170 Proceedings of Royal Society of Edinburgh. [sess. 
while the products become associative ? I find that the desired 
modification is accomplished by introducing J - 1 before the second 
and third sets. The rules then become 
i 2 = + 1 i 2 =+ 1 k 2 = + 1 
V = s/~ 1ft jh — J - li ki = jETij 
ji— - J -Ik kj — - J - li ik— - f - 1 j. 
As the quaternion ij k are quadrantal unit- vectors, they can be 
analysed into J - li 0i J—lj 0 , J - 1 k 0) where i Q j Q k 0 are unit- vectors. 
The quaternion rules, modified for order, then become 
(J^ - 1 
U-K'U-Vo=-J-ih (J- iio)(V^*o)= -V-l»o 
o) = V-V» 
(V - i/o)W - it.) = (V^i^oXv/ - i/„) = J~K 
(~u 0 )U~ ik)=J~jo- 
These rules are in perfect harmony with the vector rules when 
made associative as above ; for, on dividing the left hand by J - 1 
J - 1 , and the right hand side by the equivalent -, they yield 
*o 2 =l io 2 =l *o 2 =l_ 
^°.?o ~ \J ~~ 1*0 jfo \J K Ki 0 = lj i 0 
JoK— ~ ~~ 1*0 kpjo ~ \! 1% ifo — — \j — ljo* 
Let p denote any real unit axis; then p 2 =l. Similarly for 
any imaginary unit axis {J - 1 p) 2 = - 1. It is evident that p 2 =l 
is in nature a principle of reduction. But there is also the principle 
of reduction p/p = 1 or J — lp/J - lp= 1. This latter is a more 
absolute principle, and the reduction specified can be made at any 
time; whereas the former is legitimate only under certain con- 
ditions. The rules of the form ij = J - Ik are also principles of 
reduction of a relative nature. 
A more general statement of these rules is as follows : — For any 
two real unit axes /3 and y. 
j3y = cos /3y + sin fiy J - 1 /3y 
where fty denotes in the simplest case the axis perpendicular to 
P and y, but more correctly the axis conjugate to the plane of 
