . . ' 
day 1 
dx 
-j-fw(a 3 —r 2 ) 
d 2 W 
dx 2 
(47) 
Again, differentiating (44) with regard to z } writing down dy/dx by symmetry, and 
adding the two together, we have from (46) 
KT= 3k 
-a~ 
M 2 W dW dW 
. r j ' 1 ' 
1 
doxU 
dx 
■x- 
dz 
-\-ioj(a 2 —r 2 ) 
d 2 W 
. . (48) 
From (47) and (48) the other stresses may be written down by cyclic changes of 
letters. 
Now let us suppose that the incompressibility is small compared with the rigidity. 
