[BAKER ] BASE OF QUATERNION ANALYSIS 19 
vector. Thus in the form afy, we may suppose / to operate on y, and 
a to operate on the result; or we may suppose a to operate on the 
operator f, and the result to act on y. With the convention just named 
it is a matter of demonstration that these 
two modes of dealing with the form are 
equivalent. 
It is now a matter of demonstration that 
the versor AB followed by the versor BC is 
equivalent to the versor AC If, however, 
we accept this as a convention (and it seems 
most natural to do so), we can then show 
that the order in which we combine succeed- 
D ing operations is immaterial. 
With the definitions first given we readily obtain the forms 
afi = ab (—cos 6+ e sin 0) 
A 

and =f = : (cos 6 + esin 6) 
a 
In this latter we agree that @ has 
been converted into 5 when it has 
been converted into ON! + N'B. 
Here, however, our conventions 
and definitions cease. 
The truth of 
k (+ j) = ki + kj 
is at once evident from the figure, 
but of course is a matter of proof 
and not of convention. 
If a, fi, y be in one plane we 
have 


a (B+ y) = af + ay 
c For 
APE 
a.OC (—cos AOC + e sin AOC), 
afi = a.OB (—cos AOB + esin AOB). 
ay =a.BC (—cos ATC + esin ATC). 
and by projections the sum of the latter 
is equal to the first. 
0 T fi A If a, fi, y be not in one plane we 
have 
a (B + y) = a.0C (— cos 6, + e, sin 6,). 
af = a.OB (— cos 6, + e, sin 6,). 
ay = a. BC (— cos 0, + e, sin 6,). 

