190 
MR. Gr, H. DARWIN" ON THE STRESSES 
V =-P +2v 7H. 
m_ (da. dy' 
\dz Ax i 
> 
(4) 
And on putting co infinite in (1) we have 
' ^ + 2 V _ (f+1) (2i +3) 9 1 dWj _ 
2(2i-f-l) J dx 2 i + 
d 
— , , _ _p2i+3—/ 
2(i-l) 2(2i + l) ; dx 2i+l dx y ' > 
and symmetrical expressions for /3 and y. 
• • (5) 
The hydrostatic pressure might have been found from this general solution for the 
case of incompressibility, but in order to do so it would have been necessary to go back 
to the equations of equilibrium of the solid, and I prefer to deduce it from Sir William 
Thomson’s solution in the more general case. 
Since 
d 
and since 
. r 2 it 3 ^( W ‘ r zi ~ l ) —- ( 2i + i ) xw >+ q (]x 
(i + L)(2f+ 3) + 2 i= (2 i+ l)(i + 3) 
we may write a as follows :— 
a; 
2 Iv 
i(i + 2) 
2 ixWi 
(6) 
In order to find the stresses P, Q, &c., we must now evaluate C ~> &c. 
clx clz d/x 
Differentiating (6) with regard to x ,— 
■ • (?) 
Differentiating with regard to 2 ,— 
2 Ivy= ((*+3)i4 
dz [ ?,—l 
and by symmetry 
d*Wi . n \. dWi , oX dW{ 
2i ix— -(iff-3 )z 
dxdz 
dz 
dx 
( 8 ) 
2/u-i 
dy [ i(i + 2) a ,. . •*! d 2 W ; . „ f . dW ; dW{ 
H 
i — 1 
a®-(i+3)e 
dx 
dz 
• » 
• (9) 
Adding (8) and (9) together and dividing by 2, we have 
(10) 
Hence from (3) (4) (7) and (10) w T e have, 
