18 ROYAL SOCIETY OF CANADA 
tion of a quadrantal versor being accepted. The annexed figure explains 
this. 
If, however, in aa or a’, the a 
to the right represents a vector and 
that to the left an operator, we must 
for this case invent a fresh defini- 
tion, and our definition is that aa@ or 
a? is —a?, suggested manifestly by 
the fact that this is what a? 
when both a’s are operators. 
Our definitions up to the present 
leave perfectly unintelligible such a 
combination of symbols as af, where 
a and f are not at right angles to 
each other, whether a and f are both operators acting on a suppressed 
vector, or a an operator acting on the vector B. 
Lives 

We must therefore again have re- 
Mea course to our privilege of defining, and 
we select as our definition either of the 
B two following, both definitions giving 
the same result : 
(1). @ as an operator on / is equiv- 
alent to the sum of its components as 
operators, 1.e., is equivalent to the sum 
ZS ; ; 
of the results obtained from the opera- 
tors ON and VA. The effect of each of these as operators has already 
been defined. 
(2). @ as an operator on / is equivalent to the sum of the operations 
of aon the components of /, 1.e., is equivalent to the sum of the results 
obtained from the operation of a on ON! and N'B. The effect of a 
operating on each of these has al- 
ready been a matter of definition. 
This definition, indeed, is the same as 
saying that the conventional form 
AB + BC = AC shall be accepted 
(AB and BC being perpendicular to 
each other), not only when standing 
singly, but alsojwhen operated on by 
a quaternion whose axis is along AB. 0 
Even here our conventions are not ended, for in order to preserve 
harmony between the two roles in which our symbols appear,—as mag- 
nitudes and as operators,—we agree that an operator may act on an 
operator and in doing so the second operator may be regarded as a 
œ 
Etre on enerezeuriner reseseearuras 
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