STRAINED ELASTIC SOLIDS 439 



^ + 5^ = (an + danW + (ai2 + Ba^iW + (ai3 + Sa^)^', 

 5rj = (rt2i + 5a2i)^' + (a22 + 5022)77' + (023 + 5a23).C', 

 5f = (a^l + Sflsi)^' + (032 + ^a^^W + {a^s + 6033)^'.* 

 Hence 



5^ = dan-^' + 5ai2-77' + 5ai3-f', 

 67? = 6021-^' + 5022-77' + 5a23-r', 

 5r = Sasi-^' + 5a32-77' + 5a33-f'. 



Now we need to express these variations in terms of ^, and this 

 is easily done; for, on solving equations (33) for ^', 17', f' in terms 

 of ^, we find that 



^ H ^* 



, A,, 



where Ara is the first minor (vv^th its correct sign) of ars in the 

 determinant H. 



We write for convenience hrs for Ars/H, and in consequence 

 we have the following three results 



8^ = (hnSan + 6i25ai2 + hsdais)^,) 



8r] = (6ii5a2i + 6120022 + &i35a23)^, f (34) 



5f = (6ii5a3i + 6i25a32 + 6135033)^.] 



It is easy to see that the coordinates of f/5 for the local axes at 

 Ps are just — (^ + 5^), —677, —8^. Thus it appears that the 

 rectangular element has had its edge parallel to OX elongated 



* Observe that Pd and Qs are positions in the slightly altered state of 

 strain of the same original points P', Q' in the state of reference. 



