ternlons 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 abc with 

 another triangle c'ba', vertically opposite and equal thereto, 

 and such that the cgmmon corner b bisects each of the two 

 arcs 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 

 c, b, a, by the two following equations between geometrical 

 quotients, or quaternions, 



a' __ b b __ c 

 b ~ a' c' "" b ' 



for then, by the definition of multiplication of such quotients 

 here proposed, we have the two difi'erent results, 



c t) _ c b c a' 

 a a a be' 



and although these two resulting quaternion products have 

 equal moduli and equal amplitudes, yet they have in general 

 different axes, because the arcs 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 T^vo^Qrixe?, of that operation 

 do always so extend ; and that the commutative and associative 



