
SECTION III., 1901. aga Trans. R. S. C. 
UL—The Principles at the Base of Quaternion Analysis. 
By Professor ALrrep BAKER, M.A., of University of Toronto, 
(Read May 23, 1901.) 
The digits we use in ordinary arithmetic have two meanings. They 
represent magnitude, and they represent also operations performed on 
magnitude, 
The symbols we employ in quaternion analysis in like manner have 
two functions, each with a more strongly marked individuality than we 
ascribe to the symbols of arithmetic. 
Corresponding to the use of digits in arithmetic to represent magni- 
tude we have the vector of Quaternions; and corresponding to the use 
of digits in arithmetic to represent Operations we have the quaternion 
operator. 
These two qualities of the quaternion symbol are of course quite dis- 
tines. Yet in organizing a workable analysis it is evidently desirable 
that the laws governing the combinations of the symbols, whichever 
function they are performing, should not result in analytical contradic- 
tions. Thus the form af, whether it signifies the operation (on a sup- 
pressed vector) which f represents followed by the operation which « 
represents, or the operation a performed on the magnitude /, should, 
when translated into another form, be represented by one form and not 
by two. 
We detine a vector as the representative of transference through a 
given distance in a given direction ; or, more generally, as the represen- 
tative of something which has magnitude and direction; and we accept 
the convention AB + BO = AC where “ = ” may be best interpreted or 
read, perhaps, as “ is equivalent to”. 
We define a quadrantal versor as one 
which has the power of turning a vector 
through 90°. Thus ki = j, where # is the 
axis which does the turning, or about which 
the turning takes place, and is written before 
the vector on which it operates. 
This definition, however, leaves perfect- 
ly unintelligible such a combination as a or 
aa, unless «x here are both operators acting 
on a suppressed vector perpendicular to a, in 
which case we must plainly have a? = —q? 
to be in accord with well-recognized analytical forms, the previous defini- 
Sec. III., 1901. 2. 

