502 
Proceedings of Royal Society of Edinburgh. [sess. 
so that 
2 Y ep = ap + Y aC^p) + ( 1 — c) Y. Y p . 
Now Hamilton (in giving his cubic) showed that 
( 7/2 2 — Eh) Y ap = YfxTjCt p + Y a^hp 
so we have 
2 Yep = s(m 2 Yap - VSTja p) + (1 - cJV.YaEfia p ; 
and, as this is true for all values of p , 
2e = s(m 2 a — tTTja) + (1 — c)Yat7 x a , 
the second term disappearing when the rotation is about one of the 
axes of the pure part of the strain. Again 
2C7p = 2 cTStf + S^Y ap — Y a&xp) — ( 1 — c) { S^aSap + aSa^p } 
is obviously self -conjugate.] 
8. An instantaneous, and (at first sight) apparently quite 
different, solution of (5) is obtained by multiplying each side 
into the reciprocal of its conjugate. For we thus have a case 
of (1) in the form 
But this equation, which would assign to x any value commutative 
with is very much more general than (5) from which it is 
derived. [This is an excellent example of the necessity for caution 
already pointed out.] 
To analyse this solution, with the view of restricting it, note 
that by Hamilton’s method we have at once 
where m is the product of the scalar roots of <j> ; a a unit vector, 
and e a scalar constant, both definite. 
m(g>' 1 - cj) x ) = 2Y£7e = 2eY67-a , suppose, 
m 
m 
I P + 
2 em 
m 
ET J Yat 2 r*p - —ZSiaSaZS-ip 
m 
