110 Proceedings of Royal Society of Edinburgh. [sess. 
Now 
8 X (o= -S P ' 1 Sa>V 1 (29). 
Hence we see from equation (7) that in any expression, linear both 
in V 1 and Sp\, such as the last expression for Sw, we may substi- 
tute instead of these two £, S x £ respectively. Hence 
8w-lf»S8 x W x , - 1 { (30). 
This expresses Sw in terms of the variation of strain. It is con- 
venient to modify the last equation by means of equation (6). 
Applying the rule given in connection with that equation, and 
changing £, x -1 £ into X -1 £j£ respectively, we get 
Sw = — raSSxx -1 ^^ (31). 
Suppose, now * that the strain x is made up of a pure strain if/, 
followed by a rotation of q( )q _1 , so that 
X w = qif/wq -1 (32). 
Then it is easy to prove that 
8 X o> = 2WSg r (Z~ 1 *X <0 
Substituting in the last expression for Sw, we get 
Siv/m = - 2S NSqq^X^t ~ • 
The first term on the right is zero, * . * V£<££= 0. That 8q should 
disappear from the expression for Sw is, of course, what we should 
expect. It is, however, well to show that this follows as a mathe- 
matical consequence of our fundamental assumptions. We now 
have 
Siv= - mSSifrx’^q'^q 
= - mSSif/if/^iq^^q-^Cq 
[by substituting for x -1 in terms of if/ and q] 
= - mS8ifnf/- 1 Zq- 1 <l>(q£q- 1 )q 
[by substituting £, q^q -1 for q ~ 1 £q,£ respectively] 
= - mSSi^i/' -1 ^^ 
* See Tait’s Quaternions, 3rd ed., § 381, where it is shown how to deter- 
mine both r]/ and q in terms of x • 
