350 Proceedings of the Royal Society 
because <£cr is perpendicular to y, and so o- must be as it is parallel 
tO <j t>(T. 
If we treat (27) by S( )X we get, (A. being different from y) 
SA(</> — A)cr = 0 , 
namely 
(29) So-(4>'-A)A = 0. 
Thus, if we call N a scalar, we get by (28) and (29) 
No - = V. y(f> — li )\ . 
By the second equation (23) the second member in the value of No- 
becomes 
(30) Rtr = (wi 2 — cjj - 7i)V y \ . 
We will now consider the value of li. The relation (27) when 
treated by </> gives 
<£ 2 <r = Jicficr = h 2 a . 
Thus by (24), the factor o- being suppressed, we get : 
(h 2 - mji + mf = 0 . 
Calling h' and h" the two roots, we have 
h' + h" = m 2 , 
and calling <r', c r" the corresponding directions of cr, the expression 
(30), in which we change the sign of JST, will give now 
NV = (</>-&>> 
XV' = (<£-A>, 
where 
Syw = 0 . 
These are the vectors, namely, cr', cr", which satisfy 
= ll(T . 
When <f> is self-conjugate they are necessarily real, because then 
the roots h', h", are real. We may show this in the following way. 
We have for the roots : 
2 h = m 2 ± R , 
R 2 = ml — 4 m 1 
= S 2 ( aai + ppj + 4SVa/5Ya 1 ^ 1 , 
by the expressions (12) and (12 bis). 
