90 G. FLEURI. 



AOB = a (cos a + A sin a) 



since AOD = T. AOD . U. AOD 



We have therefore for a quaternion the forms 



a n + ia 1 + ja^+ka 8 analogous to x + iy and 

 a (cos a + A sin a) analogous to a (cos a + i sin a) 

 Subtraction — Inverse operation of addition. 



Multiplication. — Two quaternions being given and rendered 

 collinear, turn them round their common origin O and alter the 

 lengths of their vectors, if necessary, so that the initial vector of 

 the multiplier coincides with the terminal vector of the multipli- 

 cand. Then, they occupy the respective positions BOC_AOB 



(Fig. 15) 



AOO is said to be the product BOC ■ AOB 



(the multiplier being first) 



and that definition is easily extended to any 



number of quaternions. 



The operation is uniform but not commutative 

 However it is associative and distributive and satisfies the equalities 



AOB . 1 = AOB AOB .0=0 

 Demonstrations of those properties are found in any treatise on 

 quaternions.* 



Describing a sphere from O as centre with the unit as radius, 

 we see that the multiplication of quaternions comes to 

 First, a multiplication of tensors 

 Second, a spherical addition of their argument.! 



* One of the best demonstrations of the law of associativity is the one 

 due to Mobius, and based on composition of rotations. 



f A spherical addition, that is to say an addition of arcs of great circles 

 on the sphere corresponds to a plane addition of vectors. Two arcs of 

 circles are equal when they are of equal lengths, on the same great circle 

 and of same direction. The sum of two arcs of circles u-f/3 is obtained 

 as follows : Take any of the two points of intersection of the circles on 

 which the arcs a and (3 are situated ; let us denote it by P. Displace a 

 and /3 on their respective circles, the former so that its terminal point 

 and the latter so that its starting point be at P. The arc of great circle 

 joining the starting point of a to the terminal point of /3 in their new 

 positions represents, by definition the sum a + (3. 



