500 Proceedings of Royal Society of Edinburgh. [sess. 
and (4) becomes 
CTwc o~ l tt = tf 2 = oTWco , 
from which (coupled with the results of (1)) the above conclusions 
are obvious. 
6. The form 
X$X=<t> .... (5) 
also admits of simple treatment. Its conjugate is 
X0Y = $ * 
Now we can always write 
cf> = 1T) + Ye , with <£' = £>- Ve , 
and the equations above become, by addition and subtraction, 
X £T X ' = ^, x V€ x' = Ve - 
Put the first of these in the form 
X^(o 1 . id 
where and w 2 are, so far, arbitrary. As each side is the product 
of a strain and its conjugate (because the conjugate of a pure 
rotation is its reciprocal), we may at once write 
X^wi = C7ho 2 
or X = 
where to = o) 2 o) 1 -1 is still arbitrary. To determine it, the second 
equation above, viz. 
x y £x ' = y £ 
gives me = \ € 
where m is the product of the numerical (scalar) roots of x 1 
obviously unit in this case, as there is no change of volume. This 
gives 
< office — 
so that the axis of w is cde, but the angle of rotation remains 
undetermined. 
The direct algebraic verification of this solution is troublesome, 
