FKOM NUMBER TO QUATERNION. 91 



Owing to the non-commutativity of spherical addition, quatern- 

 ion multiplication is not generally commutative, as already seen. 

 However when the quaternions are coplanar, the spherical addition 

 becomes a circular addition and the multiplication of quaternions 

 the multiplication of plane complex quantities which is commutative 



The definition of multiplication applied to quadrantal versors 

 shows immediately that 



A 2 = — 1 



and 



generally 











A4n = 



1 







A&i + l 



= A 







A^ + 2 



= —1 







A4n + 3 



= —A 



and 



in particular 









»■ =P 



= k 2 : 



also 





ijk = - 



-1 





with 



ij = k 





— 1 



= —k 



jk — i kj = — i 



ki —j ik = — j 



the fundamental equations of the quaternion calculus in the case 

 where the vectors of reference i, j, k have the positions given 

 (Fig. 16) as it is always considered in mechanics. 



Dealing with coplanar quaternions, we find 

 "* de Moivre's formula 



[a (cos a + A sin a)] n = a n (ces na + A sin no) 



Now let us consider a series of vectors 



«1 ^2 V n 



Q = v 1 +v 2 + +v n + 



Fig. 16. , • ' i • ■ 7 



where v = ai + oj +ck 

 that series is said to be convergent when the sum of its n first 

 terms has a limit for n infinitely great, that is to say when the series 

 A = a 1 +a 2 + ... + a n + ... 



C = c x +c 2 + ...+c n + ... 

 are separately convergent. 



