490 PROFESSOR C. NIVEN ON THE 
Hence the above result furnishes the following 
— K.-H, +J,4+1=MK—H4+3+1)) 
and by puttng A=B=...=0, this also, | ; ee 0'9) 
K,=j7+h+k—-1 j 
§ 18. We are now in a position to find the value of the work W,, which 
corresponds to the compound strain, in terms of its components ; to do this 
write W,=W,+w, and analyse w into the terms of the first, second... 
degrees in the secondary strains ; thus— 
W=W,+W.+W3+... 
On substituting for J,, H,, K, their values, we can readily find w,, w.,.... 
If we confine ourselves to the case where 
1=4(mJ{+nH,) 
we shall have W,=4{2mI, 9+ n(hJ +24 + Ji)} 
W,=4(m~# +NJ,) 
Now w, may be written in the form 
5) (DA+()B+ ()C +2(d)D + 2@)E+ 2(/)F} 
where n(a)=2(mJ, +n) a+n(e—ca+f?—ab) (32.) 
nf) =2AmI, +n) f +n(de—c/) , 
These expressions may be simplified by supposing the axes of co-ordinates 
to be those of the primary quasi-strain ellipsoid; and if the primary stress be 
homogeneous throughout the solid, these axes will have the same direction 
throughout. In this case we shall write 
(a) =P’, (6) =Q*,(¢) =’, @=()=(/)=0 
P?=2CJ,+1)o—@b+ac),... . : -. | ees 
w,=5(P?A + Q’B+R°C). 
The simplified value of w, has been already found. 
The expressions found above (32) may be used to furnish the expressions 
for the stresses given in my paper at art. 10, by observing that 
WdV=WdV +58V 
where dV and 86V are corresponding elements of the solid in its free state, and 
after the primary strain has been produced. 
(31.) 
We may also apply the method given in art. 7 to find expressions for the z | 
stresses, whatever may be the nature of the function which expresses W, in P| : 

