654 The Principles of the Calculus of Quaternions. 



cc^=zm^ot}. Thus the associative law gives 



a^ /S = m Sa2 . /5 = am ^a/3 + nWccYa/S . = a . «^. . (0) 



VttV6</3 



It is geometrically evident that the vector VaVa/S is 

 coplanar with a and and at right angles to a, while it make^ 



the angle 0+ ^ with /S. Its length is equal to a% sin S. 



Resolving yS along this vector and a, we have by equation (0), 



m 



Sa^ . h cos fe f I") =n^a% sin 0, 



or simply 



m= +?i' 



(D; 



Thus m is a j^ositive quantity unless we suppose n to be 

 imaginary, and we may without loss of generality take it to be 

 equal to unity. The number n is then + 1, and taking n= + 1, 

 we have the quaternion relation 



afi = ^ccl3 + Ya^ (E) 



Hence follows in particular the fundamental formula 



i^=f = k'^ = ijk=-l. 



The foregoing analysis establishes the proposition that 

 quaternions is tlie only system of vector analysis ivhich is at 

 once distributive and associative in multiplication of vecto-rs 

 when we avoid introducing v — 1 into a product of real 

 vectors. I think that originators of new methods should 

 consider very carefully whether any advantage they may 

 gain outweighs the disadvantage of depriving their calculus 

 of the simplicity of either of these properties. 



