504 Proceedings of Royal Society of Edinburgh: [sess. 
Thus the directional roots of f are treated alike by and by <£, 
and must therefore belong to So those of x belong to 
< jxf Thus we are again conducted to the previous result; hut 
this third method gives us great additional information as to the 
intrinsic nature of the strains involved, and the relations which 
exist among them. 
10. It is, of course, only in special cases that simple methods 
like these can be applied to linear vector-function equations of a 
little greater complexity. But when they are applicable they 
often give singularly elegant solutions. As an instance take the 
equation 
^iX + X^'A .... (7), 
or, as it may obviously be written, 
x" 1 <£i + <£ 2 x~ 1 =x'tyx -1 - 
Let a be a directional root of <£., , then at once 
<£iX a + ( JX a = > 
or 
X a = (cj> l + g)- 1 if,a. 
If the roots of <£ 2 be unequal, the three equations of this form 
completely determine y . 
11. Again, let 
$\X + X<k = <l>3X<t>i + 'P • • • (8). 
E g v etc., are roots of <£ 2 , this gives three equations of the form 
W> i +^i)x a i = ^3x(^4 a i) + • 
If the values of a be unequal, we can of course find the coefficients 
in 
“I - “I - Cjftg 
^4^2 "h ^2 a 2 "b ■ • 
4>4 a 3 = 
Then, putting A 1 for ycq, etc., we have finally 
4> ?j + , < 7[)X 1 = cqA 1 + 4- Ag + cf)^ l \f / a | . 
The three equations of this form give A 1} etc., that is, ya ]5 etc., and 
thus x is found in terms of its effects on three known vectors. 
