96 Proceedmgs of Royal Society of Edinburgh. [jan. 6, 
These roots are evidently, all of them, scalars, of the form 
A + B^-1, adniitting J-l to be a scalar, in consequence of 
what is admitted in the theory of biquaternions. 
We cannot, therefore, accept the expression II. for the repre- 
sentation of the real versors^ depending on S, which form the 
second members of the expression of the ^ power of g, and 
we are consequently led to admit the form I. as being the true 
expression of that power of £. 
Generalising this result, we adopt the principle in virtue of 
/I \ w _1 
which must be represented by \g”"j , and not by (g”)”"’ 
We may apply this principle even to the case ?7^ = 2, 7^ = 2, when 
we get 
(2-)2=S, 
and we have to reject (g^)^ 
Should - converge towards an incommensurable value t (like that 
of a surd, &c.), we may still apply the principle by forbidding the 
separation of t into 2 x when g^ would not be admitted equal to 
(g^)^; but we might write (g^) =^\ 
As to the point of view under which the symbolic square (IJp)^ 
of a unit-vector Up may be looked upon, I would refer the reader 
to my paper printed in the volume xxvii. part ii. pages 175 to 
202, of the Trans. Roy. Soc. Edin., and particularly to the view 
taken there of treating ii,ij, ih:, ji, &c., &c., each as a symbol by 
itself, to be determined by the condition that the products of two 
vectors should remain unaltered, whatever the system of the 
directions may be by which the components of the vector factors 
are reckoned. 
3. On Vanishing Aggregates of Determinants. By 
Thomas Muir, LL.D. 
1. In a paper‘d communicated to the Berlin Academy on 27th 
July 1882, Kronecker pointed out that certain sets of minors of 
* Kronecker L., “ Die Subdeterminanten symraetrischer Systeme,” Sitzungsb. , 
d. Tc. Akad. d. Wiss., 1882, pp. 821-824. 
