130 Proceedings of the Royal Society of Edinburgh. [Sess. 
which vanishes when the six vectors lie on a quadric cone (see Tait, Quater- 
nions, Art. 261 of the 3rd edition). In other words, if p be regarded as 
variable, the vanishing of (3) defines a quadric cone through the other five 
vectors. The order of vectors in (3) may be easily remembered if wo first 
establish the cyclic order 
a/3y8epa ...... (4) 
and then note that, in (3), each of the three expressions in parentheses 
is formed as if the second V stood for an omitted vector : we do not have 
Va/3yS but Va/3VSe, etc. 
We note also that the three groups of four vectors in parentheses 
begin respectively with a, f3, y, the first three vectors as laid down in (4). 
If we had begun the groups with /3, y, and S, leaving the cyclic order 
unchanged, thus : 
S.V(Vf3yVep)V(Vy8Vpa)Y(V8eVa[3) ... (5) 
the sign of the result would have been changed, for 
V(VSeVa/b= — V(Va/m e ) .... (6) 
but the expression is otherwise unaltered. 
To get further light on the changes of sign produced by change in 
the order of vectors, we may expand (3) by repeated use of (2) and 
obtain easily 
Sya/3S8e/3Sy8pSaep — SyS/3Sae/3SyapS8ep . . (7) 
identically equal to (3). Now by inspection (7) changes sign when y and 
e are interchanged, or if /3 and p are interchanged, or if a and S are inter- 
changed. Similarly, we may show that (3) changes sign when any two 
vectors are interchanged. 
It follows that all expressions similar to (3) which can be made from 
the six vectors are equal except in sign. To determine in any given case 
whether the sign is the same as that of (3), we have first to count the 
number of interchanges needed to bring the cyclic order into agreement 
with (4), and second to note whether, after these interchanges have been 
made, the first vector in the first group is of odd or of even number, when 
a is counted as number one : if even, an additional change of sign must 
be counted. 
4. Expressions formed by selecting Six out of Seven Vectors. 
Let us now adjoin a seventh vector cr and lay down the cyclic order 
afiySeporoL ...... (8) 
It is evident that by dropping any one of the seven vectors we may form 
expressions like (3) out of the remaining six. If it be agreed that the 
