
STRESSES DUE TO COMPOUND STRAINS. 483 
we arrive at the following :— 
n(~—aD,)u,=m. apo 
n(y—bD,)v,=m. b9g0 ; : nae (419) 
n(b—cD,)w,=m. cre, 
and on the elimination of u,v, w, , the following equation for 
Qy2 2 2 202, 
a*p bq ey n 
~—aD, * p—0D, 1 $—eD,~ my’ 

(22.) 
It appears from this result that there are three values of V’ for each plane 
wave, and the wave which spreads out from a centre will, therefore, possess 
three sheets. 
But if the solid be incompressible, and the primary stress be finite, we shall 
have M= o , m= o, and aiso J,=0; and in this case the equation which gives 
V reduces to 
a2p? bg? Coy? 
v—ab, 4 v—oD, ~—oD,—° 
and the equations which give w, % w, require us to suppose 0=0, mé@=finite. 
: , d d L : é 
Now 6=0 is equivalent to os + bay + cz =0 ; and at first sight it seems 
difficult to understand the significance of this condition. We must remember, 
however, that the word “incompressible,” as here used, is a relative term, and 
denotes that when the strains are indefinitely small, the stresses in such a solid 
required to produce a very small cubic compression, are indefinitely great com- 
pared to those necessary to produce a shear of the same magnitude. This 
implies that m is indefinitely great compared to n; and the preceding investi- 
gation shows that if such a solid, having previously undergone considerable 
strain, be still further subjected to stresses of small magnitude, the distortions 
: du dv dw 
which they produce are such as to make a7~ + Ee ; 
With these explanations we now proceed to consider the case where a—1 , 
b—1,c—1 are such that their squares and products may be neglected. I 
shall also suppose that mJ,=0. Let therefore a=1+a, then P?’=a(a—1)=a, 
and similarly Q?=0—1=f, R’=c—1=y; thus D,=ap’?+8q7’+yr’, D,=1+D, 
and aD, =(1+a)(1+D,)=1+a+D,,.. 
+a) 4 Gaus) srl Sel tla 8 
; : 2 V 2 
The equation for V becomes a le ipa i a. Lois 
2 
It appears therefore that ey is of the order a, 8B, y, and we may write 
