XXXIV 
ternions is therefore seen, in this as in other ways, to be not 
in general a commutative operation, or the result depends, in 
general, essentially on the order in which the factors are taken. 
The same remarkable conclusion follows from the compa- 
rison of the lately mentioned spherical triangle aBe with 
another triangle c’Ba’, vertically opposite and equal thereto, 
and such that the cgmmon corner B bisects each of the two 
ares C’c, a’A, joining the two pairs of corresponding corners ; 
which other triangle may represent the directions of three 
lines c’, b, a’, related to the system of the three former lines 
e, b, a, by the two following equations between geometrical 
quotients, or quaternions, 
ly Pile Ai 
Tigh eh AT By” 
for then, by the definition of multiplication of such quotients 
here proposed, we have the two different results, 
GC 
x D = v3 
and although these two resulting quaternion products have 
equal moduli and equal amplitudes, yet they have in general 
different axes, because the ares ac and a’c’, though equally 
long, are parts of different great circles, and are therefore 
situated in different planes. However, in that particular but 
useful and often occurring case, where the two factors have one 
common axis, the order of those factors becomes indifferent ; 
and if attention be paid to positive and negative signs, it may 
be said that coaxal quaternions may be multiplied together, in 
either order, by adding their amplitudes, multiplying their 
moduli, and retaining their common axis. In general, it may 
be proved, from the views here given of multiplication and addi- 
tion, that, although the commutative property of ordinary mul- 
tiplication does not usually extend to operations on quaternions, 
yet the distributive and associative properties of that operation 
do always so extend ; and that the commutative and associative 
