172 PHILLIPS AND MOORE. 



planes kij is invariant. A particular linear function of considerable 

 interest is the vector of the dyadic. 



<^^= 2 2 sin qa ka (24) 



obtained by replacing each dyad of by the vector product of its two 

 factors. The rotation cannot be determined uniquely when the vector 

 is known, however, for the angles qi, might be replaced by their 

 supplements without changing the vector of the dyadic. 



7. Commutative rotations and star product. Consider two 

 rotations 



Since <^i and <^2 are sums of powers of I -Mi and I -Mi they will be 

 commutative if I • Mi and / • M^ are commutative, that is if 



^ = (I-Mi)-(I-M2) - {I -Mo- I- My) = 0. (25) 



Represent Mi and M^ in the symbolic form 



Mi= AB, i¥2= CD. 



Then 



I-Mi= BA - AB, Ii\h= DC - CD, 



and 



^ = U-D) (BC- CB) + {B-C) {AD-DA) - {A-C) (BD-DB) - 



{B-D)(AC -CA) 



= /• [(A-D) (C X 5) + (B-C) {D X A) - {A-C) {D X B) - 



{B-D){CX A)] 



= -/•[(/• MO •(/•M2)]x (26) 



where [ ]x signifies the vector of the dyadic contained in the brackets. 

 Let 



Mi*M,= - [{I-Mi)-{I-M,)]^ = [(/•i¥2)-(/-Mi)]x 



= (A-C) {BXD)-\- {B-D) {AXC)- (A-D) {B X C) - 



{B-C){AXD). (27) 



We will call this the star product of Mi and M2. It is evidently 



